Stabilization of the response of cyclically loaded lattice spring models with plasticity

Gudoshnikov, Ivan; Makarenkov, Oleg

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. S8

This paper develops an analytic framework to design both stress-controlled and displacement-controlled T-periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t↦(e(t), p(t)), where e$$(t) is the elastic elongation and p$$(t) is the relaxed length of spring i, defined on [t₀, ∞) by the initial condition (e(t₀), p(t₀)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the elastic component t↦e(t) always converges to a T-periodic function as t →∞. The achievement of this paper is in spotting a class of loadings where the Krejci’s limit doesn’t depend on the initial condition (e(t₀), p(t₀)) and so all the trajectories approach the same T-periodic regime. The proposed class of sweeping processes is the one for which the normals of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any d different facets of the moving polyhedron C(t) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem.

Classification : 37J25, 49K24, 74C05
Keywords: Elastoplasticity, Moreau’s sweeping process, quasistatic evolution, periodic loading, stabilization

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[52] E. Svanidze, T. Besara, M. F. Ozaydin COCV_2021__27_S1_A10_097f0882d-6d28-4a97-854a-642d692984facocv19019510.1051/cocv/202005710.1051/cocv/2020057 A fixed point algorithm for improving fidelity of quantum gates* 0000-0002-5458-182X Pereira da Silva Paulo Sergio 1** Rouchon Pierre 2 Silveira Hector Bessa 3 1 Polytechnic School – PTC, University of São Paulo (USP), São Paulo, SP, Brazil. 2 Centre Automatique et Systèmes, Mines ParisTech, Paris, France. 3 Departamento de Automação e Sistemas (DAS), Universidade Federal de Santa Catarina (UFSC), Florianópolis, SC, Brazil. **Corresponding author: [e-mail] SupplementS9 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF)Full (DJVU)

This work considers the problem of quantum gate generation for controllable quantum systems with drift. It is assumed that an approximate solution called seed is pre-computed by some known algorithm. This work presents a method, called Fixed-Point Algorithm (FPA) that is able to improve arbitrarily the fidelity of the given seed. When the infidelity of the seed is small enough and the approximate solution is attractive in the context of a tracking control problem (that is verified with probability one, in some sense), the Banach Fixed-Point Theorem allows to prove the exponential convergence of the FPA. Even when the FPA does not converge, several iterated applications of the FPA may produce the desired fidelity. The FPA produces only small corrections in the control pulses and preserves the original bandwidth of the seed. The computational effort of each step of the FPA corresponds to the one of the numerical integration of a stabilized closed loop system. A piecewise-constant and a smooth numerical implementations are developed. Several numerical experiments with a N-qubit system illustrates the effectiveness of the method in several different applications including the conversion of piecewise-constant control pulses into smooth ones and the reduction of their bandwidth.

Controllability quantum control right-invariant systems Lyapunov stability Banach Fixed-Point Theorem 93D05 93D30 8108 Conselho Nacional de Desenvolvimento Científico e Tecnológico http://dx.doi.org/10.13039/501100003593 305546/2016-3 Fundação de Amparo à Pesquisa do Estado de São Paulo http://dx.doi.org/10.13039/501100001807 18/17463-7 idline ESAIM: COCV 27 (2021) S9 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S9 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020057 www.esaim-cocv.org A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES Paulo Sergio Pereira da Silva1, , Pierre Rouchon2 and Hector Bessa Silveira3 Abstract. This work considers the problem of quantum gate generation for controllable quantum systems with drift. It is assumed that an approximate solution called seed is pre-computed by some known algorithm. This work presents a method, called Fixed-Point Algorithm (FPA) that is able to improve arbitrarily the fidelity of the given seed. When the infidelity of the seed is small enough and the approximate solution is attractive in the context of a tracking control problem (that is verified with probability one, in some sense), the Banach Fixed-Point Theorem allows to prove the exponential convergence of the FPA. Even when the FPA does not converge, several iterated applications of the FPA may produce the desired fidelity. The FPA produces only small corrections in the control pulses and preserves the original bandwidth of the seed. The computational effort of each step of the FPA corresponds to the one of the numerical integration of a stabilized closed loop system. A piecewise- constant and a smooth numerical implementations are developed. Several numerical experiments with a N-qubit system illustrates the effectiveness of the method in several different applications including the conversion of piecewise-constant control pulses into smooth ones and the reduction of their bandwidth. Mathematics Subject Classification. 93D05, 93D30, 8108. Received November 11, 2019. Accepted August 18, 2020. 1. Introduction Quantum control theory is now an important subject whose introduction is presented in some textbooks [1, 3]. Many important problems of quantum control are essentially versions of open loop motion planning problems over the unitary Lie-group U(n), which can be considered by controllability theory, Lie Group decompositions, and optimal control theory [3]. In general, controllability theory provides only existence theorems. It is true that decomposition theory may furnish piecewise-constant control pulses as a solution of motion planning, but this solution is restricted to small dimensions n of systems state. Optimal control may furnish fast solutions in the context of state steering and quantum gate generation [13, 14]. However, when the dimension of the state The first author was partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq), Brazil, Project 305546/2016-3, and by Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP), Brazil, Project 18/17463-7. Keywords and phrases: Controllability, quantum control, right-invariant systems, Lyapunov stability, Banach Fixed-Point Theorem. 1 Polytechnic School ­ PTC, University of Sao Paulo (USP), Sao Paulo, SP, Brazil. 2 Centre Automatique et Systemes, Mines ParisTech, Paris, France. 3 Departamento de Automacao e Sistemas (DAS), Universidade Federal de Santa Catarina (UFSC), Florianopolis, SC, Brazil. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 P.S.P. DA SILVA ET AL. n is large, optimal control may be impossible to be implemented due to complexity issues [23]. For quantum systems, for instance N-coupled qubits, the dimension of the state is 2N , which gives a propagator that is a 2N × 2N complex matrix. This fact justifies the need of new methods, like numerical optimal control, Lyapunov methods and so on. Lyapunov stabilization [6, 7, 11, 12, 15, 24­27] may be applicable for large n, but they generate slow solutions (in the sense that that the final time must be large) when compared to the ones that are generated by optimal control, when this last approach is applicable. Hence the study of numerical methods that can be applied for large n and also produces a fast solution is a relevant field of research, indeed. In this paper, a piecewise-constant implementation of an algorithm means that the control pulses to be generated are piecewise-constant functions of time, whereas a smooth implementation means that the control pulses to be generated are smooth functions of time. The Krotov method [23], GRAPE [4, 8], and RIGA [19] in the piecewise-constant setting and GOAT [10], CRAB [20] or RIGA in the smooth setting, are efficient methods for solving the quantum gate generation problem. A comparison between the methods of Krotov, GRAPE, CRAB and GOAT for small dimensions (two and three qubits) is presented in [21]. A full comparison between the known methods (Krotov, GRAPE, CRAB, GOAT and RIGA) that includes high-dimensional systems is yet to be done, although some comparisons between the piecewise-constant version of RIGA and GRAPE were performed in [19] showing that RIGA is indeed a promising method. A smooth MATLABr implementation1 of RIGA is available in [16]. The basic problem of quantum gate generation considers controllable right-invariant systems with m inputs u = (u1, . . . , um) Rm and state X(t) evolving on the unitary Lie group U(n), that is, the set of n-square complex matrices X such that X X = I, where X is the conjugate transpose of X and I is the identity matrix. Here u(n) denotes its Lie-algebra formed by the n-square anti-Hermitian complex matrices, that is u(n) if = -. In this paper, kXk stands for the Frobenius norm of a complex matrix. In the context of coherent control, this question relies on an open loop steering problem. The models of such quantum systems are of the form X(t) = - H0 + m X k=1 uk(t)Hk ! X(t) = S0X(t) + m X k=1 uk(t)SkX(t), X(0) = I (1.1) where X U(n) is the state (propagator), S0 = -H0, Sk = -Hk u(n), and uk(t) R are the controls. The Quantum Gate Generation Problem is defined by: Problem 1.1. (Quantum Gate Generation Problem) fix Xgoal U(n) associated to the desired goal quantum gate Xgoal U(n) to be generated. Fix a desired final time Tf > 0. One says that this problem is solvable if it is possible to compute open-loop controls u : [0, Tf ] Rm such that system (1.1) is steered from X(0) = I to some X(Tf ) U(n) for a desired final fidelity. In quantum control, a fidelity function F : U(n) [0, 1] is a measure of the precision of the solution of a given problem2 . In this paper, the infidelity function I : U(n) [0, 1] will be defined by I( e X) = 1 - 1 n ktrace( e X)k 2 (1.2) and the fidelity function is given by I( e X) = 1 - F( e X). The gate fidelity and infidelity are defined respectively by F(X goalX(Tf )) and I(X goalX(Tf )), as considered in [9]. In this paper, it is assumed that an approximate solution of the Problem 1.1 is given. This work presents a method, called Fixed-Point Algorithm (FPA) that is able to improve the fidelity of the quantum gate generation. Before describing the FPA, let us give a context of the previous contributions of the authors to this problem. Inspired by the Coron's return method [2], the authors have developed a method of quantum gate generation 1Mainly based on the same ideas of Section 3.2 for integrating numerically the quantum system. 2The fidelity function normally is related to the probability of the correctness of the result after a measure. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 3 for driftless systems [24]. However, this method is not iterative, and hence the solutions are slow, that is, a large final time Tf that depends on the rate of convergence of the Lyapunov tracking control is needed. In [25], the method of [24] was generalized for systems with drift. However, it also generates slow solutions when compared for instance to GRAPE [8, 9]. In [19], we have introduced an iterative method, called RIGA, that produces results that are compatible with GRAPE, with the advantage that RIGA may also generate smooth controls. Note that GRAPE is conceived in a way that it can only generate piecewise-constant controls, and this may be a disadvantage for the physical implementation, since the small discontinuities of the pulses can be filtered by the communication channel that links the control pulse generator to the quantum device, generating distortions that will produce a fidelity degradation. In the present paper, it is shown that RIGA may be combined with FPA in order to shorten the runtime of the overall computations (see Tab. 1 of Sect. 4.2). Section 4.3 shows how to convert piecewise-constant control pulses (that could be generated by GRAPE) to smooth ones, improving the fidelity. Section 4.5 shows that the FPA can be used for shortening the bandwidth of a pre-computed control, also improving the fidelity. These potential applications enhances the interest of the FPA in quantum control. The main ideas of the algorithm will be presented now. The original approximated solution will be called the seed of the FPA (see Def. 2.1). Under certain conditions, the FPA is a virtually exact method in the sense that the final Frobenius error ` = kX(Tf ) - Xgoalk that is obtained in the step ` is such that lim` ` = 0 (exponentially). One first obtains, by right translation of the approximate solution, a new trajectory X(t) such that its final condition X(Tf ) is exactly equal to Xgoal. The idea is then to consider the trajectory X(t) as a reference trajectory and to apply a stabilizing feedback of the corresponding tracking problem, as done in previous works of the authors [24, 25]. The novelty of this work is that we consider the construction of an adequate right-translation R U(n) that corrects the action of the feedback in order to obtain an exact generation of quantum gate. The existence of this exact correction R is based on the Banach Fixed-Point Theorem. The FPA is an algorithm that obtains an approximation R` U(n) of this exact correction, where R0 is the identity matrix and R` = G(R`-1), where the map G is the flow of a closed loop stabilized system. Such map G will be a contraction and R` will converge exponentially when: (a) the error = 0 of the seed is small enough; and (b) the seed is attractive in the context of the tracking control. This second assumption (b) is shown to be satisfied with probability one with respect to the choice of the seed u(t) in a sense to be precised later. This technique may be viewed as a "fine tuning" of other techniques, and it produces small corrections of the original input u(t) of the approximate solution that are able to obtain the desired final precision. It will be clear that it suffices to show that such map G is a contraction and the proof of Banach Fixed-Point will assure the exponential convergence of R` to R. It will be shown that R` can be computed by successive numerical integrations of a closed-loop dynamics in the interval [0, Tf ]. Normally, the solution that is generated by the seed must have a good fidelity in order to assure the con- vergence of FPA described in Algorithm 1. In fact what is shown in this paper is a local result, just because the map G is only a "local contraction" and the radius of this domain cannot be estimated a priori. However, we shall present an iterative version of FPA (see Algorithm 2). For this algorithm, even if the map G is not a contraction, iterative applications of FPA can produce the desired fidelity. By comparing with RIGA [19], is not difficult to show that, if in each step of Algorithm 2 only one step of FPA is executed, then Algorithm 2 reduces3 to RIGA. Otherwise, if various steps of FPA are executed, then Algorithm 2 behaves as an "accelerated RIGA", as shown in the numerical experiments of Section 4.2. However, as the Lyapunov-based feedback that is related to our choice of the Lyapunov function (see (2.1)) is not bounded, if the fidelity of the seed is too low, Algorithm 2 may generate a huge control effort (eventually giving rise to numerical problems). In this case the combination of RIGA and FPA is indicated because RIGA always works "far" from the singular points of the Lyapunov function. Two different numerical implementations of FPA are considered. The first one considers piecewise-constant control pulses, and it is compatible with the solutions provided by GRAPE, Krotov and the piecewise-constant 3Without the protection against singular points of the Lyapunov function that RIGA provides. 4 P.S.P. DA SILVA ET AL. version of RIGA. The second one considers smooth inputs, and it is compatible with GOAT, CRAB and the smooth version of RIGA. Numerical experiments are presented here for an example that is also considered in [9, 19]. It consists of a coupled chain of N qubits and the goal is to implement a Hadamard gate for N = 3, 4, . . . , 10. The seed is first generated with the algorithm RIGA4 The FPA is then executed in order to improve the fidelity of the seed. The examples of Section 4 illustrates that the FPA may be useful in the following four scenarios: ­ First generate a piecewise-constant pulse that is an approximation of the solution of the problem and then use a piecewise-constant version of FPA in order to improve its fidelity. ­ First generate a continuous pulse by some method and then use a smooth version of FPA in order to improve its fidelity; ­ First generate a piecewise-constant pulse and then use a smooth version of FPA in order to convert it into a smooth pulse with the desired fidelity; ­ When the seed has a "large" bandwidth, the FPA may be combined with a pre-filtration of the seed, and this process may be iterated, shortening the bandwidth of the control pulses, while preserving, or even improving, the desired fidelity. Since the FPA produces small corrections of the control pulses, if the seed respects some restrictions of amplitude, bandwidth or other ones, these restrictions will be preserved up to small changes in the final solution. This is an important issue that may be included in the scenarios of applicability of the FPA. The idea is to generate the seed with all the necessary restrictions which may lead to a slow convergence and some difficulties to attain the desired fidelity. In this case, the application of FPA in combination to some method may shorten the total runtime of this combination with respect to the use of this method alone. This fact is illustrated in some numerical examples for the combination RIGA-FPA. A MATLABr implementation of FPA is available in [16]. This work is organized as follows. Section 2 presents the FPA. Section 3 discusses the two different imple- mentations of FPA. Section 4 presents the examples. Some numerical experiments cover the different scenarios of the applications of the FPA. Section 5 presents the convergence theorems concerning the FPA. Finally, Section 6 presents the concluding remarks. For convenience, many proofs were deferred to the Appendices. 2. Description of the fixed-point algorithm (FPA) In order to give a complete description of the FPA, it will be necessary to recall some results of [25]. 2.1. The closed loop error system Consider the Lyapunov function V: W U(n) [0, ) [25]: V( e X) = -Tr ( e X - I)2 ( e X + I)2 ! 0, (2.1) where W is the open subset of U(n) defined as W = {W U(n) | det(I + W) 6= 0}. (2.2) It was shown in [25] that V( e X) corresponds to a notion of distance between e X W and the identity matrix I. This notion of distance then naturally defines a (not bounded) notion of distance between two matrices X1, X2 of U(n) by considering the map dist : U(n) × U(n) [0, ) {} defined by dist(X1, X2) = V(X 1X2) when X 1X2 W and dist(X1, X2) = when X 1X2 6 W. 4First results of the combination RIGA-FPA where reported in the conference paper [17]. The smooth implementations of RIGA and FPA are available in [16]. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 5 Consider the reference system (on U(n)) X(t) = S0X(t) + m X k=1 uk(t)SkX(t), X(0) = X0, (2.3) As in [25] it will be considered that a reference input u : [0, Tf ] Rm is given and the corresponding solution X : [0, Tf ] U(n) of (2.3) is generated. Consider the (tracking) error matrix e X(t) = X (t)X(t) U(n). The dynamics of e X(t) is given by e X(t) = m X k=1 e uk(t)e Sk(t) e X(t), e X(0) = e X0, (2.4) where e uk(t) = uk(t) - uk(t) and e Sk(t) is given by (2.5c). Assume that e X(0) W. Consider the (smooth) feedback-law5 e uk(t) = e Uk(X(t), e X(t)) = Ktrace h Z e X(t) e Sk(t) i , (2.5a) uk(t) = e uk(t) + uk(t), (2.5b) e Sk(t) = X (t)SkX(t), (2.5c) where K is the chosen feedback gain6 , and Z is the map defined by Z( e X) = e X( e X - I)( e X + I)-3 . (2.5d) This control law produces the closed-loop dynamics7 (2.3)­(2.4)­(2.5). One may show that V = - Pm k=1 4 K e u2 k(t) 0. Several results of stability of this tracking error dynamics are studied in [24, 25] and those results are instrumental for the convergence results of the Section 5. 2.2. The fixed-point algorithm (FPA) The input data of FPA is called seed, and it may be produced by any known method. Definition 2.1. The pair {u(·), Xf } is called seed whenever the input u : [0, Tf ] Rm with u(t) = (u1(t), . . . , um(t)) is such that the solution of system (1.1) with X(0) = I and u(t) = u(t) produces X(Tf ) = Xf . Let > 0. One says that the seed is a -approximated solution of Problem 1.1, if the final condition Xf is such that kX(Tf ) - Xgoalk . The FPA is now presented: Algorithm 1. Fixed Point Algorithm (FPA) % BEGIN FPA % BEGIN STEP 0 - Integrate the open loop system Fix a desired infidelity I. Given a seed {u(·), Xf } (see Def. 2.1), integrate numerically the open loop system (2.3) with X0 = X(0) = (Xf ) Xgoal obtaining8 the reference trajectory X : [0, Tf ] U(n). Initialize R-1 = R0 = I. Choose Nfixed > 2 (maximum number of steps). Set ` = 0 and Infidelity = I(X goalXf ). 5It is shown in [24] that W is a positively invariant set for the time-varying closed-loop system (2.4)­(2.5). 6For simplicity, the feedback gains are supposed to be equal for all k = 1, . . . , m. 7The closed-loop system is also equivalently written as (1.1)­(2.3)­(2.5). 8The Proposition 5.5 assures that X(Tf ) = Xgoal. 6 P.S.P. DA SILVA ET AL. % END STEP 0 - Integrate the open loop system WHILE Infidelity > I and ` < Nfixed % BEGIN Step ` - integrate closed loop system Increment ` e X(0) = X 0R`-1 Integrate numerically system (2.4)­(2.5) with initial condition e X(0) R` = e X(Tf ) IF kR` - R`-1k kR`-1 - R`-2k. THEN (Contraction test failed) EXIT WHILE ELSE (Contraction test passed) u(·) = u(·) + e u(·) Infidelity = I(R `-1R`) END % END Step ` - Integrate closed loop system END % END While % END FPA Remark 2.2. It will be shown that the solution of system (1.1) with input u(·) that is generated in the step ` of Algorithm 1 is the solution of system (2.3)­(2.4) the in closed loop with R`-1-corrected feedback (5.6). Then, by Proposition 5.11, the infidelity of the step ` of the FPA is I(R `-1R`). Under mild assumptions, Theorem 5.9 imply that kR` - R`-1k converges exponentially to zero, and so it will be easy to conclude from and Proposition 5.6 that the infidelity also converges to zero exponentially under the same conditions. If the FPA has improved the original infidelity of the reference trajectory, one may accept a feedback correction R` coming from a map that is not a contraction (i.e., the "contraction test" has failed at some step `). Such a situation has taken place in some of the examples of Section 4. If the "contraction test" has failed, them Algorithm 1 finishes, just because new iterations of FPA will not improve the fidelity. Then it is possible to repeat the FPA for u(·) = u(·), as done in Algorithm 2 below9 . When Xf is not available, the open loop integration is performed with X(0) = I as in the implementation of the smooth version of FPA. Algorithm 2. (Iterated FPA) % BEGIN ALGORITHM 2.4 Fix a desired infidelity I. Assume that the seed {u(·), Xf } (see Def. 2.1) is such that Infidelity = I(X goalXf ) > I. WHILE Infidelity > I Execute FPA with seed {u(·), Xf } obtaining a control pulse u(·). Compute Infidelity = I(X goalX(Tf )). Set u(·) = u(·) Compute Xf = X(Tf ). END % end While % END ALGORITHM 2 9It is easy to show that, if the repetition of FPA fail in the contraction test in the first iteration every time, then Algorithm 2 reduces to RIGA. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 7 3. The numerical implementations of FPA This section will discuss the two ways of implementing the FPA, the piecewise-constant (discontinuous) and the smooth implementations. Since the heart of the FPA relies in the way that the open- and closed-loop systems are simulated, this section will be devoted to this aspect. In both implementations, the interval [0, Tf ] will be divided in Nsim equal parts, and the time t will be discretized at instants ts = s, s = 0, 1 . . . , Nsim, where = Tf /Nsim. By simplicity, the reference inputs uk(ts) and the input uk(ts) will be denoted respectively by: uk(ts) = uks (3.1a) uk(ts) = uks . (3.1b) In the piecewise-constant case, this means that uk(ts + ) = uks , for [0, ), s = 0, 1, . . . , Nsim - 1. (3.2) In the smooth case, for [0, ] the following linear interpolation is used as an approximation for the open-loop simulations: uk(ts + ) = - uks + uks+1 , s = 0, 1, . . . , Nsim - 1. (3.3) The discrete representation of the seed of the algorithm is the set of control inputs {uks , k = 1, . . . , m, s = 0, 1, . . . , Nf } that was produced by some algorithm, which produces a final condition X(Tf ) = Xf . These control inputs are interpreted respectively as (3.2) with Nf = Nsim - 1 in the piecewise-constant case, and as (3.3) with Nf = Nsim in the smooth case. The step ` of the FPA produces the inputs {uks , s = 0, 1, . . . , Nsim - 1}, interpreted as piecewise-constant or smooth inputs (approximated by (3.3)), according to the case. By simplicity, denotes also Xs = X(ts), Xs = X(ts) for s = 0, . . . , Nsim. In all implementations, a gain K > 0 is chosen. 3.1. The piecewise-constant implementation The piecewise-constant implementation of FPA is based on the piecewise-constant implementation of RIGA [18]. Denote Ts = S0 + Pm k=1 uks Sk and Ts = S0 + Pm k=1 uks Sk where Sk, k = 1, . . . , m are the system matrices. The exponential of a matrix will be computed using a 4th-order Pade approximation: exp(S) (I + S/2 + 2 S2 /12)(I - S/2 + 2 S2 /12)-1 (3.4) The open loop simulation reads: % BEGIN Step 0 ­ open-loop simulation10 X0 = X f Xgoal FOR s = 0, 1 . . . Nsim - 1 Xs+1 = exp Ts Xs END % END Step 0 The closed loop simulation corresponds to the simulation of system (2.4) in closed loop with the R`-1- corrected feedback (5.6), and this reads: % BEGIN Step ` ­ Closed-loop simulation X0 = I 8 P.S.P. DA SILVA ET AL. e X0 = X 0 FOR s = 0, 1 . . . Nsim - 1 e Xs = X sXs FOR k = 1, . . . , m uks = uks + Window(s)KTr h Z e XsR`-1 X sSkXs i END Xs+1 = exp (Ts) Xs END R` = X Nsim+1XNsim+1 % END Step ` - Closed Loop Simulation Recall that map Z is given by (2.5d). The function Window(s) is normally equal to one for all s. In the case that one has used a window-function in RIGA (see the example in the next section), one may choose Window(s) = 1, if s = 1, 2, . . . , Nf - 1 0, if s = 0 and s = Nf (3.5) where Nf = Nsim - 1. This will assure that the generated input will never be modified in the frontier points of the interval [0, Tf ]. 3.2. The smooth implementation of FPA The smooth implementation of FPA is based on the smooth implementation of RIGA. Both implementations are available in [16]. Using the same notation (3.1), recall that the control inputs are supposed to be smooth and in the open loop integration they will be approximated to piecewise-linear functions, which means that (3.3) holds. The idea of this implementation is to develop a 4th-order Runge­Kutta integration scheme for both closed-loop and open-loop cases. Since U(n) is not a Euclidean space, there is no sense in applying a Runge­ Kutta method directly to the dynamics11 (2.3)­(2.4). The idea is based on the method that is proposed in [5], and relies on a12 smooth-map W that is similar to the homographic function that was considered in [24] for defining the Lyapunov function V given by (2.1). Let W : W U(n) u(n) defined by e X 7 ( e X - I)( e X + I)-1 , where u(n) is the set of anti-Hermitian complex matrices such that ( - I) is invertible13 . The inverse of W is the smooth map X : W such that X(W) = -(W - I)-1 (W + I). From ([24], Eq. (9)), it is clear that V( e X) = kW( e X)k2 . Some standard computations show easily that, if e X(t) U(n) is a solution of a differential equation d dt e X(t) = e S(t) e X(t), with e S(t) u(n), then: W(t) = - 1 2 (W(t) - I)e S(t)(W(t) + I) (3.6) The equation (3.6) may be integrated numerically, instead of (1.1), with the advantage that u(n) is a Euclidean space, and so the Runge-Kutta method may be applied in a natural way. Define the map F : × u(n) by: F(W, e S) = - 1 2 (W - I)e S(W + I) (3.7) 11The Runge­Kutta method in this case will generate non-unitary matrices. 12The Cayley transformation considered in [5] is given by -W. 13Recall that W is the set of complex matrices e X such that ( e X +I) is invertible. It is easy to show that (W( e X)-I) = -2( e X +I)-1 and so (W( e X) - I) is always invertible, with inverse -1 2 ( e X + I). A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 9 then equation (3.6) reads W(t) = F(W(t), e S(t)). Each Runge­Kutta step of the open loop integration of (3.6) considers the interval [ts, ts+1] with W(ts) = 0 (corresponding to the identity in U(n)), with posterior correction by right-invertibility. Note that W(t) may not leave the region , otherwise it will reach a singularity (and numerical problems will certainly occur). The simulation of the closed loop system does not have this problem, since W is invariant for the closed loop system just because the Lyapunov function is non-increasing [25]. So W(t) = W( e X(t)) will never leave u(n). The details are presented in Appendix E. 4. Examples This section will present some numerical experiments considering a coupled chain of N qubits for which the Hamiltonian is given by: H(t) = J0 "N-1 X s=1 (s) z (s+1) z # + J " N X k=1 uxk (t)(k) x + uyk (t)(k) y # + Jgug(t)I, (4.1) where x, y, z are the Pauli matrices. The artificial input ug(t) is only a global phase input that may be disregarded in the application of the generated input, needed because our theory is developed in U(n) and not in SU(n). This example is a benchmark proposed in ([24], Eq. (9)), for testing a implementation of GRAPE. The goal is to generate a Hadamard gate. The Hadamard gate for N = 1 is defined by: H(1) = 1 2 1 1 1 -1 Then, one may obtain Xgoal = H(N) inductively by the Kronecker product H(N) = H(1) H(N - 1). The system (4.1) is considered with J0 = J = Jg = 2100 MHz. The final time Tf = 2N × 10-9 (s) where N is the number of qubits. The dimension of the Hilbert space is n = 2N and m = 2n + 1. The experiments have considered the infidelity function given by (1.2). The value of Nsim = 10N is considered in [9]. In this work, the value Nsim = 20N is chosen in all numerical experiments. The feedback gains K = 1 J K(N) where K(N) is the Nth entry of the vector [10, 10, 10, 2, 2, 1, 0.5, 0.25, 0.125, 0.0625]. The values of gains K were chosen with the aid of an option of the piecewise-constant MATLABr R2015a implementation of RIGA [18]. The same gains K are used both in the smooth and the piecewise-constant implementations of FPA. The smooth MATLABr implementation of RIGA and FPA is available in [16]. All numerical experiments were done with MATLABr R2015a running on a PC having a Intel(R) Core(TM) i7-8700 CPU @ 3.20\,GHz, 3.19\,GHz processor, with 16G of RAM and in a Windows 10 environment. 4.1. Results with the piecewise-constant implementation of FPA This subsection reports the results of the piecewise-constant implementation of FPA. The seed of FPA was generated by the piecewise-constant version of RIGA for N = 8 qubits obtaining a solution of an infidelity that is close to 0.005. The RIGA was executed with a Hamming-like window function (see Sect. E.2), which is an option for generating control pulses that are null at t = 0 and t = Tf . The window function used by FPA is given by (3.5), chosen in order to preserve the restrictions of the seed at the endpoints of the interval [0, Tf ]. Algorithm 2 was executed for a desired infidelity given by 0.001. The RIGA was also used for generating a solution with infidelity 0.001 (without the FPA). The results are presented in Table 1, in which TRIGA is the runtime of RIGA, TF P A is the runtime of FPA. IRIGA and IF P A are respectively the infidelity of the solutions produced by RIGA and FPA. To prevent from possible numerical errors of the algorithms, these infidelities are computed by an open-loop simulation using the MATLABr function expm instead of the Pade approximation (3.4), blue as a post-computation for precision evaluation (expm is not used in the method itself). The Table 1 also presents the number NRIGA of steps of RIGA, the number Ncalls of times that the FPA was executed in Algorithm 2. Furthermore, NF P A is the total sum of the number of steps of FPA that were 10 P.S.P. DA SILVA ET AL. Table 1. Results of the simulations in the piecewise-constant case. N IRIGA NRIGA IF P A Ncalls NF P A TF P A TRIGA+ (qubits) (s) TF P A (s) 8.0 0.00491 78.0 9.94 × 10-4 3.0 52.0 2311.0 5766.0 8.0 9.97 × 10-4 191.0 0 0 0 0 9922.0 Figure 1. This figure illustrates the numerical experiment for N = 8 qubits in the piecewise- constant case. The seed of FPA was produced by RIGA with an infidelity that is close to 0.005. Top: plot of the control pulses produced by FPA with an infidelity that is less than 0.001. Bottom: plot of the function max{|uk(t) - uk(t)| : k = 1, 2, . . . , m}, where u(t) is the seed input and u(t) is the solution that is produced by the FPA. executed in all calls of FPA of Algorithm 2. Note that Ncalls - 1 is the number of times that the FPA was aborted because the contraction test has failed. It is then clear that the runtime for RIGA alone is greater than the runtime of the combination RIGA plus FPA. Similar behavior has been verified for N = 2, 3, . . . , 10 qubits. Figure 1 shows the generated control pulses by the FPA for N = 8, illustrating that the FPA perform only small corrections in the control pulses. 4.2. Results with the smooth implementation of FPA The numerical experiments of this section consider the smooth implementation of the FPA. The context of these experiments are essentially the same of the piecewise-constant case, but using the smooth versions of RIGA and FPA. The results of Table 2 are presented with the same notations of the Table 1. That table allows to conclude again that the combination of RIGA with FPA leads to lower runtime for the same desired infidelity with analogous qualities of the final control pulses. Figure 2 illustrates the fact that the FPA only performs small corrections, preserving the original desired restrictions of the seed. The only difference with respect to the piecewise-constant case is the fact that the numerical experiments were performed for N = 3, 4, . . . , 10 qubits, illustrating the applicability of these algorithms for large dimensions. In fact, for N = 10 the dimension of A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 11 Table 2. Results of the simulations in the smooth case. N IRIGA NRIGA IF P A Ncalls NF P A TF P A TRIGA+ (qubits) (s) TF P A (s) 3.0 0.00489 24.0 9.67 × 10-4 2.0 23.0 0.812 1.99 3.0 9.99 × 10-4 55.0 0 0 0 0 2.59 4.0 0.00491 36.0 9.87 × 10-4 1.0 28.0 2.09 6.34 4.0 9.87 × 10-4 77.0 0 0 0 0 8.17 5.0 0.00499 266.0 9.84 × 10-4 4.0 67.0 14.9 91.4 5.0 9.95 × 10-4 477.0 0 0 0 0 135.0 6.0 0.00469 48.0 9.71 × 10-4 2.0 31.0 27.0 81.8 6.0 9.87 × 10-4 84.0 0 0 0 0 90.8 7.0 0.00484 93.0 9.26 × 10-4 1.0 11.0 46.3 496.0 7.0 9.73 × 10-4 130.0 0 0 0 0 632.0 7.0 0.00484 93.0 5 × 10-7 16 9067 9.16 × 104 9.31 × 104 8.0 0.00485 75.0 0.0010 3.0 43.0 1566.0 4444.0 8.0 9.98 × 10-4 148.0 0 0 0 0 5166.0 9.0 0.00494 189.0 0.0010 6.0 109.0 3.37 × 104 9.87 × 104 9.0 9.97 × 10-4 362.0 0 0 0 0 1.24 × 105 10.0 0.00493 42.0 0.0010 5.0 82.0 1.86 × 105 2.9 × 105 10.0 9.96 × 10-4 128.0 0 0 0 0 3.14 × 105 system matrices are n = 210 = 1024 and u(n) has dimension 220 . To prevent from possible numerical errors of the algorithms, the infidelities IRIGA and IF P A that are presented in Table 2 are (re)computed by an open-loop simulation using 4th-order Runge-Kutta method with an step /2 instead of = Tf Tsim . Remark 4.1. When the seed u that is generated by RIGA does not respect restrictions, the runtime of the combination of RIGA-FPA is not that different from the runtime of RIGA alone. For instance, for N = 8 qubits, the RIGA was executed without the Hamming-like window option14 in order to give an infidelity that is close to 0.005. Then the FPA was executed, produced a final infidelity of 0.001, in the same context of Table 1. Figure 3 shows the obtained control pulses. The runtime of RIGA-FPA was only 3% smaller than the runtime of RIGA alone. This behavior is also found in all other numerical experiments with N = 3, 4, . . . , 9, when RIGA is performed without this window option, both in the smooth and in the piecewise-constant implementation. However, the use of a Hamming-like window is highly recommended in practical applications. Another numerical experiment was done for the particular case where N = 7 qubits. We have generated the seed of Algorithm 2 with RIGA furnishing an infidelity approximately equal to 0.005 and we have executed Algorithm 2 with a desired infidelity of 5 × 10-7 , corresponding to line 11 of Table 2. After one day of runtime corresponding to 9067 steps of FPA, Algorithm 2 was able to produce the desired infidelity, indeed. It must be stressed that the desired infidelity must be compatible with the precision of the numerical integration15 . If it is not the case, it would be necessary to increase Nsim (since the numerical integration step is = Tf Nsim ). As the runtime of each step is proportional to Nsim, it is expected that runtime will increase accordingly. 4.3. Conversion of piecewise-constant control pulses into smooth ones of same fidelity In this numerical experiment, done with N = 6 qubits, a piecewise-constant pulse {uks : k = 1, . . . , m, s = 0, 1, . . . , Nsim - 1} was generated by the piecewise-constant version of RIGA, assuring a final fidelity that is 14This window option of RIGA multiplies the gain K of the feedback by a Hamming-like function Window(t). This assures that the control pulses will be zero at the frontier points of [0, T-f ]. This options leads to a small bandwidth of the control pulses. 15The implementation of [18] gives the infidelity between Xf computed with step and Xf computed with step 2 . This infidelity must be compatible with the desired infidelity. 12 P.S.P. DA SILVA ET AL. Figure 2. This figure illustrates the numerical experiment for N = 10 qubits in the smooth case. The seed of FPA was produced by RIGA with an infidelity that is close to 0.005. Top: plot of the control pulses produced by FPA with an infidelity that is less than 0.001. Bottom: plot of the function max{|uk(t) - uk(t)| : k = 1, 2, . . . , m}, where u(t) is the seed input u(t) is the solution that is produced by the FPA. smaller than 0.001. This pulse was interpreted as continuous one by the smooth version of FPA simply by linear interpolation (3.3), considering ukNsim = ukNsim-1 for k = 1, . . . , m. Under this assumption, the (smooth) open- loop simulation has produced an infidelity equal to 0.0105. Perhaps a reasonable conversion to a continuous signal could be obtained by filtering the control pulses, but this question will not be addressed here, since the smooth version of FPA is able to convert this signal to a continuous signal with any desired infidelity, chosen here to be less than 0.001 (to be precise, equal to 0.00094). The FPA has taken only 14 seconds to accomplish this task in 5 steps. Figure 4 shows the original piecewise-constant control pulses along with the superexposed smooth control pulses. Again, the FPA is able to perform small corrections to the original seed in order to obtain the desired fidelity. 4.4. The FPA preserves the bandwidth of its seed The computation of the spectra of the seed and of the control pulses that are produced by FPA shows that it produces only small changes in the spectrum of the seed. Furthermore, these changes are concentrated in low frequencies. In particular the FPA preserves the original bandwidth of the seed u(t). For instance, in Figure 5, the absolute value FFT's of control signals of the smooth case with N = 6 qubits are depicted. This plot shows the average (in k) of the absolute value of the FFT's, that is, it is a plot of 1 m Pm k=1 kFFT(uks )k and of 1 m P k=1 1m kFFT(uks )k. The same qualitative behavior is found in all numerical experiments of this paper. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 13 Figure 3. This figure illustrates the numerical experiment for N = 8 qubits in the smooth case. The seed of FPA was produced by RIGA with an infidelity that is close to 0.005 without the option of a Hamming-like window. Top: plot of the control pulses produced by FPA with an infidelity that is less than 0.001. Bottom: plot of the function max{|uk(t) - uk(t)| : k = 1, 2, . . . , m}, where u(t) is the seed input u(t) is the solution that is produced by the FPA. 4.5. The FPA for shortening the bandwidth The result depicted in Figure 5 suggests the following idea. Given a seed (see Def. 2.1) formed by control pulses with a large bandwidth, but with the desired infidelity, then one may construct a filter and execute the following algorithm16 : Algorithm 3. (FILTERED FPA) % BEGIN ALGORITHM 3 Fix a desired final infidelity I and an intermediary infidelity I1 > I. Fix a number of steps Ntimes. Given a seed {u(·), Xf } (see Def. 2.1) with small infidelity I(X goalXf ) but "large" bandwidth, execute the following FOR ` = 1 to Ntimes Filter u(·), obtain ufiltered(·) Set u(·) = ufiltered(·) Execute Algorithm 2 for the filtered signal u for the desired infidelity I1. Algorithm 2 furnishes u(·) Set u(·) = u(·) END Execute Algorithm 2 for the filtered signal u for the desired infidelity I. Algorithm 2 furnishes u(·) corresponding to the desired infidelity I % END ALGORITHM 3 16It is not the aim of this paper to discuss Algorithm 4.2 very deeply, but only to establish a potential application of the FPA for reducing the bandwidth of the control pulses respecting the desired fidelity. 14 P.S.P. DA SILVA ET AL. Figure 4. This figure superexposes the original piecewise-constant control pulses with the smooth control pulses that are obtained by the FPA, assuring a final infidelity of 0.00094. Figure 5. This figure shows the spectra (FFT) of the seed u(·) and the solution u(·) that is produced by FPA in the smooth case with N = 6 qubits. Due to the known symmetry of FFT, only the first Nsim/2 points of the FFT are plotted. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 15 Figure 6. This figure shows the spectra (FFT) of the seed produced by RIGA with the legend "RIGA". Note that the seed contains high frequencies components, that were completely eliminated by Algorithm 3, furnishing the solution with legend "filter+FPA". It must be pointed out is that the filtering in each step ` cannot increase the infidelity too much, otherwise the FPA works badly, and may present numerical instabilities since the closed loop simulation may attain points that are close to the frontier of W. In fact, a limitation of FPA is the fact that its seed must have a sufficient small infidelity. To test this idea, the chosen parameters are N = 4 qubits, I1 = 0.005,I = 0.001, and the low pass filter is a discrete one, that will filter the discrete signals uk(ts) for s = 0, 1, . . . , Nsim. The transfer function of the filter is (z)3 (z-1+)3 with = 0.2. Recall that = Tf Nsim is the sampling period. The effect of this filter corresponds approximately to have three poles in the frequency / = 0.2Nsim Tf . The RIGA normally furnishes a control pulse with "small "bandwidth. To produce a seed with a "large" bandwidth, one has chosen the parameter T of RIGA (T is the fundamental frequency of the seed of RIGA [19]) twenty times lower than the chosen one in all other simulations. This will produce a seed for the RIGA which is a sum of M = 11 harmonics of the angular frequencies 2 T , and these frequencies appear in the seed u of Algorithm 4.2 (see the FFT of u in Figure 6, with the legend "RIGA"). The value of Nsim in this numerical experiment is Nsim = 40N = 160. To avoid an undesirable effects at the endpoints of the interval [0, Tf ] since the filtering process do not preserve the null control at t = Tf , the Hamming-like window function (E.2) was used for modulating the gain K of the FPA in the closed loop simulation. The final result is shown in Figure 6 with the legend "filter+FPA". In fact, all the high-frequency components of the control pulses are almost completely eliminated after Ntimes = 10 iterations of Algorithm 3, respecting the desired infidelity of 0.001. 5. Convergence properties of the FPA The idea behind the construction of the right translation R of our main convergence result is illustrated in a very simple example. For this consider, the scalar system: 16 P.S.P. DA SILVA ET AL. x(t) = u(t) (5.1) where x(t), u(t) R. One will consider the tracking problem of a smooth trajectory x(t) that is obtained with x(0) = x0 and a given smooth reference input u(t). The standard method is to consider the error e x(t) = x(t) - x(t). So, the corresponding error dynamics is given by e x(t) = e u(t) where e u(t) = u(t) - u(t). Choose some > 0. Taking e u(t) = e U(e x(t)) = -e x(t) one will get e x(t) = -e x(t) (5.2) and hence e x(t) = F(t, e x0) = exp(-t)e x0 which assures exponential convergence for the tracking problem. Now, assume that one desires a finite time steering, namely e x(t) = 0 at t = Tf . For this, fix some r R. Consider that r-translated state e x1(t) = e x(t) + r and apply the control law e u(t) = e U(e x1(t)) = -e x1(t) to system (5.2). Note that the closed loop dynamics may be rewritten in the following form e x1(t) = -e x1(t) In particular (5.2) is invariant by r-translation, and so e x1(t) = F(t, e x0 + r) Assume that r is such that F(e x0 + r, Tf ) = r This means that x1(Tf ) = r, and so e x(Tf ) = 0. In other words, applying e u(t) = e U(e x1(t)) to (5.1), this control law will steer this system from x(0) = e x0 to x(Tf ) = 0. Clearly, for a fixed e x0 and Tf , the map G : R R defined by G(r) = F(e x0 + r, Tf ) = (x0 + r) exp(-Tf ) is a contraction. This follows easily from the fact that |G r | = | exp(-Tf )| < 1. In particular the Banach Fixed- Point Theorem always assures the existence of (a unique) fixed point r. Figure 7 illustrates this construction by considering x0 = 1, Tf = 1, and = 1. This simple idea will be directly translated to the more complex closed loop dynamics (2.4)­(2.5). For this, fix (a finite) c > 0 and consider the "closed ball"17 in U(n) with radius c: B V c (I) = {X U(n) | dist(X, I) c}, (5.3) 17It is shown in [25] that B V c (I) is indeed a compact set in U(n). A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 17 Figure 7. This figure illustrates the trick for constructing a finite time control law for steering the system from e x(0) = e x0 to e x(Tf ) = 0. Note that the trajectory e x1(t) = F(Tf , e x0 + r) is the r-translation of trajectory e x(t). Furthermore, r is a fixed-point of the map r 7 G(r) = F(Tf , e x0 + r). where "dist" is the notion of distance on U(n) induced by the Lyapunov function V in (2.1) (see the beginning of Sect. 2.1). It is clear that B V c (I) W since c > 0 is finite. In this work one will also consider the usual closed ball of radius > 0 induced by the Frobenius norm: B(I) = {X U(n) | kX - Ik }. (5.4) The following definition is instrumental in the context of the work [19]: Definition 5.1. A reference trajectory X : [0, Tf ] U(n) generated by a reference input u : [0, Tf ] Rm is said to be -attractive on B V c (I), where 0 < < 1 and c > 0, when the closed-loop system (2.3)­(2.4)­(2.5) satisfies18 dist( e X(Tf ), I) dist( e X(0), I), for all e X(0) B V c (I). (5.5) It is only the reference input that determines whether a reference trajectory is -attractive or not. The following properties stated in the next proposition shows the right-invariance of -attractive trajectories. Proposition 5.2. [19] If a reference trajectory X(t) that is generated by a solution of (2.3) with reference inputs u : [0, Tf ] Rm and initial condition X(0) = X0 U(n) is -attractive in B V c (I), then for every X(0) U(n) and every r > 0, the corresponding solution is also -attractive in B V r (I) (with the same (0, 1)). Furthermore, this is equivalent to say that this property is invariant by right-translation of the reference trajectory X(t). 18Since V( e X(t)) is non-decreasing and B V c (I) is compact, then e X(t) is well-defined and remains in B V c (I) W for t [0, Tf ]. 18 P.S.P. DA SILVA ET AL. The following proposition is proved in [19] and it shows that -attraction occurs with probability one with respect to the choice of the jet of the reference input. Before stating this result19 , we shall define some notations in the context of jet bundles [22]. Definition 5.3. Given a smooth map u : [0, Tf ] R Rm , the M-degree polynomial approximation of u around x0 given by its Taylor expansion at t is denoted by u. Two maps u : [0, Tf ] Rm and v : [0, Tf ] Rm are M-equivalent if their M-degree polynomial Taylor approximations at t coincides, that is, u = v. This equivalence relation induces the set Jt(Rm , [0, Tf ]) of all equivalent classes of smooth maps in C ([0, Tf ], Rm ). The equivalent class corresponding to a representative u will be denoted by [u] and is called the M-jet of u. As the set RM [t] of polynomials of degree M in the variable t may be identified with RM+1 (by choosing some ordering of their real coefficients), then the M-jet [u] may be canonically identified with a vector in RM+1 m . Proposition 5.4. [19] Fix r > 0 and choose t [0, Tf ]. One says that [u] (RM+1 )m is an admissible M-jet if every smooth map v : [0, Tf ] Rm inside the equivalent class [u] generates a -attractive reference trajectory in B V r (I) for some (0, 1) ( may depend on v). Then, there exist M N large enough such that, a randomly chosen M-jet [u] (RM+1 )m is admissible with probability one. The key ingredient of the proof of the convergence of the FPA is the Banach Fixed-Point Theorem applied to the flow of a closed-loop system. It is assumed that a seed is given. A reference trajectory X is then defined in the context of the next proposition: Proposition 5.5. Let X : [0, Tf ] U(n) be the solution of (1.1) that is obtained with X(0) = I and the application of a -approximated seed u (see Def. 2.1). Then, by right-translation, one may construct a reference trajectory X : [0, Tf ] U(n) with X(0) = X0 = X(Tf ) Xgoal that is generated by the input u(t) = u(t). By construction, X(Tf ) = Xgoal and kX0 - Ik . Proof. By right-invariance of (1.1), since X0 = X0R1, with R1 = X(Tf ) Xgoal, then X(Tf ) = X(Tf )R1 = Xgoal. Since the Frobenius norm is right- and left- invariant by multiplications by unitary matrices, kX0 - Ik = kX(Tf ) Xgoal - Ik = kX(Tf )[X(Tf ) Xgoal - I]k = kXgoal - X(Tf )k . Fix Tf > 0 and consider a reference trajectory X : [0, Tf ] U(n) with X(Tf ) = Xgoal, as in the context of the last proposition. Now, fix R U(n), and consider system (2.3)­(2.4) in closed-loop with the R-corrected feedback e u1k(t) = e Uk(X(t), e X(t)R) = Tr h Z e X(t)R e Sk(t) i , (5.6) where e Sk(t) is defined by (2.5c). Note that20 , by restricting to the initial instant t0 = 0, the (time-varying) closed-loop system (2.4)­(5.6) admits a well-defined flow F : [0, Tf ] × W U(n) for which e X(t) = F(t, e X0) [25]. Take the right-translation e X1(t) = e X(t)R. (5.7) By right-invariance of system (2.4), the dynamics of e X1(t) is time-varying and given by: e X1(t) = m X k=1 e u1k(t)e Sk(t) e X1(t), e X1(0) = e X0R (5.8a) e u1k(t) = fkTr h Z e X1(t) e Sk(t) i , (5.8b) 19The original statement of [19] is different. It has been modified here for the sake of conciseness. 20When system (2.3) is disregarded from the closed-loop system (2.3)­(2.4)­(5.6), it turns out that the "reduced" closed-loop system (2.4)­(5.6) is time-varying. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 19 that is, such system corresponds to same equations of the original closed-loop system (2.4)­(2.5), but with e X replaced by e X1 and with the right-translated initial condition e X1(0) = e X0R. Since e X(t) = F(t, e X0), it is immediate to show that e X1(t) = F(t, e X0R). In particular, by assuming that R U(n) is a fixed point of the map21 GTf , e X0 (R) = F(Tf , e X0R), (5.9) that is, GTf , e X0 (R) = R, then it follows that e X1(Tf ) = R. By (5.7) one has e X(Tf ) = I, which in turn implies X(Tf ) = X(Tf ) e X(Tf ) = Xgoal. Then it is easy to see that the step ` of Algorithm 1 corresponds to the numerical integration of system (5.8) with e X(0) = e X0R`-1 . In other words: Proposition 5.6. The construction of R` U(n) of the Algorithm 1 corresponds to the iteration R` = GTf , e X0 (R`-1). Definition 5.7. Fix Tf > 0 and consider a reference trajectory X : [0, Tf ] U(n) which is the solution of (2.3) for arbitrarily fixed reference inputs. One says that the flow F (restricted to t0 = 0) of the (time-varying) closed-loop system (2.4)­(5.6) is a local radial contraction (with respect to the Frobenius norm) if there exist 0 < < 1 and r > 0 such that22 kF(Tf , e X0) - Ik k e X0 - Ik, for all e X0 B V c (I). (5.10) Condition (5.10) above is (locally) ensured by the next proposition if the reference trajectory is -attractive. Theorem 5.8. Let X : [t0, Tf ] U(n) be a given reference trajectory (solution of (2.3)). Assume that X is -attractive in B V r (I) in the sense of Definition 5.1. Hence for every such that 2 (, 1), there exists c > 0 small enough such that (5.10) holds for all e X0 B V c (I). Proof. See Appendix C. The existence of a fixed point R of GTf , e X0 in (5.9) is (locally) assured by the result below, whose proof is deferred to Appendix A: Theorem 5.9. Fix Tf > 0. Consider a reference trajectory X : [0, Tf ] U(n) which is the solution of (2.3) for given reference inputs such that (5.10) holds for some 0 < < 1 and c > 0. Fix any < < 1. Then, there exist 1, 2 > 0 such that, for every fixed e X0 B1 (I), one has that the map GTf , e X0 : B2 (I) B2 (I) defined by (5.9) is well-defined and is a -contraction. In particular, for every fixed e X0 B1 , the Banach Fixed-Point Theorem implies that the map GTf , e X0 admits a unique fixed point R in B2 (I). Definition 5.10. When a fixed point R U(n) of GTf , e X0 exists, then it will be called an exact correction. If R U(n) is such that kR1 - Rk (Frobenius norm) with R1 = GTf , e X0 (R), then R will be called an -approximation of the exact correction. Proposition 5.11. Fix Tf > 0 and consider that the reference trajectory X : [0, Tf ] U(n) is the solution of (2.3) for arbitrarily fixed reference inputs. Assume that X(0) is chosen so that X(Tf ) = Xgoal. Let R be an -approximation of the correction. Then, the closed-loop system (2.4) with the R-corrected feedback (5.6) will produce a final state X(Tf ) such that kX(Tf ) - Xgoalk . Now let R1 = GTf , e X0 (R). Then the infidelity I(X goalX(Tf )) = I(R1R ) = I(R R1). 21At this point, we are not being completely precise, since the domain of the map GTf , e X0 is not defined. 22Recall that the "closed ball" B V c (I) is defined by (5.3) based on the notion of distance induced by the Lyapunov function V, whereas the usual closed ball induced by the Frobenius norm is given in (5.4). 20 P.S.P. DA SILVA ET AL. Proof. Let e X1(t) = e X(t)R, and recall that R1 , GTf , e X0 (R) = e X1(Tf ) and e X(0) = X (0) (see Prop. 5.5). In particular, since the Frobenius norm is invariant by left- and right-multiplication by a unitary matrix, one has k e X(Tf ) - Ik = k e X1(Tf )R - Ik = kR1R - Ik = k(R1 - R)R k = kR1 - Rk . Similarly, kX(Tf ) - Xgoalk = kX (Tf )[X(Tf ) - Xgoal]k. Since X(Tf ) = Xgoal, one gets kX(Tf ) - Xgoalk = k e X(Tf ) - Ik . This means that the final error will be less than or equal to . Now, by the same reasoning, it follows that X goalX(Tf ) = X (Tf )X(Tf ) = e X(Tf ) = e X1(Tf )R = R1R . For a fixed e X0, Algorithm 1 of the previous section is a numerical method for determining an -approximation of the exact correction, whose existence is ensured by Theorem 5.9. The well-known proof of the Banach Fixed- Point Theorem finds the unique fixed-point R of a contraction G by taking the limit of the sequence R` as ` , with R`+1 = G(R`). Note that Algorithm 1 is essentially the computation of R` in this way. Furthermore, if G is a -contraction, then `+1 = kR`+1 - R`k = kG(R`) - G(R`-1)k kR` - R`-1k. In particular, ` ` 0. In particular, from Proposition 5.11, the convergence of the FPA will assure an exponential convergence of the final error of the quantum gate. 6. Conclusions This paper presented an iterative Lyapunov-based algorithm for improving the precision of quantum gate generation for systems with drift in a given fixed finite time Tf in the context of coherent (open-loop) control. When FPA converges, this convergence is exponential, and so the fidelity can be exponentially improved by FPA. The FPA will converge only when its seed (the original approximated solution) is already a good approximation of the quantum gate generation and it is also a -attractive trajectory in the sense of Definition 5.1. When the FPA does not converge, which is detected when the contraction test of Algorithm 1 fails, the Algorithm 2 is still able to reduce the infidelity to desired one. This situation was found in the examples of this work. The FPA may be combined with any technique for generating its seed, for instance, the Krotov method, [23], GRAPE [8], CRAB [20] or RIGA [19] in the piecewise-constant setting, and with GOAT [10] or RIGA in the smooth setting. An interesting feature is that FPA promotes only very small corrections in the inputs that were previously computed for its seed (see Figs. 1 and 2 respectively for the piecewise-constant and the smooth cases). It is important to stress that the RIGA [19] is already a method that produces exponential convergence of the solution23 . Normally, RIGA has a convergence rate that is compatible with the one of FPA (see Rem. 4.1). However, when restrictions on the inputs are included (for instance the input format of Figs. 1 and 2, beginning and finishing with zero amplitude) this makes the convergence of RIGA (alone) slower than the combination RIGA-FPA (see Tabs. 1 and 2 respectively for the piecewise-constant and the smooth cases). Although the smooth version of the FPA is meant to be used with a continuous seed, one may start with a piecewise-constant set of control pulses and use the FPA to convert it to a smooth set of control pulses with the desired fidelity. An important feature of both RIGA and FPA is the presence of the simultaneous qualities: (a) they can consider high dimensional systems in implementations running fast in affordable personal computers; (b) they can generate smooth control pulses with small bandwidth; (c) they are rather flexible to be applied easily to any quantum system of the form (1.1). The FPA may be used also for shortening the bandwidth of control pulses, preserving or improving fidelity, as suggested by the numerical experiments regarding Algorithm 4.3. Appendix A. Proof of Theorem 5.9 Along this proof, one will consider U(n) Cn×n as a real manifold. For this one will consider the standard projection : C R2 such that a + b 7 (a, b). Analogously, the inverse map $ : R2 C2 is defined by (a, b) 7 a + b. Given a complex matrix A + B Cn×n , such projection may be extended to a map A + B 7 (A, B) Rn×n ×Rn×n . In this context, one shall identify R2n2 = Rn×n ×Rn×n = Cn×n . Note that the Euclidian 23At least when Tf is large enough. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 21 norm on R2n2 induces the Frobenius norm on Cn×n . An open ball of Cn×n with center X and radius > 0 with the Frobenius norm will be denoted by BF (X) (and will be identified with the corresponding ball of R2n2 with the Euclidean norm). The ball B(X) U(n) stands for BF (X) U(n). Closed balls will be denoted as usual, for instance B(X). One will consider the restriction F(t, e X0) to the initial instant t0 = 0 of the flow of the time-varying system (2.4)­(2.5) as a map F : [0, Tf ] × W R2n2 . One will denote the singular-value norm of a linear operator L : Rp Rq by kLk. The proof of Theorem 5.9 relies on the result below, which in turn is proved in Appendix B. Lemma A.1. Under the assumptions of Theorem 5.9, if one chooses any < < 1, then there exists > 0 such that (Frobenius norm): kF(x1) - F(x2)k kx1 - x2k, for all x1, x2 B(I), where F : W U(n) R2n2 is defined by F( e X) = F(Tf , e X). Now, for showing Theorem 5.9, fix < < 1, and take = () > 0 as in Lemma A.1 above. As GTf , e X0 (R) = F( e X0R), now one will construct 1 > 0 and 2 > 0 such that: (a) e X0R B(I) for all R B2 (I) and e X0 B1 (I); (b) F( e X0R) B2 (I) for all R B2 (I) and e X0 B1 (I). In order to construct 1 and 2 assuring the properties (a) and (b), let R = R - I and X = e X0 - I. Note that k e X0R - Ik = kX + R + XRk kXk + kRk + kXkkRk 1 + 2 + 12. Let 1 = b2, where b (0, 1) will be chosen afterwards. Then, condition (a) is implied by following condition: 1 + 2 + 12 = 2[1 + b + b2] < , which in turn is equivalent to 2 < 1+b+b2 . Assuming that 2 < 1 and b < 1, then in order to ensure that e X0R B(I), it suffices to choose 2 min{ 3 , 1} Now, to assure that F( e X0R) B2 (I), note by Lemma A.1 (applied with x2 = I and x1 = e X0R) that kF( e X0R) - Ik k e X0R - Ik. Hence, kF( e X0R) - Ik 2[1 + b + b2]. It is thus clear that if one chooses 2 such that 2[1 + b + b2] 2, (A.1) then it will be true that F( e X0R) B2 (I). Simple manipulations show that (A.1) is equivalent to have: b (1 - )/[(1 + 2)] = . As (, 1), one may choose b = min{1, }. This shows that one may construct 1 and 2 such that (a) and (b) hold. Now, recall that GTf , e X0 (R) = F( e X0(R)). Given e X0 U(n) and W Cn×n , note that k e X0Wk = trace(W e X 0 e X0W) = trace(W W) = kWk. Hence, to complete the proof of Theorem 5.9, note that kF( e X0R1)- F( e X0R2)k k e X0R1 - e X0R2k = k e X0(R1 - R2)k = kR1 - R2k. Appendix B. Proof of Lemma A.1 The present considers the same notations and the identification R2n2 = Cn×n described in the beginning of Appendix A. The proof of Lemma A.1 is based on the following mathematical analysis results: 22 P.S.P. DA SILVA ET AL. Lemma B.1. Let F : U Rp Rq be a smooth map, where U Rp is an open subset. Let x0 U and assume that F x x0 = < . Fix any (, ). Then, there exists > 0 such that, for every x1, x2 B(x0) U, one has kF(x1) - F(x2)k kx1 - x2k. Proof. The proof is left to the reader. It is in fact an easy consequence of the convexity of a ball, the continuity of the singular-value norm, and the mean value theorem. Lemma B.2. Let H : U Rp Rq be a smooth map, where U Rp is an open set, and let u0 U. Assume that there exist , > 0 such that B(u0) U and kH(u) - H(u0)k ku - u0k, for all u B(u0). Then H u |u0 . Proof. Note that the directional derivative H u |u0 · h is given by limt0 H(th+u0)-H(u0) t . For |t| small enough, one gets kH(th+u0)-H(u0) t k ||khk, and so the result easily follows since the singular-value norm of a matrix is its greater amplification factor. Assume that exp : Cn×n Cn×n stands for the exponential map and log : BF 1 (I) Cn×n Cn×n stands for the logarithm map, which is well-defined and smooth inside the ball centered at the identity matrix and with unit radius with respect to the Frobenius norm. One will abuse notation and consider the exponential as a map from R2n2 to R2n2 , and the logarithm as a map from BF 1 (I) R2n2 to R2n2 . As exp() = I + + O(), it is easy to show that exp u 0 = I, the 2n2 × 2n2 identity matrix. By the inverse mapping theorem, it is clear that log x I = I. As kIk = 1, a double application of Lemma B.1 allows one to show the following proposition: Proposition B.3. Fix any a (1, ). Then, there exists > 0 such that, for the ball BF (I) R2n2 and the open set U = log(BF (I)) (containing the null matrix), one has k exp(u1) - exp(u2)k aku1 - u2k, u1, u2 U R2n2 , (B.1a) k log(x1) - log(x2)k akx1 - x2k, x1, x2 BF (I) R2n2 . (B.1b) Proof. First apply Lemma B.1 to the maps exp and log, obtaining inequalities (B.1a) and (B.1b) on the open balls BF 1 (0) and BF 2 (I), respectively. Then, take > 0 small enough so that BF (I) BF 2 (I) log-1 (BF 1 (0)). By construction, (B.1b) holds on BF (I), and (B.1a) holds on U = log BF 1 (I) . Proof. (of Lem. A.1) Fix (, 1). Let a1 = p /. Fix any 1 < a < a1 and apply Proposition B.3 for such value of a > 1. Now, note that24 one can always take a smaller value of > 0 in Proposition B.3 in a way that B(I) B V c W, where W is defined in (2.2). This follows from ([25], Ineq. (C1) in Appendix 3). Let F : W U(n) R2n2 be defined by F( e X) = F(Tf , e X). From the assumptions of Theorem 5.9, one obtains kF( e X) - Ik k e X - Ik, e X B(I) U(n) In particular, as < 1, one has F(B(I)) B(I). Let U = log(B(I)) and U = u(n) U. Clearly, exp(U) BF (I) U(n) = B(I). Then one may define the map H : U B(I) by H(u) = F exp(u). By definition, kH(u) - Ik = kF(exp(u)) - Ik k exp(u) - exp(0)k aku - 0k, u U. From Lemma B.2, one has k H u 0 k a. Let 1 = a1 > a. By Lemma B.1, there exists an open neighborhood V U of the null matrix such that kH(u1) - H(u2)k 1ku1 - u2k, u1, u2 V. 24Once again one is abusing notation through the identification R2n2 = Cn×n. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 23 Since V u(n) is an open set containing the null matrix, it follows that exp(V) is an open set of U(n) containing I. Choose > 0 small enough such that B(I) exp(V) exp(U) B(I). Let x1, x2 B(I), and u1 = log(x1) and u2 = log(x2). Then, by Proposition B.3, one gets: kF(x1) - F(x2)k = kH(u1) - H(u2)k 1k log(x1) - log(x2)k aa1kx1 - x2k, x1, x2 B(I). Note that aa1 a2 1 = . This shows Lemma A.1. Appendix C. Attractiveness and local radial contractions The proof of Theorem 5.8 is based on the following lemma. Lemma C.1. Fix > 0. Let X U(n) and let X = U diag(exp(1), . . . , exp(n))U be its eigenstructure, where U U(n), and i (-, ], i = 1, . . . , n. Let H : U(n) R be defined by H(X) = Pn i=1 2 i and G : U(n) R be defined by G(X) = kX - Ik2 . Then, there exists c1 > 0 such that, for every X B V c1 (I), one has: (a) (1 - ) H(X) 4 V(X) (1 + ) H(X) 4 ; (b) (1 - )H(X) G(X) (1 + )H(X). Proof. (of Lem. C.1) Recall that V(X) = Pn i=1(tan i 2 )2 [24]. Let O denote a function such that limx0 O(x) |x| = 0. Since tan(x) = x + O(x), it follows easily that tan2 (/2) = 2 /4 + P(), where P() is such that lim0 P() (2/4) = 0. In particular, V(X) = n X i=1 2 i 4 1 + P(i) 2 i 4 . Fix > 0, and take r small enough such that |i/2| r implies that P(i) 2 i 4 . By the continuity of tan2 (/2) and the fact that this function is an increasing map when restricted to the interval (-, ], then one take c1 small enough such that tan2 (/2) < c1 implies that |/2| < r. Now, if V(X) c1 it is clear that P(i) 2 i 4 , and this shows (a). From ([25], Proof of Lem. C2) one has G(X) = Pn i=1(2 sin i 2 )2 . Hence, in a similar fashion, one may show (b) (with a common c1) by using the fact that (2 sin 2 )2 = 2 + Q() where lim0 Q() 2 = 0. Proof. (Of Thm. 5.8) One denotes here e X1 = F(Tf , e X0), where F is the flow of the closed-loop system (2.4) restricted to the initial instant t0 = 0. Since F(Tf , I) = I one will assume that e X0 6= I, otherwise (5.10) is trivial. The assumption is equivalent to say that V( e X1) V( e X0) for all e X0 B V c1 (I). Note that (5.10) is equivalent to say that G( e X1) G( e X0) 2 for all e X1 B V c1 (I). From the part (a) of the last lemma, it is easy to show that, for all > 0 one may construct c1 such that (1 - )H( e X1) (1 + )H( e X0) V( e X1) V( e X0) (1 + )H( e X1) (1 - )H( e X0) , (C.1) 24 P.S.P. DA SILVA ET AL. for all e X0 B V c1 (I). Using the part (b) of the last lemma, one may obtain a similar inequality that replaces V( e X1) V( e X0) by G( e X1) G( e X0) . Note that, if x, y are real numbers in [a, b], then |x - y| (b - a). Then, from these two inequalities, it is easy to verify that V( e X1) V( e X0) - G( e X1) G( e X0) (1 + )H( e X1) (1 - )H( e X0) - (1 - )H( e X1) (1 + )H( e X0) 4H( e X1) (1 - 2)H( e X0) . In particular, G( e X1) G( e X0) V( e X1) V( e X0) + 4H( e X1) (1 - 2)H( e X0) . After using the left side of (C.1), one gets G( e X1) G( e X0) V( e X1) V( e X0) 1 + 4(1 + ) (1 - 2)(1 - ) . One concludes the proof by taking small enough such that h 1 + 4(1+) (1-2)(1-) i < 2 / (recall that 2 / > 1 by assumption). So G( e X1) G( e X0) 2 . Appendix D. Some properties of the Lyapunov function and the map Z Proposition D.1. Fix c > 0 and consider the ball B V c (I) defined by (5.3). Then (a) There exists k1, k2 > 0 such that k1k e X - Ik2 V( e X) k2k e X - Ik2 , e X B V c (I). (b) There exists k3, k4 > 0 such that k3k e X - Ik2 kZ( e X)k2 k4k e X - Ik2 , e X B V c (I) (c) There exists c1 > 0 such that kZ( e X)k2 c1V( e X), e X B V c (I). Proof. See ([19], Appendix 6). Appendix E. Smooth implementations of FPA As said in Section 3.2, for the open loop simulation, each step of 4th-order Runge-Kutta in the interval [ts, ts+1] relies on the right-invertibility of system (2.3). For fixed s {0, 1, . . . , Nsim}, denote W(t) = W(X(t)X s ). The simulation of W(t) in the interval [ts, ts+1] using (3.6) will always consider the initial condition W0 = 0 at t = ts(corresponding to the identity in U(n)). The final value of each step will be corrected by right- invertibility, accordingly. In other words, denoting Ws+1 = W(ts+1), then Xs+1 = X(Ws+1)Xs. Let [0, ], Define S(ts + ) = S0 + Pm k=1 uk(ts + )Sk, where Sk are the system matrices for k = 0, 1, . . . , m and uk(ts + ) is computed by linear interpolation as in (3.3). The open loop simulation reads: % BEGIN Step 0 ­ Open-loop simulation X0 = I W0 = 0 FOR s = 0, 1 . . . Nsim - 1 % BEGIN RUNGE KUTTA 4th-order k1 = F(W0, S(ts)) A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 25 k2 = F(W0 + k1 2 , S(ts + 2 )) k3 = F(W0 + k2 2 , S(ts + 2 )) k4 = F(W0 + k3, S(ts + )) Ws+1 = W0 + 1 6 (k1 + 2k2 + 2k3 + k4) % END RUNGE-KUTTA 4th-order Xs+1 = X(Ws+1)Xs END Xf = XNsim FOR s = 0, 1 . . . Nsim Xs = XsX f Xgoal END % END Step 0 ­ Open-loop simulation There are different approximations used in the open-loop and the closed loop simulations. For the open-loop, the smooth control u(t) is approximated by the linear interpolation (3.3), whereas the closed-loop integration considers that the feedback is smooth, but the values of X(t) in 2.5 are conveniently interpolated, as it will be stated. In order to avoid cumulative errors, the value of Xf will not be deduced from e X(Tf ), and that is the reason why Xf must be computed above25 in each step 0 of the FPA. The values of Ws+1 that are previously obtained in the open loop simulation are used in the closed loop simulation, since the intermediary value of X(ts + ) for = /2 is needed. The interpolation is done in the space u(n) using these values of Ws+1 in the following way: X(ts + 2 ) = X 1 2 (W0 + Ws+1) Xs = X 1 2 Ws+1 Xs Let e Sk(ts + ) = X(ts + ) SkX(ts + ). The computation of the feedback-law is given by e uk(ts + , W) = Window(ts + )Ktrace h ZW (W)e Sk(ts + ) i (E.1) which is done easily with ZW (W) = 1 4 W(W + I)(W - I) being the expression of (2.5d) as a function of W. Define T(ts + , W) = Pm k=1 e uk(ts + , W)e Sk(ts + ). The function Window(ts + ) for [0, ] is obtained by linear interpolation from the values of Window(ts), s = 0, 1, . . . , Nsim analogously to (3.3). One may take these values of Window(ts) all equal to one. When it is imperative that the values of control pulses will not be modified by FPA at t = 0 and t = Tf , one may take the same window function (3.5), but with Nf = Nsim and for simplicity Window(s) stands for Window(ts), for s = 0, 1, . . . , Nf . Sometimes it is useful to use a Hamming- like window function given by (E.2) (that is also used in RIGA). It may let the convergence of FPA slower than the one with the unitary window but it produces a smaller bandwidth and it respects common restrictions of null control at the endpoints of the interval [0, Tf ]. Window(t) = 1 2 1 - cos 2 t Tf (E.2) The closed-loop simulation reads: % BEGIN Step ` ­ Closed-loop simulation e X0 = X 0 W0 = W( e X0R`-1) 25This is not the case for the implementation of the piecewise-constant case. 26 P.S.P. DA SILVA ET AL. FOR s = 0, 1 . . . Nsim - 1 % BEGIN RUNGE KUTTA 4th-order e uks = e uk(ts + , Ws) % Save the value of the feedback control k1 = F(Ws, T(ts)) % (save the values of e uk(ts, Ws))) k2 = F(Ws + k1 2 , T(ts + 2 )) k3 = F(Ws + k2 2 , T(ts + 2 )) k4 = F(Ws + k3, T(ts + )) Ws+1 = Ws + 1 6 (k1 + 2k2 + 2k3 + k4) % END RUNGE-KUTTA 4th-order END e ukNsim = e uk(tNsim , WNsim )) % Save the Last value of the feedback control R` = X(WNsim ) % END Step ` - Closed loop simulation References [1] S. Cong, Control of Quantum Systems: Theory and Methods. John Wiley & Sons (2014). [2] J.-M. Coron, Control and Nonlinearity. American Mathematical Society (2007). [3] D. D'Alessandro, Introduction to Quantum Control and Dynamics. Chapman & Hall/CRC, Boca Raton (2008). [4] P. de Fouquieres, S.G. Schirmer, S.J. Glaser and I. Kuprov, Second order gradient ascent pulse engineering. J. Magn. Reason. 212 (2011) 412­417. [5] F. Diele, L. Lopez and R. Peluso, The Cayley transform in the numerical solution of unitary differential systems. Adv. Comput. Math. 8 (1998) 317­334. [6] D. Dong and I.R. Petersen, Quantum control theory and applications: a survey. IET Control Theory Appl. 4 (2010) 2651­2671. [7] Symeon Grivopoulos and Bassam Bamieh. Lyapunov-based control of quantum systems, in 42nd IEEE CDC, Vol. 1 (2003) 434­438. [8] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbruggen and S.J. Glaser, Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reason. 172 (2005) 296­305. [9] N. Leung, M. Abdelhafez, J. Koch and D. Schuster, Speedup for quantum optimal control from automatic differentiation based on graphics processing units. Phys. Rev. A 95 (2017) 042318. [10] S. Machnes, E. Assemat, D. Tannor and F.K. Wilhelm, Tunable, flexible, and efficient optimization of control pulses for practical qubits. Phys. Rev. Lett. 120 (2018) 150401. [11] M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential. Ann. Inst. Henri Poincare (C) Non-Linear Anal. 26 (2009) 1743­1765. [12] M. Mirrahimi, P. Rouchon and G. Turinici, Lyapunov control of bilinear Schrodinger equations. Automatica 41 (2005) 1987­ 1994. [13] J.P. Palao and R. Kosloff, Quantum computing by an optimal control algorithm for unitary transformations. Phys. Rev. Lett. 89 (2002) 188301. [14] J.P. Palao and R. Kosloff. Optimal control theory for unitary transformations. Phys. Rev. A 68 (2003) 062308. [15] Y. Pan, V. Ugrinovskii and M.R. James, Lyapunov analysis for coherent control of quantum systems by dissipation, in 2015 American Control Conference (ACC) (2015) 98­103. [16] P.S. Pereira da Silva and P. Rouchon, RIGA and FPA, quantum control with smooth control pulses [source code] (2019). [17] P.S. Pereira da Silva, P. Rouchon and H.B. Silveira, Geracao rapida e virtualmente exata de portas quanticas via metodos iter- ativos do tipo Lyapunov, in Proc. CBA'2018 - Congresso Brasileiro de Automatica: CBA'2018, Brazilian Control Conference. Joao Pessoa, Brazil (2018). [18] P.S. Pereira da Silva, H.B Silveira and P. Rouchon, RIGA, a fast algorithm for quantum gate generation [source code] (2019). [19] P.S. Pereira da Silva, H.B. Silveira and P. Rouchon, Fast and virtually exact quantum gate generation in U(n) via iterative Lyapunov methods. To appear in: Int. J. Control (2019). https://doi.org/10.1080/00207179.2019.1626023 [20] N. Rach, M.M. Muller, T. Calarco and S. Montangero, Dressing the chopped-random-basis optimization: a bandwidth-limited access to the trap-free landscape. Phys. Rev. A 92 (2015) 062343. [21] B. Riaz, C. Shuang and S. Qamar, Optimal control methods for quantum gate preparation: a comparative study. Quantum Inf Process 18 (2019) 100. [22] D.J. Saunders, The Geometry of Jet Bundles. Vol. 142 of London Mathematical Society Lecture Note Series. Cambridge University Press, London (1989). [23] S.G. Schirmer and P. de Fouquieres. Efficient algorithms for optimal control of quantum dynamics: the Krotov method unencumbered. New J. Phys. 13 (2011) 073029. [24] H.B. Silveira, P.S. Pereira da Silva and P. Rouchon, Quantum gate generation by T-sampling stabilization. Int. J. Control 87 (2014) 1227­1242. A FIXED POINT ALGORITHM FOR IMPROVING FIDELITY OF QUANTUM GATES 27 [25] H.B. Silveira, P.S. Pereira da Silva and P. Rouchon, Quantum gate generation for systems with drift in U(n) using Lyapunov- Lasalle techniques. Int. J. Control 89 (2016) 1­16. [26] N. Yamamoto, K. Tsumura and S. Hara, Feedback control of quantum entanglement in a two-spin system. Automatica 43 (2007) 981­992. [27] J. Zhang, Y.-x. Liu, R.-B. Wu, K. Jacobs and F. Nori, Quantum feedback: theory, experiments, and applications. Phys. Rep. 679 (2017) 1-60. [1] S. Cong, Control of Quantum Systems: Theory and Methods. John Wiley & Sons (2014). [2] J.-M. Coron, Control and Nonlinearity. American Mathematical Society (2007). [3] D. D’Alessandro, Introduction to Quantum Control and Dynamics. Chapman & Hall/CRC, Boca Raton (2008). [4] P. De Fouquieres, S. G. Schirmer, S. J. Glaser and I. Kuprov, Second order gradient ascent pulse engineering. J. Magn. Reason. 212 (2011) 412–417. [5] F. Diele, L. Lopez and R. Peluso, The Cayley transform in the numerical solution of unitary differential systems. Adv. Comput. Math. 8 (1998) 317–334. [6] D. Dong and I. R. Petersen, Quantum control theory and applications: a survey. IET Control Theory Appl. 4 (2010) 2651–2671. [7] Symeon Grivopoulos and Bassam Bamieh. Lyapunov-based control of quantum systems, in 42nd IEEE CDC, Vol. 1 (2003) 434–438. [8] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen and S. J. Glaser, Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reason. 172 (2005) 296–305. [9] N. Leung, M. Abdelhafez, J. Koch and D. Schuster, Speedup for quantum optimal control from automatic differentiation based on graphics processing units. Phys. Rev. A 95 (2017) 042318. [10] S. Machnes, E. Assémat, D. Tannor and F. K. Wilhelm, Tunable, flexible, and efficient optimization of control pulses for practical qubits. Phys. Rev. Lett. 120 (2018) 150401. [11] M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential. Ann. Inst. Henri Poincaré (C) Non-Linear Anal. 26 (2009) 1743–1765. [12] M. Mirrahimi, P. Rouchon and G. Turinici, Lyapunov control of bilinear Schrödinger equations. Automatica 41 (2005) 1987–1994. [13] J. P. Palao and R. Kosloff, Quantum computing by an optimal control algorithm for unitary transformations. Phys. Rev. Lett. 89 (2002) 188301. [14] J. P. Palao and R. Kosloff. Optimal control theory for unitary transformations. Phys. Rev. A 68 (2003) 062308. [15] Y. Pan, V. Ugrinovskii and M. R. James, Lyapunov analysis for coherent control of quantum systems by dissipation, in 2015 American Control Conference (ACC) (2015) 98–103. [16] P. S. Pereira Da Silva and P. Rouchon, RIGA and FPA, quantum control with smooth control pulses [source code] (2019). [17] P. S. Pereira Da Silva, P. Rouchon and H. B. Silveira, Geração rápida e virtualmente exata de portas quânticas via métodos iterativos do tipo Lyapunov, in Proc. CBA’2018 - Congresso Brasileiro de Automática: CBA’2018, Brazilian Control Conference. João Pessoa, Brazil (2018). [18] P. S. Pereira Da Silva, H. B Silveira and P. Rouchon, RIGA, a fast algorithm for quantum gate generation [source code] (2019). [19] P. S. Pereira Da Silva, H. B. Silveira and P. Rouchon, Fast and virtually exact quantum gate generation in U(n) via iterative Lyapunov methods. To appear in: Int. J. Control (2019). [20] N. Rach, M. M. Müller, T. Calarco and S. Montangero, Dressing the chopped-random-basis optimization: a bandwidth-limited access to the trap-free landscape. Phys. Rev. A 92 (2015) 062343. [21] B. Riaz, C. Shuang and S. Qamar, Optimal control methods for quantum gate preparation: a comparative study. Quantum Inf Process 18 (2019) 100. [22] D. J. Saunders, The Geometry of Jet Bundles. Vol. 142 of London Mathematical Society Lecture Note Series. Cambridge University Press, London (1989). [23] S. G. Schirmer and P. De Fouquieres. Efficient algorithms for optimal control of quantum dynamics: the Krotov method unencumbered. New J. Phys. 13 (2011) 073029. [24] H. B. Silveira, P. S. Pereira Da Silva and P. Rouchon, Quantum gate generation by T-sampling stabilization. Int. J. Control 87 (2014) 1227–1242. [25] H. B. Silveira, P. S. Pereira Da Silva and P. Rouchon, Quantum gate generation for systems with drift in U(n) using Lyapunov-Lasalle techniques. Int. J. Control 89 (2016) 1–16. [26] N. Yamamoto, K. Tsumura and S. Hara, Feedback control of quantum entanglement in a two-spin system. Automatica 43 (2007) 981–992. [27] J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs and F. Nori, Quantum feedback: theory, experiments, and applications. Phys. Rep. 679 (2017) 1–60. COCV_2021__27_S1_A11_08115f66f-02dd-4e07-bbcd-3a1898574500cocv19015010.1051/cocv/202005810.1051/cocv/2020058 A Zermelo navigation problem with a vortex singularity* Bonnard Bernard 1 0000-0002-4703-4369 Cots Olivier 2 Wembe Boris 3** 1 Inria Sophia Antipolis and Institut de Mathématiques de Bourgogne, UMR CNRS 5584, 9 avenue Alain Savary, 21078 Dijon, France. 2 Toulouse Univ., INP-ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France. 3 Toulouse Univ., IRIT-UPS, UMR CNRS 5505, 118 route de Narbonne, 31062 Toulouse, France. **Corresponding author: [e-mail] SupplementS10 © The authors. Published by EDP Sciences, SMAI 2021 2021 The authors. Published by EDP Sciences, SMAI This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Full (PDF)Full (DJVU)

Helhmoltz–Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem which can be analyzed with the technics of geometric optimal control combined with numerical simulations, the geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. The analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. The explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak.

Helhmoltz–Kirchhoff $$ vortices model Zermelo navigation problem geometric optimal control conjugate and cut loci Clairaut–Randers metric with polar singularity 49K15 53C60 70H05 idline ESAIM: COCV 27 (2021) S10 open-access yes cover_date 2021 first_year 2021 last_year 2021 transformative_agreement national-agreement-fr_2020 ESAIM: COCV 27 (2021) S10 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020058 www.esaim-cocv.org A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY Bernard Bonnard1 , Olivier Cots2 and Boris Wembe3, Abstract. Helhmoltz­Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem which can be analyzed with the technics of geometric optimal control combined with numerical simulations, the geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. The analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. The explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak. Mathematics Subject Classification. 49K15, 53C60, 70H05. Received September 24, 2019. Accepted August 14, 2020. 1. Introduction Helhmoltz and Kirchhoff originated the model of the displacement of particles in a two-dimensional fluid, see [24, 27] for the original articles and [2, 30, 34] for a modern presentation of Hamiltonian dynamics. In this model, the vorticity of the fluid is concentrated at points zi := (xi, yi), i = 1, . . . , N, with circulation parameters ki and the configuration space is R2N with coordinates (x1, y1, . . . , xN , yN ) endowed with the symplectic form := PN i=1 ki dyi dxi. The dynamics is given by the Hamiltonian canonical equations ki xi = H yi , ki yi = - H xi , (1.1) This research is supported by the French Ministry for Education, Higher Education and Research. Keywords and phrases: Helhmoltz­Kirchhoff N vortices model, Zermelo navigation problem, geometric optimal control, conjugate and cut loci, Clairaut­Randers metric with polar singularity. 1 Inria Sophia Antipolis and Institut de Mathematiques de Bourgogne, UMR CNRS 5584, 9 avenue Alain Savary, 21078 Dijon, France. 2 Toulouse Univ., INP-ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France. 3 Toulouse Univ., IRIT-UPS, UMR CNRS 5505, 118 route de Narbonne, 31062 Toulouse, France. * Corresponding author: [e-mail] c The authors. Published by EDP Sciences, SMAI 2021 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 B. BONNARD ET AL. 1 i N, where the Hamiltonian function H is H := - 1 X i<j ki kj lnkzi - zjk, (1.2) where kzi - zjk is the Euclidean distance. In this article, we consider a motionless single vortex given by equation (1.1) that we fix at the origin of the reference frame. This corresponds to set z1 = (x1, y1) at (0, 0). We consider a particle as a (point) vortex with zero circulation, setting k2 = 0 with N = 2, under the influence of the current generated by the vortex and given by the vector field with a singularity at the origin. The current is defined by (1.1) and denoted, omitting indices, F0(x, y) := X1(x, y) x + X2(x, y) y , with z := (x, y). These classical notations being not adapted for later considerations, we will denote by x := (x1, x2) the position of the particle (instead of z = (x, y)) and by k the circulation parameter of the vortex. In the following we will consider a (Zermelo) time minimization problem. To define a Zermelo navigation problem, following [15, 37] and see [12] for the optimal control frame related to Zermelo's problems, we consider the particle as the ship of the navigation problem and the control is defined by the heading angle of the ship axis. Hence the control field is given by u := umax(cos , sin ) where umax is the maximal amplitude and this leads to a control system written as: dx dt = F0(x) + u1 F1(x) + u2 F2(x), (1.3) with F1 := /x1, F2 := /x2, x = (x1, x2) and u := (u1, u2) bounded by kuk umax. We then consider the associated time minimal control problem to transfer the ship from an initial configuration x0 to a target xf , where x0 and xf are two points of the punctured plane R2 \ {0}. By a rescaling we can assume umax = 1 and denoting by g the Euclidean metric on the plane, kuk 1 bounds the control amplitude by 1 and we have two cases: the case kF0kg < 1 of weak current versus the case kF0kg > 1 of strong current. In the weak case, the time minimal problem defines a Randers metric in the plane, which is a specific Finsler metric, see [4] for this geometric frame. In the neighborhood of the vortex we have kF0kg > 1, hence, due to the singularity we have a non-trivial extension of the classical case. A neat treatment of the historical Zermelo navigation was made by [15, 37] and their study is an impor- tant inspiration for our work. Optimal control, with the Hamiltonian formulation coming from the Pontryagin Maximum Principle [32], forms the frame that we shall use in our analysis, combined with recent development concerning Hamiltonian dynamics to deal with N vortices or N bodies dynamics, see [29]. An intense research activity was led by H. Poincare on the dynamics of such systems [31] to compute periodic trajectories avoiding collisions and such techniques lead to the concept of choreography developed by [17] for the N-body problem and [14] for the N-vortex system, showing the relations between both dynamics in the Hamiltonian frame [29]. From the control point of view, there is a lot of development related to space navigation for the N-body problem, see [10], valuable in our study for ship navigation in the N-vortex problem. Optimal control problem in this area was developed for this purpose in relation with Finsler geometry, see [9] or [35] for a general setting in the planar case. More general results about vortex control may be found in [33, 36] for instance. From the time minimal point of view, using the Maximum Principle, we lift the control dynamics (1.3) by defining the pseudo-Hamiltonian H(x, p, u) := H0(x, p) + 2 X i=1 ui Hi(x, p) A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 3 where Hi(x, p) := p · Fi(x), i = 1, 2, 3, are the Hamiltonian lifts of F0, F1 and F2 and the Maximum Principle leads to analyze the extremal curves solution of the Hamiltonian vector field #-- H := H p x - H x p , defined by the maximized Hamiltonian H(x, p) := maxkuk1 H(x, p, u). In the context of geometric optimal control, see [1, 26] for a general reference, time minimal solutions are obtained by a micro-local analysis of #-- H combined with the computation of the cut point along extremal curves, that is the first point where an extremal curve ceases to be globally optimal. Fixing the initial point x(0) = x0 of the navigation problem, the set of such points is called the cut locus. Even in the Riemannian geometry, the determination of the cut locus is a very complicated problem and there are only a few results. A major recent contribution concerns the case of ellipsoids solving the Jacobi conjecture [25]. Parallel developments were obtained in the frame of space navigation where geometric analysis is combined with numerical methods, see again [9] for a general reference for such contributions. In the frame of Zermelo navigation problem with a small current, called Randers metrics, some results were obtained recently for sphere of revolutions [22]. Our aim is to extend those results for the navigation problem, with a single vortex, which combines different new phenomena in particular the existence of a singularity localized at the vortex position which leads to the strong current case and the need of extending the Finsler case [3, 4]. The ultimate goal is to analyze the regularity of the value function: xf 7 V (x0, xf , µ) where V (x0, xf , µ) is the minimum time from x0 to reach the point xf of the punctured plane, in presence of a vortex with a circulation k := 2µ. The parameter µ is introduced later for practical convenience. The regularity of xf 7 V (x0, xf , µ) is analyzed, in particular, in relation with Legendrian and Lagrangian singularities [21] associated to the Hamiltonian dynamics #-- H. Regularity of the value function in relation with the Hamilton-Jacobi-Bellman equation leads to sufficient global optimality conditions, see the seminal reference [5]. The organization of this article is the following. In Section 2, we present the existence theorem to transfer in minimum time any two points of the punctured plane. We state the Maximum Principle to parameterize the minimizers as extremal curves of a smooth Hamiltonian vector field. This leads to define a shooting method used to compute candidates as minimizers. Extremal curves are classified using generic assumptions into hyperbolic and abnormal curves, candidate as time minimizing curves and elliptic curves candidate as time maximizers. Conjugate points where an extremal curve ceases to be optimal for the C1 -topology are calculated numerically and leads to the conclusion of the absence of such points, hence, the optimality problem boils down to compute cut points in the case of an empty conjugate locus. The final Section 3 is the main contribution of this article with the existence result. Thanks to the integrability of the extremal flow due to the rotational symmetry, the micro-local classification of extremals is presented. The two important phenomena is the existence of abnormal minimizers and of a single extremal circle trajectory called a Reeb circle. Using this classification, the cut points can be computed along any extremal to determine the time minimal value function, combining geometric analysis and numerical simulations using the HamPath code. We present in detail the case where at the initial point the current is weak. This gives a nontrivial extension of the Finsler situation. 2. Existence results and Pontryagin Maximum Principle 2.1. Existence of time minimal solutions We consider a single vortex centered in the reference frame and thus the control system of our Zermelo navigation problem is given by x1(t) = - k 2 x2(t) r(t)2 + u1(t), x2(t) = k 2 x1(t) r(t)2 + u2(t), 4 B. BONNARD ET AL. with r(t)2 := x1(t)2 + x2(t)2 the square distance of the ship, that is the particle, to the origin and where k is the circulation of the vortex. This control system may be written in the following form: x(t) = F0(x(t)) + 2 X i=1 ui(t)Fi(x(t)), (2.1) with F0, F1 and F2 three real analytic (i.e. C ) vector fields, where the current (or drift) is given by F0(x) := µ x2 1 + x2 2 Å -x2 x1 + x1 x2 ã , (2.2) with µ := k/2, and where the control fields are F1 := /x1 and F2 := /x2. Considering the polar coordinates (x1, x2) =: (r cos , r sin ) and an adapted rotating frame for the control, v := u e-i , the control system (2.1) becomes r(t) = v1(t), (t) = µ r(t)2 + v2(t) r(t) . (2.3) We give hereinafter some classical definitions and refer to [8] for more details. We consider admissible control laws in the set U := {u: [0 , +) U | u measurable} , where the control domain U := B(0, umax) R2 denotes the Euclidean closed ball of radius umax > 0 centered at the origin. Since the drift introduces a singularity at the origin, we define by M := R2 \ {0} the state space, and for any u U and x0 M, we denote by xu(·, x0) the unique solution of (2.1) associated to the control u such that xu(0, x0) = x0. We introduce for a time T > 0 and an initial condition x0 M, the set UT,x0 U of control laws u U such that the associated trajectory xu(·, x0) is well defined over [0 , T], and we denote by AT,x0 := Im ET,x0 the atteignable set (or reachable set) from x0 in time T, where we have introduced the endpoint mapping ET,x0 : UT,x0 - M u 7- xu(T, x0). Then, we denote by Ax0 := T 0AT,x0 the atteignable set from x0. We recall that the control system is said to be controllable from x0 if Ax0 = M and controllable if Ax0 = M for any x0 M. Now, for a given pair (x0, xf ) M2 and some parameters umax R + and µ R, we define the problem of steering (2.1) in minimum time from the initial condition x0 to the target xf : (P) V (x0, xf , µ, umax) := inf {T | (T, u) Dx0 and xu(T, x0) = xf } , where Dx0 := {(T, u) [0 , +) × U | u UT,x0 }. We emphasize the fact that the value function V depends on the initial condition x0, the target xf and the parameters umax and µ. The first main result is the following: Theorem 2.1. For any (x0, xf , µ, umax) M2 × R × R +, the problem (P) admits a solution. Remark 2.2. Note that when µ = 0, the result is clearly true in R2 but false in M = R2 \ {0}. Up to a time reparameterization := t umax and a rescaling of µ, one can fix umax = 1 and we have the relation V (x0, xf , µ, umax) = V (x0, xf , µ/umax, 1)/umax. We thus fix from now umax = 1 and write the value function (with a slight abuse of notation) V (x0, xf , µ) := V (x0, xf , µ, 1). (2.4) We first prove that there exists an admissible trajectory connecting any pair of points in M. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 5 Lemma 2.3. The system (2.1) is controllable. Proof. Let consider a pair (x0, xf ) M2 . We introduce r0 := kx0k and rf := kxf k. From x0, we can apply a constant control v(t) = (±1, 0) (depending on whether rf is smaller or greater than r0) until the distance rf is reached and then apply a constant control v(t) = (0, sign(µ)) until the target xf is reached. Remark 2.4. The controllability gives us that the value function is finite while the existence of solutions implies that the value function is lower semi-continuous. The existence of time-optimal solutions relies on the classical Filippov's theorem [16, Thm. 9.2.i] and the main idea is to prove that the problem (P) is equivalent to the same problem with the restriction that the trajectories remain in a compact set. To prove this, we need a couple of lemmas. Let us introduce some notations for the first lemma: for a trajectory-control pair (x, u), we associate the pair (q, v) with q := (r, ) the polar coordinates and v = u e-i . We denote by qv(·, q0) the solution of (2.3) with control v such that qv(0, q0) = q0. We define for (, R, µ) R+ × R + × R and 0 R, two optimization problems: (a) The minimum time to make a complete round at a distance R to the vortex: T(R, 0, µ) := inf {T | (T, u) Dx0 and qv(T, (R, 0)) = (R, 0 + s 2)} , where s := sign(µ), x0 := (R cos 0, R sin 0) and where the control u is related to v by u = vei . (b) The minimum time to reach the circle of radius from a distance R to the vortex: Tr(, R, 0, µ) := inf {T | (T, u) Dx0 and rv(T, (R, 0)) = } . Since it is clear, due to the rotational symmetry of the problem, that T(R, ·, µ) and Tr(, R, ·, µ) are invariant with respect to 0, one can fix 0 = 0 and set T(R, µ) := T(R, 0, µ) and Tr(, R, µ) := Tr(, R, 0, µ). Besides, from the proof of Lemma 2.5, one can notice that Tr does not depend on µ, hence, one can define Tr(, R) := Tr(, R, 0). Under these considerations, we have the following comparison between T and Tr: Lemma 2.5. For any (, R, µ) s.t. µ 6= 0, 0 < R < Rµ and 0 < µ,R, with Rµ := |µ| 2 - 1 and µ,R := R Å 1 - 2R |µ| + R ã , we have 0 < µ,R < R < Rµ and T(R, µ) < Tr(, R), that is the minimum time to make a complete round at a distance R to the vortex is strictly smaller than the minimum time to reach the circle of radius < R. Proof. It is clear from (2.3) that T(R, µ) is given by the control v(t) = (0, sign(µ)). This gives by a sim- ple calculation T(R, µ) = 2R2 /(|µ| + R). It is also clear that Tr(, R, µ) is given by v(t) = (-1, 0), whence Tr(, R, µ) = R- = Tr(, R) and indeed Tr does not depend on µ. Fixing = 0, we have T(R, µ) = Tr(0, R) R = |µ|/(2 - 1) =: Rµ. Besides, we have T(R, µ) < Tr(, R) < R Å 1 - 2R |µ| + R ã =: µ,R but also, we have 0 < µ,R R < Rµ, whence the conclusion. Next an admissible trajectory x associated to a pair (T, u) Dx0 is such that x(T) = xu(T, x0) = xf . Let us fix (x0, xf , µ) M2 × R and introduce r0 := kx0k and rf := kxf k. Then: Lemma 2.6. There exists > 0 such that any optimal trajectory is contained in M \ B(0, ). 6 B. BONNARD ET AL. Figure 1. Illustration of the construction of a strictly better admissible trajectory. The vortex is represented by a red ball, while the trajectories are the solid black lines. One can see on the left, a trajectory crossing the ball of radius . This trajectory is replaced on the right subgraph by a strictly better admissible trajectory. Proof. Let consider (, R) s.t. 0 < R < min{Rµ, r0, rf } and 0 < < µ,R. Let us recall that < µ,R < R since R < Rµ and consider an admissible trajectory x intersecting B(0, ) and associated to a pair denoted (T, u). Then, there exists two times 0 < t1 t2 < T s.t. x([t1 , t2]) B(0, ). Since 0 < < R < min{r0, rf }, there exists also 0 < tin < t1 t2 < tout < T s.t. x([tin , tout]) B(0, R) and s.t. x(tin) and x(tout) belong to B(0, R) = S(0, R), the sphere of radius R centered at the origin. In all generality, one can assume that t [0 , tin) (tout , T], x(t) / B(0, R). Let consider the circular arc from x(tin) to x(tout) obtained with a control v = (0, sign(µ)), realized in a time denoted > 0. It is clear that T(R, µ) since T(R, µ) is the time to make a circular arc of angle 2. It is also clear from Lemma 2.5 and from the definition of Tr that T(R, µ) < Tr(, R) tout - tin. Let us replace the part x([tin , tout]) by the circular arc. Then, the new trajectory associated to the pair denoted (T0 , u0 ) is still admissible and is strictly better than x since T0 = T - (tout - tin) + < T. Moreover, this new trajectory is by construction contained in M \ B(0, ), as illustrated in Figure 1. Whence the conclusion. We are now in position to prove Theorem 2.1. Proof of Theorem 2.1. By Lemma 2.3, there exists an admissible trajectory x. Let T denote the first time s.t. x(T ) = xf . Let us introduce R1 := from Lemma 2.6 and R2 := r0 + T , with r0 := kx0k. By Lemma 2.6, the problem (P) is equivalent to the same problem with the additional constraint R1 r(t). Since r(t) = v1(t) and v1(t) 1, then for any t [0 , T ] we have r(t) r0 + T . The problem (P) is thus equivalent to the same problem with the additional constraints: R1 r(t) R2. The trajectories of the equivalent problem are contained in the compact set B(0, R2) \ B(0, R1). The result follows from the Filippov's existence theorem. 2.2. Classification of the extremal curves In this section, we recall concepts and results from [9]. Let x0 M and (T, u) Dx0 be an optimal solution of problem (P) with x := xu(·, x0) the associated optimal trajectory. According to the Pontryagin Maximum Principle [32], there exists an absolutely continuous function p: [0 , T] R2 satisfying the adjoint equation almost everywhere over [0 , T]: p(t) = -xH(x(t), p(t), u(t)), (2.5) A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 7 where1 H(x, p, u) := p · (F0(x) + u1 F1(x) + u2 F2(x)) is the pseudo-Hamiltonian associated to the problem (P). Besides, there exists p0 0 such that: the pair (p(·), p0 ) never vanishes (2.6) and such that the optimal control satisfies the maximization condition almost everywhere over [0 , T]: H(x(t), p(t), u(t)) = max wU H(x(t), p(t), w) = -p0 . (2.7) Definition 2.7. An extremal is a 4-uplet (x(·), p(·), p0 , u(·)) satisfying (2.1) and (2.5) ­ (2.7). It is said abnormal whenever p0 = 0 and normal whenever p0 6= 0. It is called strict if p(·) is unique up to a factor. An extremal (x(·), p(·), p0 , u(·)) is called a BC-extremal if x(0) = x0 and if there is a time T 0 s.t. x(T) = xf . Let us introduce the Hamiltonian lifts Hi(x, p) := p · Fi(x), i = 0, 1, 2, the function := (H1, H2) and the switching function defined for t [0 , T] by (t) := (z(t)) = p(t), z(·) := (x(·), p(·)). The maximization condition (2.7) implies for a.e. t [0 , T]: u(t) = (t) k(t)k = p(t) kp(t)k , whenever (t) 6= 0. Introducing the switching surface := z M × R2 (z) = 0 = M × {0} and denoting z := (x, p) M × R2 , one can define outside the Hamiltonian: H(z) := H(z, (z)/k(z)k) = H0(z) + k(z)k = H0(z) + kpk. (2.8) Definition 2.8. An extremal (x(·), p(·), p0 , u(·)) contained outside the switching surface is called of order zero. Let us recall that a switching time 0 < t < T is a time s.t. (t) = 0 and s.t. for any > 0 (small enough) there exists a time (t - , t + ) [0 , T] s.t. () 6= 0. We can show that the extremals are only of order zero, and thus are smooth: Proposition 2.9. All the extremals are of order zero, that is there are no switching times. Proof. Let (x(·), p(·), p0 , u(·)) be an extremal. If there exists a time t s.t. (t) = 0, then p(t) = 0 and we have H(x(t), p(t), u(t)) = 0 = -p0 , which is impossible by (2.6). We have the standard following result: Proposition 2.10. The extremals of order zero are smooth responses to smooth controls on the boundary of kuk 1. They are singularities of the endpoint mapping ET,x0 for the L -topology when u is restricted to the unit sphere S1 . For any Hamiltonian H(z), resp. pseudo-Hamiltonian H(z, u), we denote by #-- H(z) := (pH(z), -xH(z)), resp. #-- H(z, u) := (pH(z, u), -xH(z, u)), its associated Hamiltonian vector field, resp. pseudo-Hamiltonian vector field. With these notations, we have the following classical but still remarkable fact: Proposition 2.11. Let (x(·), p(·), p0 , u(·)) be an extremal. Denoting z := (x, p), then, we have over [0 , T]: z(t) = #-- H(z(t), u(t)) = #-- H(z(t)) = # -- H0(z(t)) + Å p(t) kp(t)k , 0 ã , (2.9) 1The standard inner product is written a · b or ha, bi. 8 B. BONNARD ET AL. that is the extremals are given by the flow of the Hamiltonian vector field associated to the maximized Hamiltonian H. Proof. Since the extremal is of order zero, the control t 7 u(t) is smooth and the adjoint equation (2.5) is satisfied all over [0 , T]. Besides, denoting (with a slight abuse of notation) u(z) := (z)/k(z)k, we have: H0 (z) = H z (z, u(z)) + H u (z, u(z)) · u0 (z) = H z (z, u(z)) + (z)T Å I2 k(z)k - (z)(z)T k(z)k3 ã | {z } =0 · 0 (z) = H z (z, u(z)) = H0 0(z) + Å 0, p kpk ã . This proposition shows the importance of the true Hamiltonian H which encodes all the information we need and gives a more geometrical point of view: we will thus refer to trajectories as geodesics. Besides, from the maximum principle, optimal extremals are contained in the level set {H 0}. Let z(·, x0, p0) := (x(·, x0, p0), p(·, x0, p0)) be a reference extremal curve solution of z = #-- H(z) with initial condition z(0, x0, p0) = (x0, p0) and defined over the time interval [0 , T]. Lemma 2.12. One has x(t, x0, p0) = x(t, x0, p0) and p(t, x0, p0) = p(t, x0, p0). Thanks to this lemma, and since p never vanishes, we can fix by homogeneity kp0k = 1. We thus introduce the following definition that gives us a way to parameterize the extremals of order zero. Definition 2.13. We define the exponential mapping by expx0 (t, p0) := et #-- H (x0, p0), (2.10) where et #-- H (x0, p0) is the solution at time t of z(s) = #-- H(z(s)), z(0) = (x0, p0). It is defined for small enough nonnegative time t and we can assume that p0 belongs to S1 . Definition 2.14. Let z(·) := (x(·), p(·)) be a reference extremal of order zero, defined on [0 , T]. Let H be the Hamiltonian defined by (2.8). The associated geodesic x(·) is called exceptional if H = 0, hyperbolic if H > 0 and elliptic if H < 0, along the reference extremal z(·). Remark 2.15. The previous definition is related to the more classical definition 2.21. Even if the elliptic geodesics are not optimal according to the PMP, they still play a role in the analysis of the optimal synthesis, in particular in the computation of the cut locus when the current (or drift) is strong, see Section 3. In Cartesian coordinates, the Hamiltonian writes H(x1, x2, p1, p2) = µ x2 1 + x2 2 (-p1 x2 + p2 x1) + » p2 1 + p2 2, and the extremals are solution of the following Hamiltonian system: x1 = -µ x2 r2 + p1 kpk , x2 = µ x1 r2 + p2 kpk , p1 = - µ r4 2 x1 x2 p1 - (x2 1 - x2 2) p2 , p2 = µ r4 (x2 1 - x2 2) p1 - 2 x1 x2 p2 . Introducing the Mathieu transformation Å pr p ã = Å cos sin -r sin r cos ã Å p1 p2 ã (2.11) A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 9 then, in polar coordinates, the Hamiltonian is given by (we still denote by p the covector in polar coordinates) H(r, , pr, p) = p µ r2 + kpkr, where kpkr := » p2 r + p2 /r2. It is clear from H that is a cyclic variable and thus the problem has a symmetry of revolution and by Noether theorem, the adjoint variable p is a first integral. This relation p = constant corresponds to the Clairaut relation on surfaces of revolution. Hence, we can fix (0) = 0 and consider p has a parameter of the associated Hamiltonian system in polar coordinates: r = pr kpkr , = 1 r2 Å µ + p kpkr ã , pr = p r3 Å 2µ + p kpkr ã , p = 0. (2.12) 2.3. C1-second order necessary conditions in the regular case Since the extremals are of order zero, one can restrict u(t) to the 1-sphere S1 . Writing u(t) =: (cos (t), sin (t)), we have with some abuse of notations H = H0 + u1H1 + u2H2 = H0 + cos H1 + sin H2, with the new control. Differentiating twice with respect to , we have H = - sin H1 + cos H2, 2 H 2 = -(cos H1 + sin H2), and since u = (cos , sin ) = /kk and = (H1, H2) never vanishes along any extremal, we have 2 H 2 = - » H2 1 + H2 2 < 0 along any extremal. Hence, the strict Legendre­Clebsch condition is satisfied, and we are in the regular case [11], but with a free final time T. Definition 2.16. Let z(·) be a reference extremal curve solution of z = #-- H(z) given by (2.9). The variational equation i z(t) = #-- H0 (z(t)) · z(t), (2.13) is called a Jacobi equation. A Jacobi field is a non-trivial solution J of (2.13). It is said to be vertical at time t if x(t) := 0 (z(t)) · J(t) = 0, where : (x, p) 7 x is the standard projection. Let z(·, x0, p0) := (x(·, x0, p0), p(·, x0, p0)) with p0 S1 , be a reference extremal curve solution of z = #-- H(z) with initial condition z(0, x0, p0) = (x0, p0) and defined over the time interval [0 , T]. Following [11], we make the following generic assumptions on the reference extremal in order to derive second order optimality conditions: (A1) The trajectory x(·, x0, p0) is a one-to-one immersion on [0 , T]. (A2) The reference extremal is normal and strict. Definition 2.17. Let z = (x, p) be the reference extremal defined hereinabove. Under assumptions (A1) and (A2), the time 0 < tc T is called conjugate if the exponential mapping is not an immersion at (tc, p0). The associated point expx0 (tc, p0) = x(tc, x0, p0) is said to be conjugate to x0. We denote by t1c the first conjugate time. The following result is fundamental, see [7]. 10 B. BONNARD ET AL. Theorem 2.18. Under Assumptions (A1) and (A2), the extremities being fixed, the reference geodesic x(·) is locally time minimizing (resp. maximizing) for the L -topology on the set of controls up to the first conjugate time in the hyperbolic (resp. elliptic) case. Algorithm to compute conjugate times Writing the reference trajectory x(t) := x(t, x0, p0) and consid- ering a Jacobi field J(·) := (x(·), p(·)) along the reference extremal, which is vertical at the initial time (i.e. x(0) = 0) and normalized by p0 · p(0) = 0 (since p0 is restricted to S1 ), then tc is a conjugate time if and only if tc is a solution of the following equation: t 7 det(x(t), x(t)) = 0. (2.14) See [11, 18] for more details about algorithms to compute conjugate times in a more general setting and [13] for details about the numerical implementation of these algorithms into the HamPath software. 2.4. The Zermelo­Caratheodory­Goh point of view From the historical point of view in the Zermelo navigation problem, Zermelo and Caratheodory use for the parameterization of the geodesics the derivative of the heading angle instead of the angle itself, see [12]. This corresponds precisely to the so-called Goh transformation for the analysis of singular trajectories in optimal control, see for instance the reference [8]. This is presented next, in relation with the problem, to derive sufficient C0 -optimality conditions under generic assumptions, see [7]. 2.4.1. Goh Transformation Restricting to extremals of order zero, the Goh transformation amounts to set (we use the same notation u for the new control but no confusion is possible): = u, that is to take as the control of the ship. Note that such a transformation transforms L -optimality conditions on the set of controls into C0 -optimality conditions on the set of trajectories. For the geodesics computations, this amounts only to a reparameterization of extremal curves of order zero. Considering the vortex problem in cartesian coordinates (x1, x2), we introduce x := (x1, x2, x3), x3 := , and the control system becomes x = F(x) + u G(x), with F(x) := Å F0(x1, x2) + cos x3 F1(x1, x2) + sin x3 F2(x1, x2) 0 ã and G = x3 . The associated pseudo-Hamiltonian reads H(x, p, u) := p · (F(x) + u G(x)), p := (p1, p2, p3), and relaxing the bound on the new control, the maximization condition implies p · G = 0 along any extremal. These extremals are called singular and the associated control is also called singular. Let us recall how we compute the singular extremals. First, we need to introduce the concepts of Lie and Poisson brackets. The Lie bracket of two C vector fields X, Y on an open subset V Rn is computed with the convention: [X, Y ](x) := X x (x) Y (x) - Y x (x) X(x), A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 11 and denoting HX, HY the Hamiltonian lifts: HX(z) := p · X(x), HY (z) := p · Y (x), with z := (x, p) V × Rn , the Poisson bracket reads: {HX, HY } := H0 X · #-- HY = p · [X, Y ](x), where #-- HX = H p x - H x p . Differentiating twice p(·) · G(x(·)) with respect to the time t, one gets: Lemma 2.19. Singular extremals (z(·), u(·)) are solution of the following equations: HG(z(t)) = {HG, HF }(z(t)) = 0, {{HG, HF }, HF }(z(t)) + u(t) {{HG, HF }, HG}(z(t)) = 0. If {{HG, HF }, HG} 6= 0 along the extremal, then the singular control is called of minimal order and it is given by the dynamic feedback: us(z(t)) := - {{HG, HF }, HF }(z(t)) {{HG, HF }, HG}(z(t)) . Plugging the control us in feedback form into the pseudo-Hamiltonian leads to define a true Hamiltonian denoted Hs(z) := H(z, us(z)), and one has: Lemma 2.20. Singular extremals of minimal order are the solutions of z(t) = #-- Hs(z(t)), with the constraints HG(z(t)) = {HG, HF }(z(t)) = 0. 2.4.2. The Case of Dimension 3 Applied to the Zermelo Problem Consider the following affine control system: x = F(x) + u G(x), with u R and x R3 , where F, G, are C vector fields. Let z(·) := (x(·), p(·)) be a reference singular extremal curve on [0 , T]. We assume the following: (B1) The reference geodesic t 7 x(t) is a one-to-one immersion on [0 , T]. (B2) F and G are linearly independent along x(·). (B3) G, [G, F], [[G, F], G] are linearly independent along x(·). From (B3), p is unique up to a factor and the geodesic is strict and moreover us can be computed as a true feedback: us(x) = - D0 (x) D(x) , where we denote: D := det(G, [G, F], [[G, F], G]), D0 := det(G, [G, F], [[G, F], F]). Moreover, let us introduce D00 := det(G, [G, F], F). In the vortex problem, one has: = - D0 (x) D(x) . 12 B. BONNARD ET AL. In our problem with the Goh extension, one orients p(·) using the convention of the maximum principle: p(t) · F(x(t)) 0 on [0 , T] and we introduce the following definition consistent with definition 2.14: Definition 2.21. Under assumptions (B1), (B2) and (B3), a geodesic is called: ­ hyperbolic if DD00 > 0, ­ elliptic if DD00 < 0, ­ abnormal (or exceptional) if D00 = 0. Note that the condition DD00 0 amounts to the generalized Legendre­Clebsch condition u d2 dt2 H u (z(t)) 0 and according to the higher-order maximum principle [28], this condition is a necessary (small) time minimization condition. See [7] for the general frame relating the optimal control problems using the Goh transformation and applicable to our study and for the following result. Theorem 2.22. Under assumptions (B1), (B2) and (B3), a reference hyperbolic (resp. elliptic) geodesic x(·) defined on [0 , T] is time minimizing (resp. maximizing) on [0 , T] with respect to all trajectories contained in a C0 -neighborhood of x(·) if T < t1c where t1c is the first conjugate time along x(·) as defined by 2.17 for the projection of x(·) on the (x1, x2) plane. In the exceptional case, the reference geodesic is C0 -time minimizing and maximizing. 2.5. Influence of the circulation 2.5.1. Influence of the Circulation on the Drift Denoting the drift (2.2) F0(x, µ) to emphasize the role of µ, one introduces for (x, µ) M × R the set F(x, µ) := ( F0(x, µ) + 2 X i=1 ui Fi(x) u := (u1, u2) U ) . (2.15) Then, we have (noticing that if u U, then -u U): 0 F(x, µ) u := (u1, u2) U s.t. F0(x, µ) = 2 X i=1 ui Fi(x) kF0(x, µ)kg = kF0(x, µ)k 1 |µ| kxk = r. This leads to introduce the following definition. Definition 2.23. The drift F0(x, µ) is said to be weak at the point x if kF0(x, µ)k < 1, strong at x if kF0(x, µ)k > 1 and moderate at x if kF0(x, µ)k = 1. Remark 2.24. Note that if the drift could have been weak at any point x M, then we would have been in the Finslerian case [3] with a metric of Randers type. However, this is not possible for µ 6= 0 (the case µ = 0 is trivially Euclidean on R2 ), since in this case, we have for any x M B(0, |µ|) 6= that the drift F0(x, µ) is not weak. Remark 2.25. One can also notice that at the initial time, the strength of the drift depends on the ratio |µ|/r0. See Figure 2 for an illustration of the different possible strengths of the drift. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 13 Figure 2. The vortex is placed at the origin and marked by a black dot, as the initial point x0 := (2, 0) in cartesian coordinates. The black circle corresponds to the set of initial directions x0 + F(x0, µ) and the thick blue vector is the initial direction associated to the initial control u(0) := (cos , sin ), with = 7/8. It is decomposed as the sum of the drift (oriented vertically) and the control field. On the left, we have a weak drift (µ = 0.5 r0) at x0, in the middle we have a moderate drift (µ = r0) and on the right a strong drift (µ = 2 r0). 2.5.2. Influence of the Circulation on the Abnormal Extremals Proposition 2.26. Let (x(·), p(·), p0 , u(·)) be an abnormal extremal, that is p0 = 0. Then, the drift is strong or moderate all along the geodesic. Proof. Since the abnormal extremal is of order 0, then all along the extremal we have H(x(t), p(t)) = p(t) · F0(x(t), µ) + kp(t)k = 0. So, the Cauchy­Schwarz inequality gives kp(t)k = |p(t) · F0(x(t), µ)| kp(t)kkF0(x, µ)k and since kp(t)k 6= 0, the result follows. Remark 2.27. According to the previous proposition, the abnormal geodesics (that is the projection of the abnormal extremals on the state manifold) are contained in the ball B(0, |µ|). According to the PMP, p0 = 0 for the abnormal extremals while p0 < 0 for the normal extremals. In the normal case, by homogeneity, one can fix p0 = -1 and the initial adjoint vector p0 := p(0) of normal extremals lives in the one dimensional space p R2 H(x0, p) = 1 . This parameterization is very classical. Another possibility is to set k(p0, p0 )k = 1 since by the PMP, the pair (p(·), p0 ) does not vanish. Finally, in our case, since all the extremals are of order zero, that is since p does not vanish, we can also fix kp0k = 1. We consider this third possibility but in polar coordinates, that is, denoting p0 := (pr(0), p) (recalling that p is constant) we parameterize the initial adjoint vector by: p0 p R2 kpkr0 = 1 . We thus introduce [0 , 2) such that pr(0) = cos , p = r0 sin , which gives the initial control v(0) = (pr(0), p/r0) = (cos , sin ). According to the Mathieu transformation (2.11), one has in cartesian coordinates that px(0) = cos 0 cos - sin 0 sin and py(0) = sin 0 cos + cos 0 sin , so in the particular case 0 = 0, we have u(0) = (px(0), py(0)) = (cos , sin ). This parameterization has the advantage to cover the normal and the abnormal extremals. According to the PMP, we have at the initial time: H(q0, p0) = p µ r2 0 + kp0kr0 = p µ r2 0 + 1 = -p0 0, 14 B. BONNARD ET AL. Figure 3. The vortex is placed at the origin and marked by a black dot, as the initial point x0 := (2, 0) in cartesian coordinates. The drift is strong at the initial point since µ = 2 r0. The black circle represents the set of initial directions x0 + F(x0, µ) and the blue vector is the initial hyperbolic direction associated to the initial control u(0) := (cos , sin ), with = 7/8. The direction in green is elliptic while the two red directions are the abnormal directions located at the boundary of the cone of admissible directions. with q0 = (r0, 0). Introducing (with a slight abuse of notation) H() := µ sin /r0 + 1, then, the abnormal extremals are characterized by H() = 0 sin = -r0/µ. We have three cases: ­ If the drift is weak at the initial point, then this equation has no solution which explains why there is no abnormal extremals. In this case, H() > 0 for any and thus there are only hyperbolic geodesics. ­ If the drift is moderate at the initial point, that is if |µ| = r0, then the single abnormal extremal is parameterized by = /2 if µ < 0 and by = 3/2 if µ > 0, ­ In the last case when the drift is strong, then for a given µ, the equation H() = 0 has two distinct solutions a 1 < a 2 in [0 , 2). If µ < 0, then a 1 and a 2 are contained in (0 , ) while if µ > 0, then a 1 and a 2 are contained in ( , 2). We have in addition the following symmetry: a 2 = - a 1 if µ < 0 and a 2 = 3 - a 1 if µ > 0. The normal extremals solution of the PMP are parameterized by the set { [0 , 2) | H() > 0} = [0 , a 1) (a 2 , 2) while for (a 1 , a 2) we have H() < 0. One can see in Figure 3, the two abnormal directions with two hyperbolic and elliptic directions (that is resp. associated to hyperbolic and elliptic geodesics). The two abnormal directions define the boundary of the cone of admissible directions and reveal a lack of accessibility in the neighborhood of x0. 2.6. Numerical results 2.6.1. Resolution of the Shooting Equation We introduce the shooting mapping S(T, p0) := expx0 (T, p0) - xf , (2.16) A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 15 Figure 4. Examples 1 and 2. Geodesic with a strong drift at the initial point: µ = 2 kx0k. (Left: Ex. 1) x0 = (2, 0), xf = (-2, 0) and the final time is T 1.641. (Right: Ex. 2) x0 = (2, 0), xf = (2.5, 0) and the final time is T 2.821. where xf is the target and exp is the exponential mapping defined by (2.10). The shooting mapping is defined for (T, p0) (T, p0) R+ × S1 T < tp0 where tp0 R + {+} is the maximal time such that expx0 (·, p0) is well defined over [0 , tp0 ). Let (T, p0) be a solution of S = 0 such that the associated extremal is normal. The shooting mapping is differentiable at (T, p0) and if T is not a conjugate time, then its Jacobian is of full rank at (T, p0), which is a necessary condition to compute numerically the BC-extremals by means of Newton-like algorithms. We present in the following some examples of hyperbolic geodesics fixing the initial condition to x0 := (2, 0) and solving the shooting equations S = 0 thanks to the HamPath code [13], for different final conditions and for different strengths of the drift. HamPath code A Newton-like algorithm is used to solve the shooting equation S(T, p0) = 0. Providing H and S to HamPath, the code generates automatically the Jacobian of the shooting function. To make the implementation of S easier, HamPath supplies the exponential mapping. Automatic Differentiation is used to produce #-- H and is combined with Runge­Kutta integrators to assemble the exponential mapping and the variational equations (2.13) used to compute conjugate times. See [13, 18] for more details about the code. Example 1 For this first example we want to steer the particle from x0 to xf := (-2, 0) with µ := 2 kx0k (strong drift). In this case, we obtain a final time T 1.641 and the shooting equation S = 0, is solved with a very good accuracy of order 1e-12 (it is the same for the other examples, but it will not be mentioned anymore). The associated hyperbolic geodesic is portrayed in Figure 4. The point vortex is represented by a black dot as the initial condition. The initial velocity x(0) is given with the boundary (the black circle) of x0 + F(x0, µ), cf. equation (2.15). One can see that the drift is strong since x0 6 x0 + F(x0, µ). Example 2 To emphasize the influence of the final condition, let us take again µ := 2 kx0k and set xf := (2.5, 0). We can note from Figure 4 that the solution turns around the point vortex and profits from the circulation. In this case we obtain a final time T 2.821. 16 B. BONNARD ET AL. Figure 5. Examples 3 and 4. Geodesic with a weak drift at the initial point: µ = 0.5 kx0k. (Left: Ex. 3) x0 = (2, 0), xf = (-2, 0) and the final time is T 2.826. (Right: Ex. 4) x0 = (2, 0), xf = (2.5, 0) and the final time is T 0.56. Compare this to Figure 4. Examples 3 and 4 Here we want to observe what happens for a weak drift. We set µ := 0.5kx0k and present two cases with xf := (-2, 0) (cf. left subgraph of Fig. 5) and xf := (2.5, 0) (cf. right subgraph of Fig. 5). When xf = (-2, 0), the final condition is the same as in the Example 1 but since the drift is weaker, the final time is longer. This is because the particle takes advantage of the vortex circulation. On the other hand, for xf = (2.5, 0) (same final condition as Ex. 2) and considering a weak drift, then the particle does not turn around the vortex. 2.6.2. Numerical Results on the Absence of Conjugate Points According to Section 3.2, there are two types of geodesics reaching the boundary of the domain: the bounded geodesics that reach the vortex and the unbounded geodesics. There exists also a unique geodesic separating these two cases, which is called the separatrix. Fixing 0 = 0 and using the parameterization u(0) = (cos , sin ), [0 , 2), from Section 2.5.2, we define for a pair (r0, µ) R + × R , the set (r0, µ) of parameters such that the associated geodesics converge to the vortex and (r0, µ) the set of parameters such that the associated geodesics go to infinity (in norm). The sets (r0, µ) and (r0, µ) depend on the strength of the drift and the sign of µ, and they are given in Section 3.2. One can find in Figure 6 the smallest singular value, denoted min(·), of det(x(·), x(·)) over the time, see equation (2.14). For a fixed [0 , 2), we denote by t R + {+} the maximal time such that the associated geodesic is well defined over [0 , t). If there exists a time tc (0 , t) such that min(tc) = 0, then the time tc is a conjugate time. If not, the geodesic has no conjugate time. One can see from the left subgraph of Figure 6, that for any weak drift and for any (r0, µ), that there are no conjugate times and min(t) 1 when t t = +.2 From the right subgraph of Figure 6, it is clear that for any weak drift and for any (r0, µ), that there are no conjugate times and that min(t) 0 when t t. On the two figures, the red curve corresponds to the separatrix. It is clear that for the separatrix min(t) 0.5 when t t = +. Since one has similar numerical results in the strong drift case, we make the following conjecture: Conjecture 2.28. For any (x0, µ) M × R , the conjugate locus is empty. 2It is clear that t = + for any (r0, µ), since |r| 1. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 17 Figure 6. Smallest singular value with respect to time. Setting: 0 = 0, µ = 2.0, r0 = 4µ/3 (weak drift at the initial point). The set of parameters [0 , 2) is uniformly discretized in N := 1000 sub-intervals and we define 0 =: 1 < · · · < N < 2 the associated parameters. Each blue curve is the graph of the smallest singular value min(t) for one (r0, µ) N i=1{i} and for t [0 , 50] on the left subgraph, and for one (r0, µ) N i=1{i} on the right subgraph. In this case, we stop the numerical integration when r(t) 10-3 which explains why the minimal value of min is between 10-5 and 10-6 . The red curve on each plot corresponds to the separatrix. 3. Micro-local analysis and properties of the value function 3.1. Poincare compactification on S3 of the extremal dynamics and integrability results The objective of this section is to provide the geometric frame to analyze the extremals of order zero. Reparametrizing, the flow defines a polynomial vector field which can be compactified using Poincare method to analyze the behaviors of extremal curves converging either to the origin or to the infinity, see Section 2.2. Thanks to the rotational symmetry, this foliation can be integrated which is crucial to define in the next section the concept of Reeb circle. Introducing c := -p0 , then from the condition H = pµ/r2 + kpkr = -p0 , one gets r2 kpkr = cr2 - µp. Plugging this into (2.12), one obtains: r3 (cr2 - µ p) r = r5 pr, r3 (cr2 - µ p) = 3r3 - 4r, r3 (cr2 - µ p) pr = 1r2 - 2, where 1 := (2µc + p)p, 2 := 2(µp)2 , 3 := µc + p and 4 := µ2 p. The aim is to get a polynomial system to integrate, so we use the time reparameterization: dt r3(cr2 - µp) = ds. Denoting by 0 the derivative with respect to s, the system is written: r0 = r5 pr, 0 = 3r3 - 4r, p0 r = 1r2 - 2. We use Poincare compactification where the sphere is identified to x = 1 and this leads to the following dynamics: r0 = r5 pr, 0 = 3r3 x3 - 4rx5 , p0 r = 1r2 x4 - 2x6 , x0 = 0, (3.1) 18 B. BONNARD ET AL. which can be projected onto the 3-sphere r2 + 2 + p2 + x2 = 1, up to a time reparameterization. Integrating the system (3.1), one has x = c1 and we get: dr dpr = r5 pr 1r2x4 - 2x6 which can integrate by separating the variables: dr r5 1r2 x4 - 2x6 = pr dpr. Hence, we obtain: 2c6 1 4r4 - 1c4 1 2r2 = 1 2 p2 r + K1. Using quadratures from (3.1), one can obtain s = f(r), = g(r). To illustrate the integrability properties, we only give the formula in the abnormal case for the r component: Proposition 3.1. For {a 1, a 2} (see Section 2.5.2 for the definition of a 1 and a 2), one has: r(t) = Å s(t)2 - 2 cos r0 s(t) + 1 r2 0 ã- 1 2 , where s(t) := sin r0 tan Å sin r0 t + arctan - cos sin ã + cos r0 . 3.2. Micro-local analysis of the extremal solutions Since is a cyclic variable, and so p is a parameter, then we can focus the analysis on the subsystem r = pr kpkr , pr = p r3 Å 2µ + p kpkr ã , (3.2) to determine the different types of extremals. First of all, it is clear that the unique equilibrium point satisfying (pr, p) 6= (0, 0) and r 0 is given by (re, pr,e) := (2|µ|, 0), and p must satisfy sign(p) = - sign(µ). Now, introducing a(p, µ) := µ2 p2 and b(p, r0, µ) := -p Å p + 2 µ + 2 p µ2 r2 0 ã , one can define (r, p, r0, µ) := a(p, µ) Å 1 r4 - 1 r4 0 ã + b(p, r0, µ) Å 1 r2 - 1 r2 0 ã + 1 - p2 r2 0 and one has the following result: Lemma 3.2. Along any extremal parameterized by k(pr(0), p)kr(0) = 1 holds pr(t)2 = (r(t), p, r(0), µ). Proof. From (3.2), one has along the extremal (omitting the time variable): prpr = p r3 (p + 2µ kpkr) r. (3.3) A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 19 Using the parameterization kp0kr0 = 1 (with p0 := (pr(0), p) and r0 := r(0)), one has kp(t)kr(t) = - Å p0 + p µ r(t)2 ã = p µ Å 1 r2 0 - 1 r(t)2 ã + 1, (3.4) since along the extremal the Hamiltonian is constant and equal to -p0 . Putting (3.4) in (3.3), one has prpr = - Å 2a r5 + b r3 ã r. Integrating this equation, we have 1 2 p2 r = 1 2 Å a r4 + b r2 + c ã , c := pr(0)2 - a r4 0 - b r2 0 , and since pr(0)2 = 1 - p2 /r2 0, we get the conclusion. Let us fix p, r0 and µ and introduce the polynomial function of degree two P(X) := aX2 + bX + c, with := b2 - 4ac = p3 Å p Å 1 + 4 µ2 r2 0 ã + 4µ ã , its discriminant. Then, we have (r, p, r0, µ) = P(1/r2 ) and the sign of the discriminant plays a crucial role in the analysis. Remark 3.3. If p = 0 then P(X) = 1 and if µ = 0 then P(X) = -p2 X + c. The polynomial P is of degree two if and only if p 6= 0 and µ 6= 0. The case µ = 0 is the simple Euclidean case that we do not develop. If p = 0 then pr = 0, r = ±1 and this case is clear: these are the fastest geodesics converging either to the vortex (r = -1) or going to infinity (r = +1). Let us introduce now p := - 4 µ 1 + 4µ2 r2 0 , such that, assuming p 6= 0, µ > 0 and r0 > 0, we have: (p, r0, µ) < 0 if p (p , 0), = 0 if p = p , > 0 if p R \ [p , 0]. Remark 3.4. If µ < 0, then p > 0 and we have (p, r0, µ) < 0 if p (0 , p ), (p , r0, µ) = 0 and (p, r0, µ) > 0 if p R \ [0 , p ]. Remark 3.5. Note that with our parameterization, we are only interested in parameters p satisfying |p| r0. We can check that for any r0 > 0 and µ 6= 0, we have |p | r0. The particular case |p | = r0 is given by r0 = 2|µ|. Thus, setting p = r0 sin , we can find two parameters 0 < 1 2 < 2 such that p =: r0 sin 1 = r0 sin 2. This gives us sin 1 = sin 2 = - 4 1 + 42 , := µ r0 , 20 B. BONNARD ET AL. Figure 7. Case < 0. Graph of r 7 P(1/r2 ). Setting: µ = 2, r0 = 3µ, p = p /2 (p , 0). and we can see again that the ratio µ/r0 is of particular interest. Note that 1 = 2 if r0 = 2|µ|. Let us analyze the three different cases depending on the sign of the discriminant, considering that p 6= 0 and fixing the parameters µ > 0 and r0 > 0. ­ Case < 0. In this case, the polynomial P has no roots and so pr(t)2 = P(1/r(t)2 ) never vanishes. See Figure 7 for an illustrative example. Thus, pr is of constant sign such as r and so r is monotone, even strictly monotone since pr is nonzero. This case is given by p (p , 0), thus, in this case |p| |p | r0 and setting (pr(0), p) = (cos , r0 sin ), then r(t) 0 (i.e. to the vortex) when t t if cos < 0 and r(t) + when t t if cos > 0, where t R + {+} is the maximal positive time such that the associated geodesic is defined over [0 , t). Denoting := arcsin Å p r0 ã [- 2 , 0) and introducing (cf. Rem. 3.5) 1 := - and 2 = 2 + = 3 - 1, then, for ( , 1), we have r(t) 0, while for ( 2 , 2), we have r(t) +, when t t. ­ Case = 0. Since we assume p 6= 0 then (p, r0, µ) = 0 if and only if p = p . In this case P(1/r2 ) = 0 if and only if r = r := 2µ = re. See Figure 8 for an illustrative example. We have four possibilities: · If r0 > r and pr(0) > 0, that is pr(0) = cos 2, then r(t) + when t t 2 . · If r0 < r and pr(0) < 0, that is pr(0) = cos 1, then r(t) 0 when t t 1 . · If r0 > r and pr(0) = cos 1 < 0, or if r0 < r and pr(0) = cos 2 > 0, then r(t) r . · If r0 = r and pr(0) = cos 1 = cos 2 = 0, then r(t) = r for any time t and thus, in this case, the geodesic describes a circle. Remark 3.6. We can notice that (re, pr,e) = (2µ, 0) is an unstable hyperbolic fixed point and its associated linearized system has ±1/2µ as eigenvalues with associated eigenvectors (±µ(4µ2 + r2 0)/r2 0, 1). Remark 3.7. Let us consider the dynamics: = 1 r2 Å µ + p kpkr ã . A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 21 Figure 8. Case = 0. Graph of r 7 P(1/r2 ). Setting: µ = 2, r0 = 3µ, p = p . Fixing p = -4µr2 /(r2 + 4µ2 ) and using the parameterization kpkr = 1 (this amounts to consider r as the initial distance to the vortex), then we have = 0 µ + p = 0 r = 2|µ|/ 3. See also equation (3.5). ­ Case > 0. In this case, P has two roots X- < X+ given by: X± := p2 r2 0 + 2 µ2 p2 + 2 µ p r2 0 ± r0 » p3 (4 p µ2 + 4 µ r2 0 + p r2 0) 2 µ2 p2 r2 0 . By a tedious calculation, one can prove that X- 0 and X- = 0 if and only if p = -r2 0/µ < p < 0. Let us introduce r1 := 1 p X+ and r2 := 1 p X- , with the convention 1/0 = +. Then, for any r (r1 , r2), we have P(1/r2 ) < 0 and thus it clear that r0 6 (r1 , r2), since by definition P(1/r2 0) = pr(0)2 0. 1. Subcase p > 0. Since p is positive we have 0 < X- < X+ and thus 0 < r1 < r2 < +. We can easily prove that p > 0 r1 < r0 since: r1 < r0 1 < r2 0 X+ 0 < p2 r2 0 + 2 µ p r2 0 + r0 » p3 (4 p µ2 + 4 µ r2 0 + p r2 0). This can be seen in Figure 9. Since r0 6 (r1 , r2), then p > 0 r2 r0 and thus r(t) will go to + in any case. More precisely, setting (pr(0), p) = (cos , r0 sin ), then we have: ­ If (0 , /2), then r(0) > 0 and thus r2 < r0. In this case, we have r(t) > 0 for any t 0 and thus r is strictly increasing and r(t) + when t t. ­ If = /2, then r(0) = 0, that is r2 = r0, but r(t) is strictly increasing for t > 0 and we still have r(t) + when t t. ­ If (/2 , ), then r(0) < 0 and again r2 < r0. In this case, r(t) is decreasing over [0 , t], where t > 0 is the time such that P(1/r(t)2 ) = 0, that is such that r(t) = r2. Since (r2, 0) is not an equilibrium point (a necessary condition is sign(p) = - sign(µ)), and since along the geodesic holds pr(t)2 = P(1/r(t)2 ), then necessarily, r(t) is increasing over [t , t) (the sign of pr changes at t = t 22 B. BONNARD ET AL. Figure 9. Case > 0 and p > 0. Graph of r 7 P(1/r2 ). Setting: µ = 2, r0 = 3µ, p = r0/2. and no more changing may occur after), even strictly increasing for t > t and we still have r(t) + when t t. 2. Subcase p < p . Here, we have -r0 p < p < 0. Setting (pr(0), p) = (cos , r0 sin ), this case corresponds to ( 1 , 2). Let us recall that if r0 = r = 2µ, then p = -r0 (i.e. 1 = 2 = 3/2), and so if r0 = r , then this last case is empty. We thus have either r0 < r or r0 > r . Let us prove now that r1 < r . To do this, we need the first following result: p < p p r2 0 + 4 µ2 p + 4 µ r2 0 < 0. Besides, r1 < r = 2 µ 1 < 4 µ2 X+ 0 < p p r2 0 + 4 µ2 p + 4 µ r2 0 + 2 , which is true since p < 0 and p < p . We are now in position to conclude: ­ If r0 > r then necessarily r0 r2 (see Fig. 10) since r0 6 (r1 , r2) and thus r(t) + when t t for any ( 1 , 2). More precisely, r(t) is strictly increasing if (3/2 , 2), it is increasing if = 3/2 (r(0) = 0) and has one oscillation if ( 1 , 3/2). ­ If r0 < r , using the fact that |p| r0 < r = 2µ, then one can prove by a tedious calculation that r0 r1 (see Fig. 11). Then, we have r(t) 0 when t t for any ( 1 , 2). More precisely, r(t) is strictly decreasing if ( 1 , 3/2), it is decreasing if = 3/2 (r(0) = 0) and has one oscillation if (3/2 , 2). Remark 3.8. According to Section 2.5.2, the abnormal extremals are given by pa := -r2 0/µ, that is by = a 1 and = a 2 (defined in Section 2.5.2). Since (recalling that µ > 0) pa - p = - r4 0 µ(r2 0 + 4µ2) < 0, it is clear that pa < p , which gives 1 < a 1 a 2 < 2, and thus the abnormal case is contained in this last case of > 0. Besides, the abnormal extremals exist only if the drift is strong or moderate, that is if r0 µ < r . To end this remark, let us mention that if the drift is strong, then for any (a 1 , a 2) 6= the associated extremal is elliptic and from the previous analysis, one can say that all the elliptic geodesics converge to the vortex, that is r(t) 0 when t t. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 23 Figure 10. Case > 0 and p < p : r 7 P(1/r2 ). Setting: µ = 2, r0 = 3µ, p = (p - r0)/2. Figure 11. Case > 0 and p < p : r 7 P(1/r2 ). Setting: µ = 2, r0 = 4µ/3, p = (p -r0)/2. We want now to group together the geodesics converging to the vortex and separate them from the ones going to infinity. We thus introduce for r0 > 0 and µ 6= 0, the following sets contained in [0 , 2): (r0, µ) := [ , 1) if r0 2µ and µ > 0 [ , 2) if r0 < 2µ and µ > 0 ( 2 , ] if r0 2|µ| and µ < 0 ( 1 , ] if r0 < 2|µ| and µ < 0 (r0, µ) := [0 , ) ( 1 , 2) if r0 > 2µ and µ > 0 [0 , ) ( 2 , 2) if r0 2µ and µ > 0 [0 , 2) ( , 2) if r0 > 2|µ| and µ < 0 [0 , 1) ( , 2) if r0 2|µ| and µ < 0 (r0, µ) := ® { 1} if (r0 > 2µ and µ > 0) or if (r0 2|µ| and µ < 0) { 2} if (r0 2µ and µ > 0) or if (r0 > 2|µ| and µ < 0) 24 B. BONNARD ET AL. Definition 3.9. The geodesics parameterized by (r0, µ) are called separating geodesics. We can now summarize the analysis: Theorem 3.10. Let us fix x0 M and µ 6= 0. Then, to any [0 , 2) is associated a unique zero order extremal z(·) := (x(·), p(·)) solution of the Hamiltonian system z = #-- H(z) given by equation (2.9), satisfying x(0) = x0, and parameterized in polar coordinates by kp(0)kr0 = 1 (with r0 := kx0k). Besides, we have: (r0, µ) : lim tt r(t) = 0, (r0, µ) : lim tt r(t) = + with t = +, (r0, µ) : lim tt r(t) = r = 2|µ| with t = +. 3.3. Reeb foliations 3.3.1. 3-Dimensional Compact Reeb Component A concrete example of foliation of codimension 1 on the 3-sphere S3 is given by the Reeb foliation [20]. To construct such a foliation, the idea is to glue together two 3-dimensional foliations on two solid tori along their boundary. Let us recall only the construction of such a foliation on the solid torus D2 × S1 , where D2 := (x, y) R2 x2 + y2 1 = B(0, 1), following the presentation of [23]. Let f : D2 R be a smooth mapping such that the following conditions are satisfied: 1. f is of the form: f(x, y) =: (x2 + y2 ), that is in polar coordinates, f(r, ) = (r2 ), 2. f(D2 ) = {0} and f(x, y) > 0 for (x, y) 6 D2 = S2 , 3. f has no critical points on D2 . Let consider the smooth mapping F : D2 × R R defined by F(x, y, t) := f(x, y) et , where (x, y) D2 and t R. F is a submersion from D2 × R to R and defines a foliation of D2 × R whose leaves are the level sets of F: F-1 (a), a 0. This foliation is called the 3-dimensional non compact Reeb component and it is denoted by R3 in [23]. Since F(x, y, t + 1) = e F(x, y, t), then, R3 defines a foliation on the quotient manifold D2 ×S1 = D2 ×R / (x, y, t) (x, y, t+1). This foliation being called the 3-dimensional compact Reeb component and denoted R3 c. This foliation admits a unique compact leaf: the boundary of D2 × R diffeomorphic to the torus T2 = S1 × S1 . All the other leaves are diffeomorphic to R2 and are also given by the quotient of the graphs of functions of the form (in polar coordinates): (r, ) B(0, 1) 7 - ln((r2 )) + b, b R. Remark 3.11. A simple example which is detailed in [23] is given by the mapping f(r, ) = 1 - r2 . Note that it is possible to construct a Reeb component on the solid torus with a function F which is a submersion only on B(0, 1) × R. Such an example is given by the function f(r, ) = exp(- exp(1/(1 - r2 ))). This possibility will be useful in the following section to construct a 2-dimensional Reeb component from the Vortex application. Let us exhibit now a function f in the Vortex application that can be used to construct a 3-dimensional compact Reeb component and thus a Reeb foliation on S3 . To this end, we focus on the separating geodesics, that is the geodesics associated to parameters (r0, µ). These geodesics are parameterized by p = p = -4µr2 0/(r2 0 + 4µ2 ) and kp(0)kr0 = 1, where p(0) = (pr(0), p) is the initial covector. Let us fix the circulation µ > 0 and the initial distance r0 > 0. Let (r0, µ) denotes the single element contained in this set. Let us denote by (r(·), (·)) the associated separating geodesic and by its orbit, that is := {(r(t), (t)) | t (t , t)}, where t < 0 is the minimal negative time such that the geodesic is well defined by backward integration. The time t > 0 being the maximal positive time such that the geodesic is well defined by forward integration. Let us consider now an arbitrary point (r1, 1) and define as the orbit associated to the unique separating geodesic when we consider (r1, 1) as the new initial state. Then, it is clear that = since again the set A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 25 (r1, µ) has only one element. Thus, considering that at any time t, r(t) is the initial distance to the vortex, leads to reparametrize the separating geodesics by setting p = -4µr(t)2 /(r(t)2 + 4µ2 ) and kp(t)kr(t) = 1, which gives pr(t)2 = 1 - p2 r(t)2 = (r(t)2 - 4µ2 )2 (r(t)2 + 4µ2)2 . In the case 0 < r0 < 2µ, the parameter (r0, µ) is given by 2 and thus pr(t) > 0 for any time t (see Sect. 3.2). In this case, the dynamics of the separating geodesics reduces to the 2-dimensional differential equations: r = pr kpkr = pr = 4µ2 - r2 4µ2 + r2 , = 1 r2 Å µ + p kpkr ã = 1 r2 (µ + p) = µ r2 Å 4µ2 - 3r2 4µ2 + r2 ã . (3.5) Writing dt = 4µ2 + r2 4µ2 - r2 dr, then by integration we have t(r) = 4µ atanh Å r 2µ ã - r + c = 2µ ln Å 2µ + r 2µ - r ã - r + c = ln ÇÅ 2µ + r 2µ - r ã2µ e-r K Ã¥ , with a constant c := ln K. Introducing 2µ2 := r, [0 , 1], then one can define the function f(, ) := (2 ) := Å 1 - 2 1 + 2 ã2µ e2µ2 which satisfies the conditions (1)-(2)-(3) to construct a 3-dimensional compact Reeb component from F(, , t) := f(, ) et . 3.3.2. 2-Dimensional Compact Reeb Component In the Vortex application, one can find a foliation in the 2-dimensional state space given by the separating geodesics. To present this foliation, we need to introduce a generalized 2-dimensional Reeb component following the presentation of [23]. We thus first recall what is a 2-dimensional Reeb component and then we introduce the generalization. Let f : [-1 , 1] R be a smooth mapping such that the following conditions are satisfied: 4. f is of the form: f(x) =: (x2 ), 5. f({-1, 1}) = {0} and f((-1 , 1)) > 0, 6. f has no critical points on {-1, 1}. Let consider the smooth mapping F : [-1 , 1] × R R defined by F(x, t) := f(x) et , where x [-1 , 1] and t R. F is a submersion from [-1 , 1] × R to R and defines a foliation of [-1 , 1] × R whose leaves are the level sets of F: F-1 (a), a 0. This foliation is called the 2-dimensional non compact Reeb component and it is denoted by R2 in [23]. Since F(x, t + 1) = e F(x, t), then, R2 defines a foliation on the annulus [-1 , 1] × S1 = [-1 , 1] × R / (x, t) (x, t + 1). This foliation being called the 2-dimensional compact Reeb component and denoted R2 c. This foliation admits two compact leaves: the two boundaries of the annulus, diffeomorphic to S1 . All the other leaves are diffeomorphic to R and each of their two extremities wind up around one of the two 26 B. BONNARD ET AL. compact leaves. These non-compact leaves are also given by the quotient of the graphs of functions of the form: x (-1 , 1) 7 - ln((x2 )) + b, b R. Remark 3.12. A simple example which is detailed in [23] is given by the function f(x) = 1 - x2 . The maximum of this function is given by solving f0 (x) = 2x = 0, that is for x = 0. Note that it is possible to construct a Reeb component on the annulus with a function F which is a submersion only on (-1 , 1) × R considering for instance f(x) = exp(- exp(1/(1 - x2 ))). Remark 3.13. Note that we can construct a 3-dimensional Reeb component from the 2-dimensional Reeb component if f satisfies the condition (4)-(5)-(6). See [23] for details. Let us slightly generalize the construction of a Reeb component on the annulus. To do so, we consider a function f : [a , b] R smooth on (a , b), a < b in R, and such that: 7. f({a, b}) = {0}, f((a , b)) > 0. Then, F(x, t) := f(x) et is a submersion from (a , b) × R to R and defines a foliation on [a , b] × R which is called the generalized 2-dimensional non compact Reeb component and denoted R2,G . As in the two previous cases, R2,G defines a foliation on the annulus [a , b] × S1 which is called the generalized 2-dimensional compact Reeb component and denoted R2,G c . This foliation, like R2 c, has two compact leaves diffeomorphic to S1 and all the other leaves are diffeomorphic to R and given by the quotient of the graphs of functions of the form: x (a , b) 7 - ln(f(x)) + c, c R. Remark 3.14. The most important difference with the classical case is that we do not impose that f(x) is of the form (x2 ). Even in the case a = -b, if f(x) is not of the form (x2 ), then we do not have necessarily f(-x) = f(x) and if we do not have this symmetry, then it is not possible to construct a 3-dimensional Reeb component from a 2-dimensional one using the construction detailed in [23]. Let us exhibit now a function f from the Vortex application that satisfies the condition (7). We fix µ > 0. Coming back to equation (3.5), one can write d = µ r2 Å 4µ2 - 3r2 4µ2 - r2 ã dr. Integrating we have: (r) = atanh Å r 2µ ã + µ r + c = 1 2 ln Å 2µ + r 2µ - r ã + µ r + c = ln ÇÅ 2µ + r 2µ - r ã1 2 e µ r K Ã¥ , with a constant c := ln K. Introducing for r [0 , 2µ]: f(r) := Å 2µ - r 2µ + r ã1 2 e- µ r , (3.6) with f(0) = 0, then the separating geodesics inside the punctured disk of radius 2µ are exactly given by the level sets of the function F(r, ) := f(r) e . Besides, the function f satisfies the condition (7) and thus, it defines a generalized 2-dimensional compact Reeb component on the annulus [0 , 2µ] × S1 , see Figure 12. The separating geodesics inside the punctured disk of radius 2µ correspond to the non-compact leaves of the foliation while the unique circle geodesic, that is the separating geodesic with r0 = 2µ, corresponds to one of the two compact leaves. The vortex corresponds to the second compact leaf. Figure 13 shows the foliation on the punctured disk of radius 2µ in the (x1, x2)-plane of the Vortex application. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 27 Figure 12. (Left) Graphs of r (0 , 2µ) 7 - ln(f(r)) + c, c R, with f defined by equa- tion (3.6). The minimum is attained at r = 2µ/ 3 where f0 (r) = 0. (Right) Generalized 2-dimensional compact Reeb component of the annulus given by the function F(r, ) = f(r) e . The intermediate dashed circle of radius 2µ/ 3 is orthogonal to all the non-compact leaves in blue. The two compact leaves are the two red circles. Figure 13. Separating geodesics in the punctured disk of radius 2µ in the (x1, x2)-plane. The vortex is placed at the origin and represented by a red dot. The red circle is the Reeb circle of radius 2µ while the black and dashed circle is the circle of radius 2µ/ 3 where = 0 along the separating geodesics. Definition 3.15. In the Vortex application, the circle of radius 2|µ| (assuming µ 6= 0) is called the Reeb circle. 3.4. Symmetries of the extremal curves There exists three continuous symmetries and one discrete symmetry of main interest. The first continuous symmetry was considered at the beginning of the paper when we have fixed umax = 1. The second symmetry is the rotational symmetry that was mentioned at the end of Section 2.2. As a consequence of this symmetry of 28 B. BONNARD ET AL. revolution, one can easily prove that: R : V (r0, 0, rf , f , µ) = V (r0, 0 + , rf , f + , µ), where V is the value function (2.4) expressed in polar coordinates. Hence, one can fix for instance 0 = 0 and reduce the set of variables of the value function. Note that the Reeb circle is the unique geodesic invariant by the rotational symmetry. The third continuous symmetry comes from the invariance of the set of geodesics by homothety combined with time/circulation dilatation. Let us detail this third symmetry. We start by introducing the change of variables (r, ) := (r, ) for a given R +. The associated Mathieu transformation leads to p := (pr, p) = pr , p and the Hamiltonian system (2.12) in polar coordinates becomes r = r = pr kpkr = pr kpkr = pr kpkr , = = 1 r2 Å µ + p kpkr ã = 2 r2 Å µ + p kpkr ã = r2 Å µ + p kpkr ã for the state and pr = pr = p r3 Å 2µ + p kpkr ã = 2 p r3 Å 2µ + p kpkr ã = p r3 Å 2µ + p kpkr ã , p = 0, for the covector, since kpk2 r = pr , p 2 r = p2 r 2 + p2 2r2 = 1 2 kpk2 r. Introducing now µ := µ, the time reparameterization t = () =: /, and the notation (r, , pr, p)() := (r, , pr, p)(()), then the system in the time writes d d r() = d d r(()) = r(())0 () = r(t) = pr() kp()kr() , d d = 1 r2 Å µ + p kpkr ã , d d pr = p r3 Å 2µ + p kpkr ã , d d p = 0, which is equivalent to the original system (2.12). As a conclusion, the set of solutions of the problem (P) is invariant by the dilatation (r(·), (·), t, µ) 7 (r(·), (·), t, µ) := (r(·), (·), t, µ), for any R +, and we have 1 V (r0, 0, rf , f , µ) = V (r0, 0, rf , f , µ). It is thus possible to fix for instance r0 and again to reduce the set of variables of the value function. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 29 Figure 14. Geodesics in the state space in the (x1, x2)-plane. The vortex is placed at the origin and represented by a black dot. The initial point, denoted x0, is the black dot at an intersection of both geodesics. The geodesic in green is obtained by a reflection of axis Rx0 from the blue geodesic and vice-versa. Assuming that the initial point is given at time t = 0, then the plain lines are obtained for times t > 0 while the dashed parts are for negative times. On the left subgraph is represented a normal geodesic with both extremities going to infinity while on the right subgraph is represented the two abnormals (in the case of a strong drift) with a cusp at r = |µ|. Both extremities of the abnormal geodesics go the vortex. Remark 3.16. Setting := 1/r0 and considering also the invariance by rotation, we can write: V (r0, 0, rf , f , µ) = r0V (1, 0, r, , µ r0 ), with r := rf /r0 and := f - 0. This shows that the value function depends, in fact, only on the three variables r, and µ/r0. We end this section presenting the discrete symmetry on the set of extremals. Let (r(·), (·), pr(·), p) be a reference extremal on [0 , T] and introduce 0 := (0). Let consider the following discrete symmetry: z(·) := (r(·), (·), pr(·), p) := (r(·), 20 - (·), -pr(·), p). Then, z(·) is solution of z = - #-- H(z), z(0) = (r(0), 0, -pr(0), p). This means that z(·) is obtained by backward integration from the same initial condition in the state space than z(·) but with the initial covector (-pr(0), p). Let us extend the reference extremal to obtain a maximal solution of z = #-- H(z), z(0) = (r(0), 0, pr(0), p). In this case, its orbit in the state space is given by := Im(r(·), (·)) and z is also a maximal solution of its associated differential equation, and we denote by := Im(r(·), (·)) its orbit in the state space. Then, we have: Å 1 0 0 -1 ã + Å 0 20 ã = , that is is obtained by an affine reflection of axis = 0. In cartesian coordinates, one has a linear reflection of axis Rx0, see Figure 14. 30 B. BONNARD ET AL. 3.5. Properties of the value function and its level sets in the weak case 3.5.1. Continuity of the Value Function and Characterization of the Cut Points Let us introduce the mapping Vµ(x0, xf ) := V (x0, xf , µ), where V is the value function defined by (2.4). We are first interested in the continuity of xf 7 Vµ(x0, xf ) at points xf where the drift is weak, that is for target xf such that kxf k > |µ|. This situation is more general than the Finslerian case with Randers metric since the drift is not necessarily weak all along the geodesics. We have: Proposition 3.17. Let µ 6= 0 be the vortex circulation and x0 M the initial condition. Then, Vµ(x0, ·) is continuous at any point xf such that kxf k > |µ|. Proof. Since we consider that the drift is weak at the target xf then local controllability around xf holds, that is we have the following: > 0, > 0 s.t. (x1, x2) B(xf , ) × B(xf , ) : Vµ(x1, x2) . (3.7) Let us fix > 0 and consider x B(xf , ). By definition of the value function and by (3.7), we have Vµ(x0, x) - Vµ(x0, xf ) Vµ(xf , x) and Vµ(x0, xf ) - Vµ(x0, x) Vµ(x, xf ) , which prove the result. We are now interested in the construction of the optimal synthesis associated to problem (P) for given values of µ and x0, that is we want to get for any xf M the value of Vµ(x0, xf ) together with the optimal geodesic joining the points x0 and xf . To construct such an optimal synthesis, a classical approach is to compute the level sets of Vµ(x0, ·), which corresponds in Riemannian or Finslerian geometry to compute the spheres centered in x0. To compute the spheres, one preliminary work is to compute the cut locus, that is the set of cut points where the geodesics cease to be optimal. Let us introduce some notations before characterizing the cut points. Following Section 2.5.2 and fixing x0 M, we introduce the notation in polar coordinates p0() := (cos , r0 sin ), with [0 , 2) and where r0 := kx0k. Let t R+, we introduce the wavefront from x0 at time t by W(x0, t) := expx0 (t, p0()) [0 , 2) s.t. t < t , where the exponential mapping is given by (2.10) and depends on the circulation µ supposed to be given, and where we recall that t R + {+} is the maximal positive time such that the associated geodesic is defined over [0 , t). Then, we define what corresponds to the balls and spheres in Riemannian or Finslerian geometry: B(x0, t) := {x M | Vµ(x0, x) < t} , B(x0, t) := {x M | Vµ(x0, x) t} , S(x0, t) := {x M | Vµ(x0, x) = t} . We use the terminology ball and sphere by analogy with Geometry. Note that the spheres S(x0, ·) are the level sets of Vµ(x0, ·). To construct the optimal synthesis, one of the main important loci to compute is the cut locus. We define first the cut times as: tcut(x0, ) := sup t R + expx0 (·, p0()) is optimal over [0 , t] . Then, the cut locus from x0 is simply defined by: Cut(x0) := x M (t, ) R + × [0 , 2) s.t. x = expx0 (t, p0()) and t = tcut(x0, ) . A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 31 Any point of the cut locus is called a cut point. From the cut times, we define also the injectivity radius from x0 as: tinj(x0) := inf [0,2) tcut(x0, ). Remark 3.18. This time is of particular interest since if the drift is weak at the initial point x0, then expx0 is a diffeomorphism from (0 , tinj(x0)) × S1 to B(x0, tinj(x0)) \ {x0}. This is similar to the Riemannian and Finslerian cases and for times 0 t tinj(x0), we have S(x0, t) = W(x0, t). In Finslerian geometry [3] (as in Riemannian geometry [19], Prop. 2.2, Chap. 13), the cut locus is a part of the union of the conjugate locus and the splitting locus, which is defined by: Split(x0) := ¶ x M (t, 1, 2) R + × [0 , 2) 2 s.t. 1 6= 2 and expx0 (t, p0(1)) = expx0 (t, p0(2)) = x © . We have the following similar result: Proposition 3.19. Let x(·) be a geodesic. If xc := x(tc) is a cut point along x(·) such that Vµ(x0, ·) is continuous at xc, then: (a) either xc is a conjugate point along x(·), (b) or xc is a splitting point along x(·). Conversely if a) or b) holds, then there exists a time 0 t1 tc such that x(t1) is a cut point along x(·). Remark 3.20. The proposition above gives a characterization of the cut points where the mapping Vµ(x0, ·) is continuous. In Riemannian and Finslerian geometry, the mapping Vµ(x0, ·) is replaced by the distance function which is continuous with respect to its second argument and so all the cut points are either conjugate or splitting points. By Proposition 3.17, if the drift is weak at the cut points, then Vµ(x0, ·) is continuous at this point and so the characterization holds. In the strong case, we can have abnormal minimizers and so we can have cut points which are neither conjugate nor splitting points. However, if the drift is weak at the initial point x0, then there are no abnormals and we can expect that the mapping Vµ(x0, ·) is continuous at any point, even where the drift is strong. This result would be useful to conclude in the following part since we are interested in the construction of the optimal synthesis for a given initial condition x0 where the drift is weak. Note that the numerical experiments of the following section suggest that Vµ(x0, ·) is continuous at any point of M. 3.5.2. Level Sets of the Value Function and Optimal Synthesis In this section, we fix for the numerical experiments x0 := (3, 0) and µ := 0.6 kx0k. The drift is thus weak at the initial position x0. We decompose the construction of the optimal synthesis in three steps. In the first step, we compute the splitting locus. In a second time we give the cut locus and we finish by the construction of the spheres and balls. Step 1: Computation of the splitting locus. Let us introduce the mapping Fsplit(t, 1, x, 2) := (x - expx0 (t, p0(1)), x - expx0 (t, p0(2))) R4 . The splitting line is then given by solving Fsplit = 0 since we have Split(x0) = {x M | (t, 1, 2) s.t. 1 6= 2 and Fsplit(t, 1, x, 2) = 0} . Introducing y := (t, 1, x) R4 and := 2 R, we have to solve Fsplit(y, ) = 0 R4 . Numerically, we compute the splitting line with the differential path following method (or homotopy method) of the HamPath software. 32 B. BONNARD ET AL. 1 2 3 Figure 15. Four wavefronts with different scaling at times t {0.5, 2, 2.8, 3.5}. The black dot is x0. For t = 3.5, the wavefront has at least three self-intersections labelled 1, 2 and 3. Under some regularity assumptions, the set F-1 split(0) is a disjoint union of differential curves, each curve being called a path of zeros. To compute a path of zeros, we look for a first point on the curve by fixing the homotopic parameter to a certain value and calling a Newton method to solve Fsplit(·, ) = 0. Then, we use a Prediction- Correction (PC) method to compute the differential curve. The HamPath code implements a PC method with a high-order Runge­Kutta scheme with variable step-size for the prediction, hence the discretization grid of the homotopic parameter is computed by the numerical integrator. Besides, the Jacobian of Fsplit is computed by automatic differentiation combined with variational equations of the exponential mapping. Remark 3.21. If there exists several paths of zeros, then each path has to be found manually and compared. To compute a splitting curve (that is a path of zeros), we need to get a first point on the curve. This step is easy since the splitting points are self-intersections of the wavefronts. On the right subgraph of Figure 15, we can see that the wavefront3 W(x0, 3.5) has at least three self-intersections labelled 1, 2 and 3. Each self-intersection of the wavefront W(x0, 3.5) is a point of a splitting curve. We denote by 1, 2 and 3 the three splitting curves associated respectively to the self-intersections 1, 2 and 3, and we have 1 2 3 Split(x0). In Figure 16 is represented the graph of the function 2 7 t(2) along the three splitting curves for times t 3.2, the homotopic parameter 2 being strictly monotone along the splitting curves. Since 1 is below 2 and 3 on the figure,4 it is clear that only 1 has a chance to be part of the cut locus, compared to 2 and 3. Note that there may exist others splitting curves but the numerical experiments suggest that 1 is below any of them. Remark 3.22. Note that the extremities of a splitting curve are the vortex and infinity according to the numerical experiments. Hence, we cannot compute all the curve. Numerically, we use an option of the HamPath code to stop the homotopy if the curve goes out the annulus of smaller circle of radius 0.005 and of larger one of radius 100. Let us describe the evolution of the wavefronts according to Figure 16. For t = 0, the wavefront is reduced to the singleton {x0}. For times t (0 , tinj(x0)) with tinj(x0) = inf t(·) along 1, the wavefronts have no self- intersections. For t = tinj(x0), the wavefront has one self-intersection, while for t (tinj(x0) , tvor(x0)), where tvor(x0) := kx0k is the minimal time to reach the vortex, the wavefronts have two self-intersections contained in 1, see Figure 17, and up to six self-intersections in 1 2 3. Finally, for times t tvor(x0), the wavefronts have one self-intersection in 1 and three in 1 2 3, see the right subgraph of Figure 15. 3The wavefronts are computed by homotopy. We compute only rough approximations of the extremities of the wavefronts which are not closed curves, like W(x0, 3.5). Hence, there may exists others self-intersections but which are not relevant for rest of the analysis. 4We mean that the graph of 2 7 t(2) associated to 1 is below the ones associated to 2 and 3. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 33 Figure 16. Graph of the function t(2) along 1 (blue), 2 (red) and 3 (green) for times t 3.2. The balls are of three types: A, B and C. See the description detailed in the step 3. Figure 17. Three wavefronts with the same scaling and a zoom around the vortex (red dot) for t {2.86, 2.88, 2.9}. For t = 2.9 > tinj(x0) 2.889, the wavefront has two self-intersections. Step 2: Computation of the cut locus. According to Proposition 3.17, at a final point xf where the drift is weak the mapping Vµ(x0, ·) is continuous. In this case, from Proposition 3.19, if xf is a cut point, then either it is a conjugate point or a splitting point. Since there are no conjugate points from Conjecture 2.28, xf is a splitting point. Finally, from step 1, we can conclude that if xf is a cut point then it belongs to 1. In this section, we consider moreover that the drift is weak at the initial condition x0. In this case, we conjecture the following: Conjecture 3.23. If the drift is weak at x0, then Vµ(x0, ·) is continuous at any point of M. This conjecture means that the characterization of cut points from Proposition 3.19 holds in all the state space M. Hence, we have Cut(x0) 1. Now, by the converse part of Proposition 3.19 and according to the previous conjecture, we can conclude that: Cut(x0) = 1. Step 3. Computations of the spheres and balls. Figure 18 represents four balls that we can group into three categories (see also Fig. 16): ­ Type A. The ball is simply connected in R2 with a smooth boundary; ­ Type B. The ball is not simply connected in R2 ; 34 B. BONNARD ET AL. Figure 18. Four balls in the (x1, x2)-plane at times t {1.5, 2.8, 2.9, 3.5} with the initial condition x0 represented by a black dot and the vortex by a red dot. The balls B(x0, 1.5) and B(x0, 2.8) on top are of type A. The ball B(x0, 2.9) (Bottom-Left) is of type B while B(x0, 3.5) (Bottom-Right) is of type C. Figure 19. Three balls with a zoom around the vortex (red dot) at times t {2.86, 2.88, 2.9}. Compare to Figure 17. For t = 2.9 the ball B(x0, t) is of type C and has a small hole with a fetus shape. A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 35 Figure 20. Spheres S(x0, t) at times t {0.5, 1.5, 2.5, 3.5} in black together with the cut locus 1 in blue. The cut locus should go to the vortex but to gain in clarity, it is represented up to a distance of 0.12 to the vortex. The vortex being represented by a red dot while the initial condition is represented by a black dot. Figure 21. Optimal synthesis. The black curves represent the spheres of types A and C. The two blue curves is a single sphere of type B. Since the balls of type B are not simply connected and have one hole, the corresponding spheres have two connected components. ­ Type C. The ball is simply connected in R2 with a singularity on its boundary. For times t (0 , tinj(x0)], we have W(x0, t) = S(x0, t) and the ball B(x0, t) is of type A. The two balls on the top of Figure 18 are of type A. For times t (tinj(x0) , tvor(x0)), the balls have a hole around the vortex and so they are of type B. The ball at the bottom-left subgraph of Figure 18 is of type B. One can see on Figure 19, the evolution of the balls around the vortex for times close to tinj(x0). The hole appears when the ball self- intersects, like a snake biting its tail. For times t tvor(x0), the balls are simply connected in R2 since the hole 36 B. BONNARD ET AL. has vanished (the vortex is reached at t = tvor(x0)) and the spheres have a singularity at the cut point. In this case the balls have a shape of an apple and are of type C. The bottom-right subgraph of Figure 18 represents a ball of type C. One can see in Figure 20 the evolution of the spheres S(x0, t) at times t {0.5, 1.5, 2.5, 3.5} together with the cut locus 1. One can notice that the singularity on S(x0, 3.5) in contained in the cut locus. Putting all together, we have constructed the optimal synthesis which is given on Figure 21. 4. Conclusion In this article, we have analyzed the Zermelo navigation problem with a vortex singularity. We have shown the existence of an optimal solution. Thanks to the integrability properties we made a micro-local classification of the extremal solutions, showing remarkably the existence of a Reeb foliation. The spheres and balls are described in the case where at the initial condition the current is weak. This gives us the optimal synthesis for any initial condition such that kx0k > |µ|. Note that the current is not necessarily weak all along the geodesics and so the situation we have analyzed is more general than the Randers case from Finsler geometry. Still, this analysis has to be completed to the case of a strong current at the initial condition, in particular in relation with the abnormal extremals. Note that the historical work [15] is a case study of the strong current situation which can be applied to our study. A possible further extension concerns the case of several vortices by analogy with the N-body problem. Finally, in a forthcoming article, the 2D-Zermelo navigation problem will be generalized to the case of arbitrary current and metric, with the symmetry of revolution. It will cover the historical problem and the vortex case. The analysis will be based on tools from Hamiltonian dynamics from [6] and in particular the concept of Reeb graph. Acknowledgements. We are grateful to Carlos Balsa for helpful discussions about vortex theory and for drawing authors' attention to this topic, and to Ulysse Serres for helpful discussions about Zermelo-like navigation problems. References [1] A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint. Vol. 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004). [2] V.I. 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Vol. 40 of Mathematics & Applications. Springer-Verlag, Berlin (2003). [9] B. Bonnard and D. Sugny, Optimal Control with Applications in Space and Quantum Dynamics. Vol. 5 of Applied Mathematics. AIMS (2012). [10] B. Bonnard, L. Faubourg and E. Trelat, Mecanique celeste et controle des vehicules spatiaux. Vol. 51 of Mathematiques & Applications. Springer-Verlag, Berlin (2006). [11] B. Bonnard, J.-B. Caillau and E. Trelat, Second order optimality conditions in the smooth case and applications in optimal control. ESAIM: COCV 13 (2007) 207­236. [12] A.E. Bryson and Y.-C. Ho, Applied Optimal Control. Hemisphere Publishing, New York (1975). [13] J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems. Optim. Methods Softw. 27 (2011) 177­196. [14] R.C. Calleja, E.J. Doedel and C. Garcia-Azpeitia, Choreographies in the n-vortex problem. Regul. Chaot. Dyn. 23 (2018) 595­612. [15] C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2. Holden-Day, San Francisco, California (1965­1967); Reprint: 2nd AMS printing, AMS Chelsea Publishing, Providence, RI, USA (2001). A ZERMELO NAVIGATION PROBLEM WITH A VORTEX SINGULARITY 37 [16] L. Cesari, Optimization-theory and Applications: Problems with Ordinary Differential Equations. Vol. 17 of Applications of Mathematics. Springer-Verlag, New York (1983). [17] A. Chenciner, J. Gerver, R. Montgomery and C. Simo, Simple Choreographic Motions of N Bodies: A Preliminary Study, edited by P. Newton, P. Holmes, A. Weinstein. Geometry, Mechanics, and Dynamics. Springer, New York, NY (2002) 287­308. [18] O. Cots, Controle optimal geometrique: methodes homotopiques et applications. Ph.D. thesis, Universite de Bourgogne, Dijon (2012). [19] M.P. Do Carmo, Riemannian Geometry. Mathematics: Theory & applications, 2nd ed. Birkhauser (1988). [20] G. Godbillon, Feuilletages, etudes geometriques. Vol.98 of Progr. Math. Birkhauser, Boston (1991). [21] V.V. Goryunov and V.M. Zakalyukin, Lagrangian and Legendrian singularities, in Real and Complex Singularities. Trends in Mathematics, edited by J.P. Brasselet and M.A.S. Ruas. Birkhauser, Basel (2006). [22] R. Hama, J. Kasemsuwan and S.V. Sabau, The cut locus of a Randers rotational 2-sphere of revolution. Publ. Math. Debrecen 93 (2018) 387­412. [23] G. Hector, Les feuilletages de Reeb, L'Ouvert. Num. 76 special Georges Reeb. IREM de Strasbourg, Strasbourg (1994) 93­111. [24] H.V. Helmholtz, On integrals of hydrodynamics equations, corresponding to vortex motions. Russ. J. Nonlin. Dyn. 2 (2006) 473­507. [25] J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids. Manuscripta Math. 114 (2004) 247­264. [26] V. Jurdjevic, Geometric Control Theory. Vol. 52 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997). [27] G.R. Kirchhoff, Vorlesungen uber mathematische Physik. Mechanik, Leipzig, Teubner (1876). [28] A.J. Krener, The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15 (1977) 256­293. [29] K. Meyer, G. Hall and D.C. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Vol. 90 of Applied Mathematical Sciences. Springer-Verlag, New York (2009). [30] P.K. Newton, The N-Vortex Problem ­ Analytical Techniques. Vol. 145 of Applied Mathematical Sciences. Springer-Verlag, New York (2001) 420. [31] H. Poincare, OEuvres. Gauthier-Villars (1952). [32] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, translated from the Russian by K.N. Trirogoff, edited by L.W. Neustadt. Interscience Publishers/John Wiley & Sons, Inc., New York/London (1962). [33] B. Protas, Vortex Dynamics Models in Flow Control Problems. Nonlinearity 21 (2008) R203. IOP Science (2008). [34] P.G. Saffman, Vortex Dynamics. Cambridge University Press (1992). [35] U. Serres, Geometrie et classification par feedback des systemes de controle non lineaires de basse dimension. Ph.D. thesis, Universite de Bourgogne, Dijon (2006) 51­62. [36] D. Vainchtein and I. Mezi, Optimal control of a co-rotating vortex pair: averaging and impulsive control. Physica D 192 (2004) 63­82. [37] E. Zermelo, Uber das Navigations problem bei ruhender oder veranderlicher wind-verteilung. Z. Angew. Math. Mech. 11 (1931) 114­124. [1] A. A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint. Vol. 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004). [2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics. Vol. 125 of Applied Mathematical Sciences. Springer-Verlag, New York (1998). [3] D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry. Vol. 200 of Graduate Texts in Mathematics. Springer-Verlag, New York (2000). [4] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66 (2004) 377–435. [5] V. G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming method. SIAM J. Control Optim. 4 (1966) 326–361. [6] A. V. Bolsinov, and A. T. Fomenko, Integrable Hamiltonian Systems Geometry, Topology, Classification Chapman and Hall/CRC, London (2004). [7] B. Bonnard and I. Kupka, Théorie des singularités de l’application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Math. 5 (1993) 111–159. [8] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory. Vol. 40 of Mathematics & Applications. Springer-Verlag, Berlin (2003). [9] B. Bonnard and D. Sugny, Optimal Control with Applications in Space and Quantum Dynamics. Vol. 5 of Applied Mathematics. AIMS (2012). [10] B. Bonnard, L. Faubourg and E. Trélat, Mécanique céleste et contrôle des véhicules spatiaux. Vol. 51 of Mathématiques & Applications. Springer-Verlag, Berlin (2006). [11] B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control. ESAIM: COCV 13 (2007) 207–236. [12] A. E. Bryson and Y.-C. Ho, Applied Optimal Control. Hemisphere Publishing, New York (1975). [13] J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems. Optim. Methods Softw. 27 (2011) 177–196. [14] R. C. Calleja, E. J. Doedel and C. García-Azpeitia, Choreographies in the n-vortex problem. Regul. Chaot. Dyn. 23 (2018) 595–612. [15] C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Part 1, Part 2. Holden-Day, San Francisco, California (1965–1967); Reprint: 2nd AMS printing, AMS Chelsea Publishing, Providence, RI, USA (2001). [16] L. Cesari, Optimization-theory and Applications: Problems with Ordinary Differential Equations. Vol. 17 of Applications of Mathematics. Springer-Verlag, New York (1983). [17] A. Chenciner, J. Gerver, R. Montgomery and C. Simó, Simple Choreographic Motions of N Bodies: A Preliminary Study, edited by P. Newton, P. Holmes, A. Weinstein. Geometry, Mechanics, and Dynamics. Springer New York, NY (2002) 287–308. [18] O. Cots, Contrôle optimal géométrique: méthodes homotopiques et applications. Ph.D. thesis, Université de Bourgogne, Dijon (2012). [19] M. P. Do Carmo, Riemannian Geometry. Mathematics: Theory & applications, 2nd ed. Birkhäuser (1988). [20] G. Godbillon, Feuilletages, études géométriques. Vol. 98 of Progr. Math. Birkhäuser, Boston (1991). [21] V. V. Goryunov and V. M. Zakalyukin, Lagrangian and Legendrian singularities, in Real and Complex Singularities. Trends in Mathematics, edited by J. P. Brasselet and M. A. S. Ruas. Birkhäuser, Basel (2006).. [22] R. Hama, J. Kasemsuwan and S. V. Sabau, The cut locus of a Randers rotational 2-sphere of revolution. Publ. Math. Debrecen 93 (2018) 387–412. [23] G. Hector, Les feuilletages de Reeb, L’Ouvert. Num. 76 spécial Georges Reeb. IREM de Strasbourg, Strasbourg (1994) 93–111. [24] H. V. Helmholtz, On integrals of hydrodynamics equations, corresponding to vortex motions. Russ. J. Nonlin. Dyn. 2 (2006) 473–507. [25] J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids. Manuscripta Math. 114 (2004) 247–264. [26] V. Jurdjevic, Geometric Control Theory. Vol. 52 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997). [27] G. R. Kirchhoff, Vorlesungen uber mathematische Physik. Mechanik, Leipzig, Teubner (1876). [28] A. J. Krener, The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15 (1977) 256–293. [29] K. Meyer, G. Hall and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Vol. 90 of Applied Mathematical Sciences. Springer-Verlag, New York (2009). [30] P. K. Newton, The N-Vortex Problem – Analytical Techniques. Vol. 145 of Applied Mathematical Sciences. Springer-Verlag, New York (2001) 420. [31] H. Poincaré, Œuvres. Gauthier-Villars (1952). [32] L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, translated from the Russian by K. N. Trirogoff, edited by L. W. Neustadt. Interscience Publishers/John Wiley & Sons, Inc., New York/London (1962).. [33] B. Protas, Vortex Dynamics Models in Flow Control Problems. Nonlinearity 21 (2008) R203. IOP Science (2008).. [34] P. G. Saffman, Vortex Dynamics. Cambridge University Press (1992). [35] U. Serres, Géométrie et classification par feedback des systèmes de contrôle non linéaires de basse dimension. Ph.D. thesis, Université de Bourgogne, Dijon (2006) 51–62. [36] D. Vainchtein and I. Mezić, Optimal control of a co-rotating vortex pair: averaging and impulsive control. Physica D 192 (2004) 63–82. [37] E. Zermelo, Über das Navigations problem bei ruhender oder veränderlicher wind-verteilung. Z. Angew. Math. Mech. 11 (1931) 114–124. COCV_2021__27_S1_A12_0c8f99e5a-448e-4b3c-b8ae-f18704f9510c cocv200078 10.1051/cocv/202005510.1051/cocv/2020055 Local minimizers and gamma-convergence for nonlocal perimeters in Carnot groups* Carbotti Alessandro 1 0000-0002-7881-742X Don Sebastiano 2 0000-0002-1909-484X Pallara Diego 3 Pinamonti Andrea 4** 1 Dipartimento di Matematica e Fisica, Università del Salento, Via Per Arnesano, 73100 Lecce, Italy. 2 Department of Mathematics and Statistics, P.O. Box 35 (MaD), 40014, University of Jyväskylä, Finland. 3 Dipartimento di Matematica e Fisica, Università del Salento, and INFN, Sezione di Lecce, Via Per Arnesano, 73100 Lecce, Italy. 4 Dipartimento di Matematica, Università di Trento, Via Sommarive, 14, 38123 Povo, TN, Italy. **Corresponding author: [e-mail] 01 03 2021 01 03 2021 2021 cocv/2021/01 Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science OuverteSupplementS11 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF) Full (DJVU)

We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi’s rectifiability theorem holds, we provide a lower bound for the Γ-liminf of the rescaled energy in terms of the horizontal perimeter.

Carnot groups calibrations nonlocal perimeters Γ-convergence sets of finite perimeter rectifiability 53C17 22E25 49Q15 53C38 26A33 49Q05 idline ESAIM: COCV 27 (2021) S11 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S11 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020055 www.esaim-cocv.org LOCAL MINIMIZERS AND GAMMA-CONVERGENCE FOR NONLOCAL PERIMETERS IN CARNOT GROUPS Alessandro Carbotti1 , Sebastiano Don2 , Diego Pallara3 and Andrea Pinamonti4,** Abstract. We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability theorem holds, we provide a lower bound for the -liminf of the rescaled energy in terms of the horizontal perimeter. Mathematics Subject Classification. 53C17, 22E25, 49Q15, 53C38, 26A33, 49Q05. Received April 5, 2020. Accepted August 8, 2020. 1. Introduction Given an open set Rn and (0, 1), we define the nonlocal (or fractional) -perimeter of a measurable set E Rn as the functional P(E; ) := L(Ec , E ) + L(Ec , E c ) + L(E , Ec c ) (1.1) where L(A, B) := Z A Z B 1 |x - y|n+ dx dy. The notion of fractional perimeter was introduced in [9] to study nonlocal minimal surfaces of fractional type, while a generalized notion of nonlocal perimeter defined using a positive, compactly supported radial kernel was introduced in [42]. Nonlocal perimeters have been object of many studies in recent years. For example they S.D. has been partially supported by the Academy of Finland (grant 288501 "Geometry of subRiemannian groups" and grant 322898 "Sub-Riemannian geometry via metric-geometry and Lie-group theory") and by the European Research Council (ERC Starting Grant 713998 GeoMeG "Geometry of metric groups"). D.P. is member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM) and has been partially supported by the PRIN 2015 MIUR project 2015233N54. A.P. is member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM). Keywords and phrases: Carnot groups, calibrations, nonlocal perimeters, -convergence, sets of finite perimeter, rectifiability 1 Dipartimento di Matematica e Fisica, Universita del Salento, Via Per Arnesano, 73100 Lecce, Italy. 2 Department of Mathematics and Statistics, P.O. Box 35 (MaD), 40014, University of Jyvaskyla, Finland. 3 Dipartimento di Matematica e Fisica, Universita del Salento, and INFN, Sezione di Lecce, Via Per Arnesano, 73100 Lecce, Italy. 4 Dipartimento di Matematica, Universita di Trento, Via Sommarive, 14, 38123 Povo, TN, Italy. ** Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 A. CARBOTTI ET AL. are related to nonlocal (not necessarily fractional) minimal surfaces, [12, 42, 43], fractal sets, [37, 54, 55], phase transition [49], and many other problems. We refer the interested reader to [18, 51] for further applications and for a comparison with the standard perimeter. Nonlocal perimeter can also be characterized in terms of the Gagliardo­Slobodeckij seminorm in the frame- work of fractional Sobolev spaces, see [17], or in terms of Dirichlet energy associated with an extension problem for the fractional Laplacian, see [10]. The limiting behavior of fractional -perimeters as 1- and 0+ turns out to be very interesting. Davila showed in [14] that for a bounded Borel set E of finite perimeter the following equality holds: lim 1- (1 - )L(Ec , E ) = cP(E; ). (1.2) In particular, when = Rn , one has lim 1- (1 - )P(E; Rn ) = cP(E) (1.3) where P(E) denotes the classical perimeter of E in Rn and c is a positive constant depending only on n. In the subsequent paper [19] the authors studied the behavior of P(E; ) as 0+ . Finally, in [2], the limiting behavior of P(E; ) is studied in the -convergence sense, see also [47] for further extensions. Carnot groups are connected and simply connected Lie groups whose Lie algebra g is stratified, i.e., there are linear subspaces g1, . . . , gs of g such that g = g1 · · · gs, [g1, gi] = gi+1, gs 6= {0}, [gs, g1] = {0} (1.4) where [g1, gi] denotes the subspace of g generated by the commutators [X, Y ] with X g1 and Y gi. In the last few years Carnot groups have been largely studied in several respects, such as Differential Geometry [11], subelliptic differential equations [6, 26, 27, 48], Complex Analysis [50] and Neuroimaging [13]. Many key results of Geometric Measure Theory in the context of metric measure spaces are based on the notion of function of bounded variation and, in particular, on sets of finite perimeter. The local theory of perimeters in Carnot groups has then attracted a lot of interest and it is natural to address the attention to their nonlocal counterpart. In the present paper we study nonlocal perimeters coming from a positive symmetric kernel K : G R satisfying Z G min{1, d(x, 0)}K(x) dx < +, where d is the Carnot­Caratheodory distance on G, see Definition 2.2. More precisely, given two measurable and disjoint sets E and F in G, we consider the interaction functional LK(E, F) := Z E Z F K(y-1 x) dx dy and we define the nonlocal K-perimeter of a measurable set E inside an open set as in (1.1), namely PK(E; ) := LK(Ec , E ) + LK(Ec , E c ) + LK(E , Ec c ). We refer to [25] and [31] for a general overview. In the first part of the paper we provide sufficient conditions that have to be satisfied by every local minimizer of the nonlocal K-perimeter. Given a measurable set E0 and an open set in G, by a local minimizer for PK in with outer datum E0 we mean a measurable set E G such that E \ = E0 \ and such that PK(E; ) PK(F; ), for every measurable F G with F \ = E0 \ . LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 3 Our first main result, see Theorem 3.7, reads as follows. Theorem 1.1. Let E0 G be a measurable set and let G be an open set such that PK(E0; ) < +. Let E G be a measurable set with E \ = E0 \ and assume E admits a calibration (see Def. 3.5 below). Then E is a local minimizer for PK in with outer datum E0. Theorem 1.1 actually holds in a slightly more general form. Indeed, it can be proved even for a natural extension of the nonlocal K-perimeter to all measurable functions (see (3.4)). Both the proof of this theorem and the definition of calibration are inspired by the ones given in [46]. We also notice that, using the generalized coarea formula (3.14), for any local minimizer provided by Theorem 1.1, among all the minimizers, one can always find the characteristic function of a set. As a consequence of Theorem 1.1 we prove that a suitably defined halfspace H is the unique local minimizer of PK in the unit ball B(0, 1) with outer datum H \ B(0, 1). In [8] it is proved that, in the Euclidean setting, every measurable set E that is foliated by sub- and super- solutions adapted to (see Def. 3.13) admits a calibration and, if some natural geometric assumption hold, the minimizer is also unique. Our Theorem 3.17 goes exactly in this direction and follows closely [8, Thm. 2.4]. Setting K := -Q K 1/, the second part of the paper investigates the asymptotic behavior of the rescaled functionals 1 P := 1 PK as 0 in the -convergence sense. Berendsen and Pagliari showed in [5] that, in the Euclidean case, such -limit exists in L1 loc and equals the Euclidean perimeter, up to a multiplicative constant. We also mention that in [2] the authors proved that, in the Euclidean setting, the functional (1 - )P, -converges in L1 loc to the standard perimeter P, up to a multiplicative dimensional constant. For an introduction to -convergence we refer the reader to the monographs [7, 16], see also [39, 40] where some classical results in -convergence have been extended to the case of functionals depending on vector fields. The main result of the second part of the paper reads as follows (see Sect. 2 for all the missing definitions). Theorem 1.2. Let G be a Carnot group satisfying property R, let be open and bounded and assume K : G [0, +) is symmetric and radially decreasing (i.e., K(x) = e K(r), where r = kxk and e K is decreasing) and such that inf r>1 rQ+1 e K(r) > 0. Then, there exists a positive density : g1 (0, +) such that, for every family (E) of measurable sets converging in L1 () to E , one has Z (E) dPG(E; ·) lim inf 0 1 P(E; ). (1.5) Here PG(E; ·) denotes the perimeter measure of E in G, E denotes its horizontal normal (see Defs. 2.3 and 2.5) and Q is the homogeneous dimension of G. Some comments are in order. The proof of Theorem 1.2 (see the proof of Theorem 4.9) follows the ideas of [5, Sect. 3.3], where the authors prove the -convergence of the rescaled functionals to the perimeter in the Euclidean setting. Theorem 1.2 gives us an estimate on the -liminf of the functional 1 P in terms of a density , which is explicitly computed and does not depend on the points in the boundary of E, but only on the horizontal directions of its normal. For the proof of this theorem, it is essential to apply a compactness argument to families of sets with uniformly bounded K-perimeters. The compactness criterion is given in Theorem 4.4 and we believe it has its own independent interest. We also notice that, in the assumptions of Theorem 1.2, one has to restrict both the class of Carnot groups and the class of kernels. The fact that K is required to be radial and with some specific rate at infinity allows us to say that is indeed a strictly positive density (see Prop. 4.6), while the assumption on the group G to satisfy property R allows us to consider blow-ups of sets of finite perimeter. A Carnot group G satisfies property R if every set of finite perimeter in G has rectifiable 4 A. CARBOTTI ET AL. reduced boundary, i.e. it can be covered, up to a set of measure zero, by a countable union of intrinsically C1 hypersurfaces, see Definitions 2.7, 2.8 and 4.1. As an immediate consequence (see Rem. 4.1), the validity of property R ensures that at PG(E)-almost every point of p in G, the family 1/r(p-1 E) converges in L1 loc, up to subsequences, to a vertical halfspace with normal E(p). As we have already pointed out, the problem of understanding what is the regularity of the (reduced) boundary of a set of finite perimeter in the context of Carnot groups has only received partial solutions, so far. Whenever property R is not assumed, only partial results about blow-up of sets of finite perimeter are available in the literature. It is proved in [30] that, for any set E G with locally finite perimeter and for PG(E)-almost every p G, the family 1/r(p-1 E) converges in L1 loc(G) to a set of constant horizontal normal F, namely a set for which there exists g1 such that F 0 and XF = 0 for every X g1 with X, (1.6) in the sense of distributions. If in addition G has step 2, or it is of type ?, then it is proved respectively in [30] and [41] that, up to a left translation, every set of constant horizontal normal is really a vertical halfspace. On the other hand, still in [30, Ex. 3.2], it is proved that, for general Carnot groups, condition (1.6) does not characterize vertical halfspaces. The classification of sets with constant horizontal normal is a challenging problem and, as far as we know, the most general result in this direction is [4, Thm. 1.2]. We mention that in the recent paper [20] the authors show that the reduced boundary of any set of locally finite perimeter in any Carnot group has a so-called cone property that in the case of filiform groups implies rectifiability in the intrinsic Lipschitz sense. Finally a natural question one might ask is whether the -liminf estimate given by Theorem 1.2 can be complemented by a -limsup estimate. The proofs of the -limsup inequality in [5] and in [2] rely heavily upon the convergence result by Davila [14], whose extension to Carnot groups is, as far as we know, still an open problem, see [38] for some results in this directions and [33] for a Bourgain­Brezis­Mironescu­Davila result in step 2 Carnot groups. 2. Preliminaries 2.1. Carnot groups A connected and simply connected Lie group (G, ·) is said to be a Carnot group of step s if its Lie algebra g admits a step s stratification according to (1.4). For a general introduction to Carnot groups from the point of view of the present paper and for further examples, we refer, e.g., to [6, 26, 35, 50]. We write 0 for the neutral element of the group, and xy := x · y, for any x, y G. We fix a scalar product h·, ·i on g1 and denote by | · | its induced norm. We recall that a curve : [a, b] G is absolutely continuous if it is absolutely continuous as a curve into Rn via composition with local charts. Definition 2.1. An absolutely continuous curve : [a, b] G is said to be horizontal if 0 (t) g1, for almost every t [a, b]. The length of such a curve is given by LG() = Z b a |0 (t)|dt. Chow's Theorem ([6], Thm. 19.1.3) asserts that any two points in a Carnot group can be connected by a horizontal curve. Hence, the following definition is well-posed. Definition 2.2. For every x, y G, their Carnot­Caratheodory (CC) distance is defined by d(x, y) = inf {LG(): is a horizontal curve joining x and y} . LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 5 We also use the notation kxk = d(x, 0) for x G. We denote by B(x, r) = {y G : ky-1 xk < r} the open ball centered at x G with radius r > 0. It is well-known (see e.g. [44]) that the Hausdorff dimension of the metric space (G, d) is the so-called homogeneous dimension Q of G, which is given by Q := s X i=1 i dim(gi). We denote by H Q the Hausdorff measure of dimension Q associated with the metric d. The measure H Q is a Haar measure on G (see [6], Prop. 1.3.21) and we write Z f(x) dx := Z f(x) dH Q (x), for every measurable set and every measurable function f : R. We recall here the notion of exponential map. Let X g and let : [0, ) G be the unique global solution of the Cauchy problem ( 0 (t) = X((t)) (0) = 0. The exponential map exp: g G X 7 exp(X) := (1) is a diffeomorphism between the Lie algebra g and the Lie Group G, and we use the notation log: G g to denote its inverse. For any > 0, we denote by : g g the unique linear map such that X = i X, X gi. The maps : g g are Lie algebra automorphisms, i.e., ([X, Y ]) = [ X, Y ] for all X, Y g. For every > 0, the map naturally induces an automorphism on the group : G G by the identity (x) = (exp log)(x). It is easy to verify that both the families ( )>0 and ()>0 are a one-parameter group of automorphisms (of Lie algebra and of groups, respectively), i.e., = and = for all , > 0. The maps , are both called dilation of factor . Denoting by x : G G the (left) translation by the element x G defined as xz := x · z = xz, we remark that the CC distance is homogeneous with respect to dilations and left invariant. More precisely, for every > 0 and for every x, y, z G one has d(x, y) = d(x, y), d(xy, xz) = d(y, z). This immediately implies that x(B(y, r)) = B(xy, r) and B(y, r) = B(y, r). 6 A. CARBOTTI ET AL. 2.2. Perimeter and rectifiability We introduce the notions of perimeter, reduced boundary and rectifiability. Definition 2.3. Let be an open set in G and let f L1 loc(). We say that f has locally bounded variation in (f BVG,loc()), if, for every Y g1 and every open set A b , there exists a Radon measure Y f on such that Z A fY dµ = - Z A d(Y f), for every C1 c (A). We say that f L1 () has bounded variation in (f BVG()) if f has locally bounded variation in and, for every basis (X1, . . . , Xm) of g1, the total variation |DXf|() of the measure DXf := (X1f, . . . , Xmf) is finite. If E is a measurable set in , we say that E has locally finite perimeter in if E BVG,loc() and we say that E has finite perimeter in if E BVG,loc() and |DXE|() < +. In such a case, the measure |DXE| is called perimeter of E and it is denoted by PG(E; ·). We also use the notation PG(E; G) =: PG(E). The following Proposition is proved in ([28], Thm. 2.2.2) and ([32], Thm. 1.14). Proposition 2.4. Let G be an open set and let u BVG(). Then, there exists a sequence (uk) in C () such that · uk u in L1 (); · |DXuk|() |DXu|(). Definition 2.5. Let E G be a set with locally finite perimeter. We define the reduced boundary FE of E to be the set of points p G such that PG(E; B(p, r)) > 0 for all r > 0 and there exists lim r0 DXE(B(p, r)) PG(E; B(p, r)) = lim r0 DXE(B(p, r)) |DXE|(B(p, r)) =: E(p) Rm , with |E(p)| = 1. Definition 2.6. Let G be an open set in a Carnot group G. We say that a function f : R is of class C1 G if f is continuous and, for any basis X = (X1, . . . , Xm) of g1, the limit, Xif(x) := lim t0 f(x exp(tXi)) - f(x) t , exists and defines a continuous function for every i = 1, . . . , m and any x . According to this definition we also denote by Xf : Rm the vector valued function defined by Xf := (X1f, . . . , Xmf). Definition 2.7. A set G is said to be a hypersurface of class C1 G if, for every p there exists a neighborhood U of p, and a function f : U R of class C1 G such that U = {q U : f(q) = 0}, and infU |Xf| > 0, for any basis X = (X1, . . . , Xm) of g1. LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 7 Definition 2.8. Let E G be a measurable set. We say that E is C1 G-rectifiable (or simply rectifiable), if there exists a family {j : j N} of C1 G-hypersurfaces such that H Q-1 E \ [ jN j = 0, where Q is the homogeneous dimension of G and H Q-1 denotes the (Q - 1)-dimensional Hausdorff measure defined through the Carnot­Caratheodory distance. Definition 2.9. For any g1 \ {0}, we define the vertical halfspace with normal by setting H := {x G: h1 log x, i 0}, where 1 : g g1 is the horizontal projection on the Lie algebra. Notice that if x G is such that h1 log x, i > 0, then x-1 Hc . We conclude this section with the following Definition 2.10. Let 1 p and let G be an open set. We set W1,p G () := {f Lp (): Xjf Lp (), j = 1, . . . , m}. Definition 2.11. The convolution of two functions in f, g: G R is defined by (f g)(x) := Z G f(xy-1 )g(y) dy = Z G g(y-1 x)f(y) dy, for every couple of functions for which the above integrals make sense. Remark 2.12. From this definition we see that if L is any left invariant differential operator in G, then L(f g) = f Lg provided the integrals converge. Moreover, if G is not abelian, we cannot write in general f Lg = Lf g. 3. Local minimizers and calibrations Throughout this section, G denotes a Carnot group and we denote by kxk := d(0, x), where d is the CC distance introduced in Definition 2.2. We however notice that the results we obtain still hold when d(0, x) is replaced by any other homogeneous norm on G. We also fix a kernel K : G R with the following property: K 0 in G, (3.1) K(-1 ) = K() for any G, (3.2) Z G min{1, kxk}K(x) dx < +. (3.3) Define also for every measurable function u: G [0, +] and every measurable set G the functional JK(u; ) := 1 2 Z Z K(y-1 x)|u(y) - u(x)| dy dx + Z Z c K(y-1 x)|u(y) - u(x)| dy dx =: 1 2 J1 K(u; ) + J2 K(u; ). (3.4) 8 A. CARBOTTI ET AL. We also denote by Ji (E; ) := Ji (E; ) for i = 1, 2. Moreover, for every measurable and disjoint sets A, B G, we define the interaction between A and B driven by the kernel K as LK(A, B) := Z B Z A K(y-1 x) dy dx. (3.5) We set PK(E; ) := JK(E; ) =: J(E; ). Therefore, PK(E; ) = LK(Ec , E ) + LK(Ec , E c ) + LK(E , Ec c ); in particular, we have that LK(Ec , E ) = 1 2 J1 K(E; ), and LK(Ec , E c ) + LK(E , Ec c ) = J2 K(E; ). We can think of J1 K(E; ) as the local part of PK(E; ), in the sense that if F is a measurable set such that H Q ((E4F) ) = 0, then J1 K(F; ) = J1 K(E; ). It is worth noticing that for = G we get PK(E; G) = LK(E, Ec ). Remark 3.1. For every measurable set E G we notice that PK(E; ) can also be written as PK(E; ) = 1 2 Z (G×G)\(c×c) |E(y) - E(x)|K(y-1 x) dx dy. (3.6) Indeed we can write Z (G×G)\(c×c) |E(y) - E(x)|K(y-1 x) dx dy = Z (G×G)\(c×c) |E(y) - E(x)|2 K(y-1 x) dx dy = Z (G×G)\(c×c) (E(y) - E(y)E(x))K(y-1 x) dx dy + Z (G×G)\(c×c) (E(x) - E(y)E(x))K(y-1 x) dx dy = 2 Z (G×G)\(c×c) E(x)Ec (y)K(y-1 x) dx dy = 2LK(Ec , E ) + 2LK(Ec , E c ) + 2LK(E ; Ec c ) = 2PK(E; ). When G is the Euclidean space Rn , a typical example of radial kernel satisfying (3.1), (3.2) and (3.3) is given by K(x) = |x|-n- with (0, 1). We refer e.g. to [51] and references therein for an overview of the classical fractional perimeter's theory. LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 9 On the other hand, if G is a general Carnot group with homogeneous dimension Q, and k · k is a homogeneous norm on G, then, for every (0, 1), the kernel K : G R defined by K() := kk-Q- , satisfies conditions (3.1), (3.2) and (3.3). A homogeneous norm that has been considered in the literature is the one associated with the sub-Riemannian heat operator, see e.g. to [24­26] for some motivations. We here briefly describe its definition. Define the map e R : G [0, +) by letting e R(x) := - 2(-/2) Z + 0 t- 2 -1 h(t, x) dt. Here h: [0, +) × G R is the fundamental solution of the sub-Riemannian heat operator H := t + L, where L := m X i=1 X2 i denotes the sub-Laplacian associated with a basis (X1, . . . , Xm) of the horizontal layer g1. In this case one has e R(x-1 ) = e R(x) and e R(x) = --Q e R(x) for any x G and any 0, and the quantity kxk := e R(x) - 1 +Q , defines a homogeneous symmetric norm on G. In particular, the kernel K() := 1 kkQ+ satisfies conditions (3.1), (3.2), (3.3) and (4.8), and hence all the results obtained in this paper apply to the special case K = K. We next state and prove some facts that will be useful throughout the paper. Proposition 3.2. Let G be an open set and let u BVG(). Let p , r > 0 such that B(p, 2r) and let g B(0, r). Then Z B(p,r) |u(x · g) - u(x)| dx d(0, g)|DXu|(). In particular, if = G and u BVG(G), one has Z G |u(x · g) - u(x)| dx d(0, g)|DXu|(G), (3.7) for every g G. Proof. Fix a basis (X1, . . . , Xm) of g1. By Proposition 2.4 we can assume without loss of generality that u C (). Let g B(0, r) with g 6= 0 (if g = 0 the thesis is trivial) and let := d(0, g) > 0. Take a geodesic 10 A. CARBOTTI ET AL. : [0, ] B(0, r) satisfying (0) = 0, () = g and (t) = m X i=1 hi(t)Xi((t)) for a. e. t [0, ], where (h1, . . . , hm) L ([0, ]; Rm ) with k(h1, . . . , hm)k 1. Notice that, for every x G, the curve x : [0, ] B(x, r) defined by x(t) = x · (t) is a geodesic joining x and x · g, and kxk = k(h1, . . . , hm)k. Therefore, for any x B(p, r), one has |u(x · g) - u(x)| = Z 0 d dt u(x(t)) dt Z 0 |Xu(x(t))| dt. Integrating both sides on B(p, r) we get Z B(p,r) |u(x · g) - u(x)| dx Z B(p,r) Z 0 |Xu (x · (t)) | dt dx, and exchanging the order of integration we conclude that Z B(p,r) |u(x · g) - u(x)| dx Z 0 Z B(p,r) |Xu(x · (t))| dx dt, where we notice that the curve depends on g. Since (t) B(0, r) for all t [0, ] and since x B(p, r), then x · (t) B(0, 2r) for all t [0, ]. Indeed, by the triangular inequality one has d(x · (t), p) d(x · (t), x) + d(x, p) = d((t), 0) + d(x, p) r + r = 2r. Thus, we finally get Z B(p,r) |u(x · g) - u(x)| dx d(0, g) Z B(p,2r) |Xu(x)| dx d(0, g)|DXu|(). Corollary 3.3. Let u L1 (G). Then lim q0 kqu - ukL1(G) = 0. Proof. If u C c (G) the conclusion follows using (3.7). Let (uh) be a sequence in C c (G) with uh u in L1 (G) and let > 0. Fix h be big enough so that ku - uhkL1(G) 4 . Then kqu - ukL1(G) kqu - quhkL1(G) + kquh - uhkL1(G) + kuh - ukL1(G) = 2ku - uhkL1(G) + kquh - uhkL1(G) 2 + kquh - uhkL1(G) and the conclusion follows taking d(0, q) small enough to have kquh - uhkL1(G) 2 . We now give a sufficient condition on E and in order to have PK(E; ) < +. The proof is inspired by the one present in [46]. LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 11 Proposition 3.4. Let E, F G be two measurable sets with H Q (E F) = 0. Then one has LK(E, F) V (E, F) Z G min{1, d(0, )}K() d, where V (E, F) := min max PG(E) 2 , H Q (E) , max PG(F) 2 , H Q (F) . Proof. Without loss of generality we can assume V (E, F) = max PG(E) 2 , H Q (E) < +. Up to modifying E on a set of measure zero we can also assume that F Ec . Therefore we have LK(E, F) LK(E, Ec ) = 1 2 Z G Z G K(-1 )|E() - E()| d d = 1 2 Z G Z G K()|E() - E()| d d = 1 2 Z B(0,1) K() Z G |E() - E()|d d + 1 2 Z G\B(0,1) K() Z G |E() - E()|d d. (3.8) Since E has finite perimeter in G and finite volume, then E BV (G) and, by Proposition 3.2, for every G we can write Z G |E() - E()|d d(0, )PG(E). On the other hand, since H Q (E) < +, we can also write Z G |E() - E()|d 2H Q (E). Using this two facts in the last part of (3.8) gives us LK(E, F) PG(E) 2 Z B(0,1) d(0, )K() d + H Q (E) Z G\B(0,1) K() d, and therefore LK(E, F) max PG(E) 2 , H Q (E) Z G min{1, d(0, )}K() d. Now, we adapt the notion of nonlocal calibration given in [46] in the Euclidean setting. We refer to [8] to point out the link between such a notion and the notion of (local) calibration of a set. Definition 3.5. Let u: G [0, 1] and : G×G [-1, 1] be measurable functions. We say that is a calibration for u if the following two facts hold. 12 A. CARBOTTI ET AL. (i) The map F(p) = R G\B(p,) K(y-1 p)((y, p) - (p, y)) dy is such that lim 0 kFkL1(G) = 0. (3.9) (ii) for almost every (p, q) G × G such that u(p) 6= u(q) one has (p, q)(u(q) - u(p)) = |u(q) - u(p)|. (3.10) Remark 3.6. If : G × G [-1, 1] is a calibration for u: G [0, 1], then also the antisymmetric function b (p, q) := 1 2 ((p, q) - (q, p)) is a calibration for u. The proof of the following theorem follows closely the one given in ([46], Thm. 2.3). Theorem 3.7. Let G be an open set and let E0 G be a measurable set such that PK(E0; ) < + and define F := {v: G [0, 1] measurable | v = E0 on c }. (3.11) Let u F and let : G × G [-1, 1] be a calibration for u. Then JK(u; ) JK(v; ), for every v F. Moreover, if e u F is such that JK(e u; ) JK(u; ), then is a calibration for e u. Proof. We can assume without loss of generality that JK(v; ) < + for every v F. Since |v(y) - v(x)| (x, y)(v(y) - v(x)) we can write for any v F JK(v; ) a(v) - b1(v) + b0, where a, b1 and b0 are respectively defined by a(v) := 1 2 Z Z K(y-1 x)(x, y)(v(y) - v(x)) dy dx, b1(v) := Z Z c K(y-1 x)(x, y)v(x) dy dx, b0 := Z Z c K(y-1 x)(x, y)E0 (y) dy dx. By (3.10), we notice that JK(u; ) = a(u) - b1(u) + b0. It is then enough to prove that, for every v F, one has a(v) = b1(v). By Remark 3.6, we can assume that is antisymmetric. Combining this with the fact that K(-1 ) = K(), we easily get a(v) = - Z Z K(y-1 x)(x, y)v(x) dy dx. (3.12) By (3.9), for almost every x , we have lim r0 Z B(x,r)c K(y-1 x)(x, y) dy = lim r0 Z B(x,r)c K(y-1 x)(x, y) dy + Z c K(y-1 x)(x, y) dy dx = 0. LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 13 Implementing this identity in (3.12) and using the dominated convergence theorem, we get a(v) = - Z Z K(y-1 x)(x, y)v(x) dy dx = - lim r0 Z Z B(x,r)c K(y-1 x)(x, y)v(x) dy dx = Z Z c K(y-1 x)(x, y)v(x) dy dx = b1(v). We are left to prove that, if e u F is such that JK(e u; ) JK(u; ), then is a calibration for e u. Since u = e u on c we get (x, y)(e u(y) - e u(x)) = |e u(y) - e u(x)|, (3.13) for almost every (x, y) c × c satisfying u(x) 6= u(y). Since JK(e u; ) = b0, we also have that JK(e u; ) = a(e u) - b1(e u) + b0. This implies that 1 2 Z Z K(y-1 x) (|e u(y) - e u(x)| - (x, y)(e u(y) - e u(x))) dy dx + Z Z c K(y-1 x) (|e u(y) - e u(x)| - (x, y)(e u(y) - e u(x))) dy dx = 0. Since both integrands are positive, we get that (3.13) holds true for almost every (x, y) ×G with e u(x) 6= e u(y). To get (3.13) for almost every (x, y) c × it is enough to use the antisymmetry of . We now notice that the functional JK(·; ) enjoys a coarea formula. Concerning the Euclidean case, we refer the reader to ([3], Thm. 2.93) for the classical formula relating total variation and Euclidean perimeter, and to [55], where the author finds a class of functionals defined on L1 () for which a generalized coarea formula holds. Proposition 3.8. Let G be an open set and let u: [0, 1] be a measurable function. Setting Et := {g G : u(g) > t} for any t [0, 1], it holds that JK(u; ) = Z 1 0 PK(Et; ) dt. (3.14) Proof. The proof is analogous to the Euclidean case, see ([12], Lem. 6.2.). Fix x, y with x 6= y and assume without loss of generality that u(x) > u(y). Then |Et (x) - Et (y)| = 1 for any t [u(y), u(x)] and |Et (x) - Et (y)| = 0 for every t [0, 1] \ [u(y), u(x)]. Therefore, for any x, y , it holds that |u(x) - u(y)| = Z u(x) u(y) |Et (x) - Et (y)| dt = Z 1 0 |Et (x) - Et (y)| dt. Now, using Tonelli's theorem, we have that JK(u; ) = Z 1 0 1 2 Z Z |Et (x) - Et (y)|K(y-1 x) dx dy dt + Z 1 0 Z Z c |Et (x) - Et (y)|K(y-1 x) dx dy dt = Z 1 0 PK(Et; ) dt. 14 A. CARBOTTI ET AL. As an immediate consequence of Corollary 3.9 below we deduce that, if the infimum of for JK(·; ) with outer datum E0 is achieved, there is always a minimizer which is the characteristic function of a measurable set. Corollary 3.9. Let G be an open set and let v: G [0, 1] be a measurable function. Then, there exists a measurable set F such that JK(F; ) JK(v; ). Proof. Denote by Et := {g G : v(g) > t}. By the coarea formula (3.14), there exists t? [0, 1] such that JK(Et? ; ) JK(v; ), otherwise the equality in (3.14) would be contradicted. In particular, setting F := Et? , the proof is complete. In Proposition 3.10 below we show that halfspaces in Carnot group admit a calibration. In Theorem 3.11, we show that halfspaces are unique local minimizers for JK(·; ) when subject to their own outer datum and whenever is a ball centered at the origin. Proposition 3.10. For any g1 \ {0}, the map : G × G [0, 1] defined by (x, y) := sign h1 log(x-1 y), i , is a calibration for H . Proof. Denote for shortness H = H and = . Let us first prove property (ii) of Definition 3.5, namely that for almost every (x, y) G × G with H(x) 6= H(y) one has (x, y)(H(y) - H(x)) = |H(y) - H(x)|. It is not restrictive to assume that x H and y Hc . Then h1 log(x-1 y), i = -h1 log x, i + h1 log y, i < 0. Concerning property (i) of Definition 3.5 we observe that for every r > 0 and every x G one has Z G\B(x,r) K(y-1 x) sign(h1 log(x-1 y), i) - sign(h1 log(y-1 x), i) dy = 2 Z G\B(x,r)xH K(y-1 x) dy - 2 Z G\B(x,r)xHc K(y-1 x) dy = 2 Z G\B(0,r)H K(z) dz - 2 Z G\B(0,r)Hc K(z) dz = 0. The last identity comes from the fact that H Q ({x G : h1 log x, i = 0}) = 0, K(x-1 ) = K(x) and the inversion 7 -1 preserves the volume and maps H onto Hc (up to sets of measure zero). Theorem 3.11. Let H be a vertical halfspace and denote by B := B(0, 1). Then PK(H; B) < + and PK(H; B) JK(v; B), for every measurable v: G [0, 1] such that v = H almost everywhere on Bc . Moreover, if u: G [0, 1] is such that u = H almost everywhere on Bc and JK(u; B) JK(H; B), then u = H almost everywhere on G. Proof. By definition of PK we can write PK(H; B) = LK(H B, Hc Bc ) + LK(Hc B, H Bc ) + LK(Hc B, H B) < +, LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 15 since each term on the right-hand side is finite because of Proposition 3.4. By Proposition 3.10 and Theorem 3.7 we only have to show that minimizers are unique (up to sets of measure zero). Let g1 \ {0} be such that H = H and let u: G [0, 1] be such that u = H almost everywhere on Bc and JK(u; B) JK(H; B). Consider the map (x, y) = sign(h1 log(x-1 y), i) which is a calibration of H. By Theorem 3.7, is also a calibration for u and therefore sign(h1 log(x-1 y), i)(u(y) - u(x)) = |u(y) - u(x)|, for a.e. (x, y) G × G. As a consequence, the implication h1 log(x-1 y), i > 0 u(y) u(x) holds for almost every (x, y) G × G. For every t (0, 1), define the set Et := { G : u() > t}. For almost every (x, y) Et × Ec t one has u(x) > u(y) and therefore h1 log x, i h1 log y, i. By Dedekind's theorem, and up to sets of measure zero, for every t (0, 1), there exists t R such that Et { G : h1 log , i t} and Ec t { G : h1 log , i t}. This implies that for all t (0, 1) one has H Q (Et4{ G : h1 log , i t}) = 0. Combining this with the fact that u = H almost everywhere on Bc , we get that t = 0 for every t (0, 1), and therefore H Q (Et4H) = 0, t (0, 1). (3.15) Consider now a sequence (tj) in (0, 1) that converges to 0 as j +. Since u has values in [0, 1], we get { G : u() 0} = { G : u() = 0} = \ jN Ec tj , and similarly { G : u() = 1} = \ jN E1-tj . Combining this fact with (3.15), we complete the proof by observing that the identities H Q ({ G : u() = 0}4Hc ) = 0 and H Q ({ G : u() = 1}4H) = 0 hold. Now, we show another useful approach for the analysis of the minimizers of the functional in (3.4). To do this, following [8] we introduce the notion of nonlocal mean curvature and of calibrating functional; these tools allow us to prove Theorem 3.17. We more precisely clarify the relation between Theorem 3.7 and Theorem 3.17 in Remark 3.18. Definition 3.12. Let E be a set of finite perimeter in G. The nonlocal mean curvature is defined as HK(E)(x) := lim 0 Z G\B(x) (Ec (y) - E(y))K(y-1 x) dy. (3.16) More generally, for every measurable map : G R we set HK[](x) := lim 0 Z G\B(x) sign((x) - (y))K(y-1 x) dy. Notice that HK[](x) = HK({ > (x)})(x). 16 A. CARBOTTI ET AL. Definition 3.13. Let G be a bounded open set and let E be a measurable set. We say that is foliated by sub- and super- solutions adapted to E whenever there exists a measurable function E : G R such that (i) E = {E(x) > 0} up to H Q -negligible sets; (ii) The sequence of functions Fh(x) := R G\B(x,1/h) sign(E(x) - E(y))K(y-1 x) dy converges in L1 () to HK(E) as h . (iii) HK[E](x) 0 for a.e. x E and HK[E](x) 0 for a.e. x \ E. Notice that the integrals defined in (ii) are finite thanks to the assumptions on the kernel. Definition 3.14. Let G be a bounded open set and let E G be measurable. Assume that is foliated by sub- and super- solutions adapted to E and let E be the measurable function provided by Definition 3.13. Then, for every measurable set F such that F \ = E \ , we define the calibrating functional as C(F) := Z (G×G)\(c×c) sign(E(x) - E(y))(F (x) - F (y))K(y-1 x) dx dy. (3.17) Remark 3.15. Let G is a bounded open set and let E G be measurable. Assume that is foliated by sub- and super- solutions adapted to E. Then, as a consequence of (i) of Definition 3.13, it immediately follows that PK(E; ) = C(E) and PK(F; ) C(F) (3.18) for every measurable set F G with F \ = E \ . Proposition 3.16. Let G be a bounded open set with PK(; G) < and let E G be a measurable set. Moreover, assume there exists a measurable function E : G R satisfying (i) and (ii) of Definition 3.13. Then, for every measurable set F such that F \ = E \ and PK(F; G) < , we have C(F) = 2 Z F HK[E](x) dx + 2 Z E\ Z sign(E(x) - E(y))K(y-1 x) dx dy. (3.19) Proof. We introduce the auxiliary kernel e K : G [0, +) by setting e K(p) := G\B(0,)(p)K(p), p G. Recalling (3.17), we have that C(F) = lim 0 Z (G×G)\(c×c) sign(E(x) - E(y))(F (x) - F (y)) e K(y-1 x) dx dy. (3.20) Since e K is symmetric we can write Z (G×G)\(c×c) sign(E(x) - E(y))(F (x) - F (y)) e K(y-1 x) dx dy = 2 Z (G×G)\(c×c) sign(E(x) - E(y))F (x) e K(y-1 x) dx dy. (3.21) LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 17 Now, we split the second integral in (3.21) in two parts: when y F (which implies that x runs through all of G), and when y F \ = E \ (which implies that x runs through ), and this gives us 2 Z (G×G)\(c×c) sign(E(x) - E(y))F (x) e K(y-1 x) dx dy = 2 Z F Z G sign(E(x) - E(y)) e K(y-1 x) dx dy + 2 Z E\ Z sign(E(x) - E(y)) e K(y-1 x) dx dy = 2 Z F H e K [E](x) dx + 2 Z E\ Z sign(E(x) - E(y)) e K(y-1 x) dx dy. We notice that, using the notation of (3.5), one has Z E\ Z K(y-1 x) dx dy = LK(E \ , ) LK(c , ) = PK(; G) < +, and, moreover, |sign(E(x) - E(y))K(y-1 x)| K(y-1 x) for any couple (x, y) E \ × . On the other hand, we know that H e K [E] converges in L1 () to HK[E] as 0. Therefore, letting 0 and recalling (3.20), we conclude the proof. Theorem 3.17. Let G be an open set satisfying PK(; G) < + and consider a measurable set E G. Assume that is foliated by super- and sub- solutions adapted to E and let E : G R be a measurable function satisfying the assumptions of Definition 3.13. Then the following facts hold. (a) For every measurable set F G with F \ = E \ one has PK(E; ) PK(F; ). (b) If K > 0 and E is continuous and such that H Q ({E = 0} ) = 0 and if there exists R > 0 such that B(0, R), E \ B(0, R) 6= and (E)c \ B(0, R) 6= , then E is the unique measurable set satisfying (a) (up to sets of measure zero). (c) If HK[E](x) = 0 for almost every x , then C(F) = C(E) for every measurable set F G with F \ = E \ . Proof. (a) By Remark 3.15, it suffices to show that, for every measurable set F G such that F \ = E \ , one has C(F) C(E). We can also assume without loss of generality that PK(F; ) < +. Now, using the fact that (F )(E \F) = (E ) (F \ E) and that both unions are disjoint, we can express the first integral in (3.19) as Z F HK[E](x) dx = Z E HK[E](x) dx + Z F \E HK[E](x) dx - Z E\F HK[E](x) dx. (3.22) Since both F \ E and E \ F are contained in , using (iii) of Definition 3.13 we get Z F Hk[E](x) dx Z E HK[E](x) dx. 18 A. CARBOTTI ET AL. Adding to both sides Z E\ Z sign(E(x) - E(y))(F (x) - F (y)) e K(y-1 x) dx dy, and recalling that E \ = F \ , we conclude the proof of (a). (b) Assume e E G is a measurable set such that PK( e E; ) PK(F; ), for every measurable F G with F \ = E \ . Then, we have PK(E; ) = PK( e E; ). By (3.6), (3.20) and Remark 3.15 we have 1 2 Z (G×G)\(c×c) |E(y) - E(x)|K(y-1 x) dy dx = Z (G×G)\(c×c) sign(E(y) - E(x))( e E(y) - e E(x))K(y-1 x) dy dx. Since K > 0, we get E(x) > E(y) for a.e. (x, y) (( e E ) × e Ec ) ( e E × ( e Ec )). (3.23) By hypothesis, the function E takes both a positive and a negative value in B(0, R) c . Since E is continuous, for every > 0 small enough, both {- < E < 0} \ B(0, R) and {0 < E < } \ B(0, R) are nonempty open sets. Hence, since e Ec \ B(0, R) = {E 0} \ B(0, R) and e E \ B(0, R) = {E > 0} \ B(0, R), from (3.23), by letting 0, we deduce E(x) 0 for a.e. x e E and E(y) 0 for a.e. y e Ec . Since the set {E = 0} has zero measure by assumption, we deduce that e E = {E > 0} = E up to a measure zero set. (c) If HK[E](x) = 0 for almost every x , the last two integrals in (3.22) vanish, and thus (3.19) leads to C(F) = C(E), for every measurable set F G with F \ = E \ . Remark 3.18. Notice that Theorem 3.7 and Theorem 3.17 present many similarities. On the one hand, Theo- rem 3.7 holds for general measurable functions and, moreover, if a set admits sub- and super-solutions adapted to E, then (x, y) = sign(E(y) - E(x)) is a calibration for E. However, although Theorem 3.17 requires more assumptions, it gives us additional information about the local minimality of the functional C and about the uniqueness of minimizers, which was only known for the specific case of halfspaces, as shown in Theorem 3.11. 4. -convergence of the rescaled functionals In this section we analyze the -limit of the rescaled sequence 1 PK (E; ), where (E)>0 is a family of measurable sets converging in L1 () to some set E . In the study of the asymptotic behavior of the functionals, one has to deal with in the blow-up of sets of finite perimeter. In the setting of Carnot groups, one of the main and still unsolved problem concerns the regularity of the (reduced) boundary of a set of finite perimeter. The solution of this problem in the Euclidean spaces goes back to De Giorgi [15]. He proved that the reduced boundary of a set of finite perimeter in Rn is (n - 1)-rectifiable, i.e., it can be covered, up to a set of H n-1 -measure zero, by a countable family of C1 -hypersurfaces. The validity of such a result has deep consequences in the development of Geometric Measure Theory and Calculus of Variations (see e.g. the monographs [3, 22]). LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 19 The validity of a rectifiability-type theorem in the context of Carnot groups is still not yet known in full generality. However, there are complete results in all Carnot groups of step 2 (see [29, 30]) and in the so-called Carnot groups of type ?, see [41], which generalize the class of step 2 (see also [36] for a generalization of type ? condition). In these papers the authors show that the reduced boundary of a set of finite perimeter in a Carnot group of the chosen class is rectifiable with respect to the intrinsic structure of the group. Motivated by these results, we introduce the following notation, see [21] that will be used in Theorem 4.9 and Remark 4.8. Also recall Definitions 2.8 and 2.5. Definition 4.1. We say that a Carnot group G satisfies property R if every set E G of locally finite perimeter in G has rectifiable reduced boundary. As already mentioned before, property R is satisfied in Euclidean spaces, in all Carnot groups of step 2 and in the so-called Carnot groups of type ?. The first part of this section is devoted to the proof of a compactness criterion for the rescaled family 1 PK , see Theorem 4.4. The final part of this section deals with the estimate of the -liminf for the same rescaled family of functionals, in the class of Carnot groups satisfying property R. We start with the following. Proposition 4.2. Let E, F G be measurable sets. Then the following fact holds: if N G is a set with finite perimeter in G such that E N and F Nc , then lim sup 0 1 L(E, F) PG(N) 2 Z G K()d(, 0) d. Proof. Up to changing N with Nc we can assume that N BVG(G). By a change of variables and Proposition 3.2 we have 1 L(E, F) 1 Z N Z Nc 1 Q K(1/(y-1 x)) dy dx = 1 2 Z G Z G K(g)|N (xg) - N (x)| dg dx PG(N) 2 Z G K()d(, 0) d Before the proof of the compactness theorem, we remark the validity of the following fact, whose proof is an immediate calculation. We denote by JG the functional in (3.4) with kernel G and by PG the corresponding perimeter. Lemma 4.3. Let G L1 (G) be a positive function. Then, for any u L (G) it holds that Z G Z G (G G)(y)|u(xy) - u(x)| dy dx 2 kGkL1(G) JG(u; G). In particular, if we choose u = E we have Z G Z G (G G)(y)|E(xy) - E(x)| dy dx 4 kGkL1(G) PG(E). We are ready to prove the compactness result. Theorem 4.4. Let G be a bounded open set. Let (n) be an infinitesimal sequence of positive numbers and let (En) be a sequence of measurable sets in . Assume that there exists C > 0 such that 1 n Pn (En; ) C, n N. (4.1) 20 A. CARBOTTI ET AL. Then, there exist a subsequence (Enk ) of (En) and a set E of finite perimeter in such that (Enk ) converges to E in L1 (). Proof. We write E in place of En, to avoid inconvenient notation. Fix a ball B in G such that B. For any positive C c (G) \ {0} we define, for every (0, 1), the map (x) := 1 Q R G () d (1/x), and we consequently set v := E . We can therefore estimate Z G |v() - E ()| d Z G Z G (-1 )|E () - E ()| dd = Z G Z G ()|E () - E ()| dd. (4.2) Notice that, by definition of and since has compact support, the families (v) and (E ) share the same limits in L1 (G). Reasoning in a similar way on the horizontal gradient of v we get Z G |Xv()| d = Z G Z G X(-1 )E () d d Z G Z G |X(-1 )||E () - E ()| dd + Z G E () Z G X(-1 ) d d = Z G Z G |X()||E () - E ()| dd. (4.3) Notice that the identity Z G X(-1 ) d = 0, holds since C c (G) and horizontal vector fields in Carnot groups are divergence-free (see e.g. ([6], Prop. 1.3.8.)). Define now the map T(s) := ( s if |s| 1, 1 otherwise, and consider the truncated kernel G := T K. We notice that G 0 and G L (G). Moreover, by (3.3) and the fact that T(s) s for any s [0, ), we can estimate Z G |(T K)()| d Z B(0,1) d + Z G\B(0,1) K() d < , LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 21 which implies that G L1 (G). Since G L1 (G) L (G), the map G G is continuous. This is a consequence of the following estimate |(G G)(p) - (G G)(q)| Z G G()|G(p-1 ) - G(q-1 )| d kGkL(G) Z G |G(p-1 ) - G(q-1 )| d = kGkL(G)kq-1pG - GkL1(G), and Corollary 3.3. We now choose a positive C c (G) \ {0} such that G G and |X| G G. We can assume without loss of generality that v C c (B) for every (0, 1). Setting G() := -Q G(1/), and taking (4.2) and (4.3) into account we obtain Z G |v() - E ()| d Z G Z G (G G)()|E () - E ()| dd, (4.4) and Z G |Xv()| d 1 Z G Z G (G G)()|E () - E ()| dd, (4.5) where the last inequality comes from the fact that (X)() = 1 Q+1 (X)(1/), and (G G)() = 1 Q (G G)(1/). By applying Lemma 4.3 and since E for each > 0, we have Z G Z G (G G)()|E () - E ()| dd 4kGkL1(G)PG (E) 4kGkL1(G)PK (E) = 4kGkL1(G) 1 2 J1 (E; ) + J2 (E; ) = 4kGkL1(G)J(E; ). Condition (4.1) then gives M > 0 such that 1 Z G Z G (G G)()|E () - E ()| dd MkGkL1(). By the estimates (4.4) and (4.5) we get that (v) is equibounded in W1,1 G (B). Then, by the general version of Rellich­Kondrakov's Compactness theorem in metric measure spaces, (see ([34], Thm. 8.1)), up to subsequences, v converges in L1 (B) to some w. We moreover observe that (4.4) also tells us that w = e E for some e E with finite measure in B. Inequality (4.5) together with the lower semicontinuity of the total variation implies that e E has finite perimeter in B. By setting E := e E , we have that E has finite perimeter in and, by (4.2), E E in L1 (). 22 A. CARBOTTI ET AL. Remark 4.5. In case has finite perimeter and the stronger integrability condition Z G K(x)d(x, 0) dx < + (4.6) is satisfied, then Theorem 4.4 can be strengthened replacing condition (4.1) with the weaker 1 n J1 n (En, ) C, n N. Indeed, applying (i) of Proposition 4.2 with N = one gets some C2 > 0 such that 1 n J2 n (En , ) = 1 n Ln ( En , c Ec n ) 1 2 PG() Z G K(x)d(x, 0) dx C2, n N. Notice however that condition (4.6) is in contrast with (4.8) below, that will be used in Theorem 4.9. Denote for shortness B := B(0, 1). For every halfspace H G we set b(H) := inf lim inf 0 1 2 J1 (E, B(0, 1)) : E H in L1 (B(0, 1)) . (4.7) A priori, the quantity b(H) defined above might depend on the halfspace H. In the following proposition, we find sufficient conditions on the kernel in order to have a uniform positive lower bound on b. In Remark 4.7, we observe that, in free Carnot groups, the function b defined above is constant. Proposition 4.6. Assume there exists a monotone decreasing e K : [0, +) [0, +) such that K() = e K(kk) for every G and that inf r>1 e K(r)rQ+1 > 0. (4.8) Then inf{b(H) : H is a vertical halfspace} > 0. Proof. Fix a halfspace H. We first prove that b(H) > 0. By definition of b(H) and a diagonal argument, there exists a family E that converges to H in L1 (B) as 0 such that lim inf 0 1 2 J1 (E; B) = b(H). Thanks to Severini­Egorov's theorem there exists an open set A B such that H Q (B \ A) < H Q (H B) 2 (4.9) and E converges to H uniformly on A, as 0. We therefore find 0 such that sup xA |E (x) - H(x)| < 1, 0, LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 23 and hence, for every 0 we have E A = H A =: C+ . By reasoning in the same way on Ec , we may assume without loss of generality that, for every 0, we also have Ec A = Hc A =: C- . Notice that, by (4.9), we have min{H Q (C+ ), H Q (C- )} > 0. (4.10) For every 0, we have 1 2 J1 (E; B) = 1 Z E Z Ec B K(y-1 x) dy dx Q-1 Z 1/C+ Z 1/C- K(y-1 x) dy dx Q-1 e K(diam(1/C+ 1/C- ))H Q (1/C+ )H Q (1/C- ) = 1 Q+1 e K diam(C+ C- ) H Q (C+ )H Q (C- ), which, by (4.8) and (4.10), is a positive lower bound independent of . To conclude the proof of (i), it is enough to check that b is lower-semicontinuous. In fact, if this were true, by the compactness of the sphere Sm-1 , we would have that b admits a minimum, that, by the previous step would be strictly positive. Let Sm-1 such that as 0 and let H be the family of vertical halfspace associated to . Then H H in L1 (B) as 0. Fix > 0. For every > 0 we can find F converging to H in L1 (B), as 0 such that lim inf 0 1 2 J1 (F ; B) b(H) + . Considering E := F , we easily find that E H in L1 (B), as 0 and hence b(H) lim inf 0 1 2 J1 (E; B) lim inf 0 b(H) + . The thesis follows by the arbitrariness of . Remark 4.7. If G is a free Carnot group (we refer to ([53], p. 45) or ([52], p. 174) for the definition) and K is radial, then, if H1, H2 G are vertical halfspaces in G, one has b(H1) = b(H2). Indeed, let 1, 2 g1 \ {0} such that H1 = H1 and H2 = H2 . It is enough to show that b(H1) b(H2). Let E2 be a family of measurable sets in B such that E2 H2 in L1 (B) as 0. Now consider an orthogonal isomorphism T : g1 g1 such that T(2) = 1. Since G is free, the map T extends in a unique way to a Lie algebra isomorphism T : g g that induces an isometry I : G G defined by I := exp T log . We claim that I(H2) = H1. Indeed, for every G, one has h1 log , 1i =h1 log , T(2)i = hT(1 log ), 2i =h1T(log ), 2i = h1 log I(), 2i. Since K is radial and I is an isometry, it is easy to see that J1 (A; B) = J1 (I(A); I(B)). By noticing that I(B) = B and that I(E2 ) H1 in L1 (B) as 0, we have that b(H1) lim inf 0 1 2 J1 (I(E2 ); B) = lim inf 0 1 2 J1 (E2 ; B), 24 A. CARBOTTI ET AL. whence b(H1) b(H2). Remark 4.8. Let G be a Carnot group satisfying property R and let E be a set of locally finite perimeter in some open set G. Then, by ([30], Lem. 3.8), if G satisfies property R, for every p FE one has lim r0 PG(E; B(p, r)) rQ-1 = PG(HE (p); B(0, 1)) =: (E(p)). (4.11) Notice also that, since H has smooth boundary for any g, its perimeter can be explicitly computed (up to identification of G with Rn by means of exponential coordinates) to get () = H n-1 e (H B(0, 1)), (4.12) where H n-1 e denotes the (n - 1)-dimensional Hausdorff measure with respect to the Euclidean metric (see e.g. ([45], Thm. 5.1.3) and ([30], Prop. 2.22)). Theorem 4.9. Let G be a Carnot group satisfying property R, let G be open and bounded and let K : G [0, +) be a radial decreasing kernel satisfying (3.1), (3.2), (3.3) and (4.8). Then, there exists : g1 (0, +) such that, for every family (E) of measurable sets converging in L1 () to E , one has Z (E) dPG(E; ·) lim inf 0 1 P(E; ). (4.13) More precisely, for every g1, the function can be represented as: () = b(H) () , where b and are respectively defined as in (4.7) and (4.11). Proof. Fix > 0. We define the function f() := 1 2 Z Ec K(-1 ) d + 1 Z cEc K(-1 ) d, if E 1 2 Z E K(-1 ) d, if Ec , and set µ = fH Q . Notice that kµk := µ() = 1 P(E; ). Without loss of generality we can assume that there exists M > 0 such that 1 P(E; ) M, > 0. By this uniform bound and the assumptions on , we get that, by Theorem 4.4, E converges in L1 () to some set E of finite perimeter in . We set for shortness PE := PG(E; ·). Moreover, thanks to the weak* compactness of measures, we can find a positive measure such that µ * µ as 0 up to subsequences, and hence kµk lim inf 0 kµk. LOCAL MINIMIZERS AND GAMMA-CONVERGENCE 25 To prove (4.13), it is enough to show that kµk Z (E) dPE, for some : g1 (0, +) that will be determined in the sequel. Notice that, since by [1] the perimeter mea- sure is asymptotically doubling, we are allowed to differentiate µ with respect to the perimeter PE, see ([23], Thm. 2.8.17). We then aim to prove that dµ dPE (p) (E(p)), for PE-a.e. p , where dµ dPE (p) denotes the Radon­Nikodym derivative of µ with respect to PE. Fix p FE . Since G satisfies property R, by (4.11) we have dµ dPE (p) = lim r0 µ(B(p, r) PE(B(p, r)) = 1 (E(p)) lim r0 µ(B(p, r)) rQ-1 . Since µ weakly converges to µ as 0, we have that µ(B(p, r)) converges to µ(B(p, r)) for every r > 0 outside a countable subset Z (0, +) of radii. We therefore have dµ dPE (p) = 1 (E(p)) lim r0,r / Z lim 0 µ(B(p, r)) rQ-1 . By a diagonal argument, we may choose two infinitesimal sequences (j) and (rj) such that lim j j rj = 0, and so that dµ dPE (p) = 1 (E(p)) lim j µj (B(p, rj)) rQ-1 j . By making the computation explicit, we can write dµ dPE (p) = 1 (E(p)) lim j 1 jrQ-1 j 1 2 Z Ej B(p,rj ) Z Ec j Kj (y-1 x) dy dx + 1 2 Z Ec j B(p,rj ) Z Ej Kj (y-1 x) dy dx + Z Ej B(p,rj ) Z cEc Kj (y-1 x) dy dx ! , and hence, since P = J 1 2 J1 and since, for j sufficiently large, one has B(p, rj) , we get dµ dPE (p) 1 (E(p)) lim inf j 1 2jrQ-1 j J1 j (Ej ; B(p, rj) ) = 1 (E(p)) lim inf j 1 2jrQ-1 j J1 j (Ej ; B(p, rj)). 26 A. CARBOTTI ET AL. By a change of variable, since J1 is left unchanged by isometries, we have J1 j (Ej ; B(p, rj)) = rQ j J1 j /rj 1/rj p-1 Ej ; B . This implies that dµ dPE (p) 1 (E(p)) lim inf j rj 2j J1 j /rj 1/rj p-1 Ej ; B . By property R, up to choosing a further subsequence of j, we can assume that 1/rj p-1 Ej converges to HE (p) in L1 (B) as j . Then we get dµ dPE (p) 1 (E(p)) b(HE (p)), which concludes the proof. Acknowledgements. The authors warmly thank Gioacchino Antonelli, Xavier Cabre, and Valerio Pagliari for interesting conversations about the problem. References [1] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces. Set-Valued Anal. 10 (2002) 111­128. [2] L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals. Manuscripta Math. 134 (2011) 377­403. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). [4] L. Ambrosio, B. Kleiner and E. Le Donne, Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane. J. Geom. Anal. 19 (2009) 509­540. [5] J. Berendsen and V. Pagliari, On the asymptotic behaviour of nonlocal perimeters. ESAIM: COCV 25 (2019) 48. [6] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007). [7] A. Braides, -convergence for Beginners, Vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002). [8] X. Cabre, Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory. Ann. Mater. Pura Appl. (2020) 1­17. [9] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63 (2010) 1111­1144. [10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Diff. Equ. 32 (2007) 1245­1260. [11] L. Capogna, D. Danielli, S.D. Pauls and J.T. Tyson, An Introduction to the Heisenberg Group and the sub-Riemannian Isoperimetric Problem, Vol. 259 of Progress in Mathematics. Birkhauser Verlag, Basel (2007). [12] E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV -estimates for stable nonlocal minimal surfaces. J. Diff. Geom. 112 (2019) 447­504. [13] G. Citti, M. Manfredini and A. Sarti, Neuronal oscillations in the visual cortex: -convergence to the Riemannian Mumford- Shah functional. SIAM J. Math. Anal. 35 (2004) 1394­1419. [14] J. Davila, On an open question about functions of bounded variation. Calc. Var. Partial Diff. Equ. 15 (2002) 519­527. [15] E. De Giorgi, Nuovi teoremi relativi alle misure (r - 1)-dimensionali in uno spazio ad r dimensioni. Ricerche Mater. 4 (1955) 95­113. [16] E. De Giorgi and G. Dal Maso, -convergence and calculus of variations, in Mathematical Theories of Optimization (Genova, 1981), Vol. 979 of Lecture Notes in Mathematics. Springer, Berlin (1983) 121­143. [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521­573. [18] S. 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Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. Anal. 13 (2003) 421­466. [31] N. Garofalo, Some properties of sub-Laplacians. Electron. J. Diff. Equ. 25 (2018) 103­131. [32] N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Caratheodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49 (1996) 1081­1144. [33] N. Garofalo and G. Tralli, A Bourgain-Brezis-Mironescu-Davila theorem in Carnot groups of step two. Preprint available at https://arxiv.org/abs/2004.08529 (2020). [34] P. Hajlasz and P. Koskela, Sobolev met Poincare. Mem. Am. Math. Soc. 145 (2000) 101. [35] E. Le Donne, A primer on Carnot groups: homogenous groups, Carnot-Caratheodory spaces, and regularity of their isometries. Anal. Geom. Metr. Spaces 5 (2017) 116­137. [36] E. Le Donne and T. Moisala, Semigenerated step-3 Carnot algebras and applications to sub-Riemannian perimeter. Preprint available at https://arxiv.org/abs/2004.08619 (2020). [37] L. Lombardini, Fractional perimeters from a fractal perspective. Adv. Nonlinear Stud. 19 (2019) 165­196. [38] A. Maalaoui and A. Pinamonti, Interpolations and fractional Sobolev spaces in Carnot groups. Nonlinear Anal. 179 (2019) 91­104. [39] A. Maione, A. Pinamonti and F. Serra Cassano, -convergence for functionals depending on vector fields. II. Convergence of minimizers. Forthcoming. [40] A. Maione, A. Pinamonti and F. Serra Cassano, -convergence for functionals depending on vector fields. I. Integral representation and compactness. J. Math. Pure Appl. 139 (2020) 109­142. [41] M. Marchi, Regularity of sets with constant intrinsic normal in a class of Carnot groups. Ann. Inst. Fourier (Grenoble) 64 (2014) 429­455. [42] J.M. Mazon, J.D. Rossi and J. Toledo, Nonlocal perimeter, curvature and minimal surfaces for measurable sets. J. Anal. Math. 138 (2019) 235­279. [43] J. M. Mazon, J.D. Rossi and J.J. Toledo, Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable sets. Frontiers in Mathematics. Birkhauser/Springer, Cham (2019). [44] J. Mitchell, On Carnot-Caratheodory metrics. J. Diff. Geom. 21 (1985) 35­45. [45] R. Monti, Distances, Boundaries and Surface Measures in Carnot-Caratheodory Spaces. Ph.D. thesis (2001). Available at cvgmt.sns.it/paper/3706/ [46] V. Pagliari, Halfspaces minimise nonlocal perimeter: a proof via calibrations. Ann. Mater. Pure Appl. 199 (2020) 1685­1696. [47] A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV-functions and the Bourgain-Brezis-Mironescu formula. Adv. Calc. Var. 12 (2019) 225­252. [48] A. Sanchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78 (1984) 143­160. [49] O. Savin and E. Valdinoci, -convergence for nonlocal phase transitions. Ann. Inst. H. Poincare Anal. Non Lineaire 29 (2012) 479­500. [50] E.M. Stein and R. Shakarchi, Complex Analysis, Vol. 2 of Princeton Lectures in Analysis. Princeton University Press, Princeton, NJ (2003). [51] E. Valdinoci, A fractional framework for perimeters and phase transitions. Milan J. Math. 81 (2013) 1­23. [52] V.S. Varadarajan, Lie groups, Lie Algebras, and their Representations. Reprint of the 1974 edition. Vol. 102 of Graduate Texts in Mathematics. Springer-Verlag, New York (1984). [53] N.T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Vol. 100 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1992). [54] A. Visintin, Nonconvex functionals related to multiphase systems. SIAM J. Math. Anal. 21 (1990) 1281­1304. [55] A. Visintin, Generalized coarea formula and fractal sets. Jpn. J. Ind. Appl. Math. 8 (1991) 175­201. [1] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces. Set-Valued Anal. 10 (2002) 111–128. [2] L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals. Manuscripta Math. 134 (2011) 377–403. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). [4] L. Ambrosio, B. Kleiner and E. Le Donne Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane. J. Geom. Anal. 19 (2009) 509–540. [5] J. Berendsen and V. Pagliari, On the asymptotic behaviour of nonlocal perimeters. ESAIM: COCV 25 (2019) 48. [6] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007). [7] A. Braides, Γ-convergence for Beginners, Vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002). [8] X. Cabré, Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory. Ann. Mater. Pura Appl. (2020) 1–17. [9] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63 (2010) 1111–1144. [10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Diff. Equ. 32 (2007) 1245–1260. [11] L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson, An Introduction to the Heisenberg Group and the sub-Riemannian Isoperimetric Problem, Vol. 259 of Progress in Mathematics. Birkhäuser Verlag, Basel (2007). [12] E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV -estimates for stablenonlocal minimal surfaces. J. Diff. Geom. 112 (2019) 447–504. [13] G. Citti, M. Manfredini and A. Sarti, Neuronal oscillations in the visual cortex: Γ-convergence to the Riemannian Mumford-Shah functional. SIAM J. Math. Anal. 35 (2004) 1394–1419. [14] J. Dávila, On an open question about functions of bounded variation. Calc. Var. Partial Diff. Equ. 15 (2002) 519–527. [15] E. De Giorgi Nuovi teoremi relativi alle misure (r − 1)-dimensionali in uno spazio ad r dimensioni. Ricerche Mater. 4 (1955) 95–113. [16] E. De Giorgi and G. Dal Maso Γ-convergence and calculus of variations, in Mathematical Theories of Optimization (Genova, 1981), Vol. 979 of Lecture Notes in Mathematics. Springer, Berlin (1983) 121–143. [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. [18] S. Dipierro, A comparison between the nonlocal and the classical worlds: minimal surfaces, phase transitions, and geometric flows. Preprint available at (2020). [19] S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0. Discrete Contin. Dyn. Syst. 33 (2013) 2777–2790 (2019). [20] S. Don, E. Le Donne, T. Moisala and D. Vittone, A rectifiability result for finite-perimeter sets in Carnot groups. Preprint available at (2019). [21] S. Don and D. Vittone, Fine properties of functions with bounded variation in Carnot-Carathéodory spaces. J. Math. Anal. Appl. 479 (2019) 482–530. [22] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL (2015). [23] H. Federer, Geometric Measure Theory. In Vol. 153 of Die Grundlehren der mathematischen Wissenschafte. Springer-Verlag New York Inc., New York (1969). [24] F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups. Math. Z. 279 (2015) 435–458. [25] F. Ferrari, M. Miranda Jr. D. Pallara, A. Pinamonti and Y. Sire, Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete Contin. Dyn. Syst. Ser. S 11 (2018) 477–491. [26] G. B. Folland, A fundamental solution for a subelliptic operator. Bull. Am. Math. Soc. 79 (1973) 373–376. [27] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mater. 13 (1975) 161–207. [28] B. Franchi, R. Serapioni and F. Serra Cassano Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math. 22 (1996) 859–890. [29] B. Franchi, R. Serapioni and F. Serra Cassano Rectifiabiliy and perimeter in the Heisenberg group. Math. Ann. 321 (2001) 479–531. [30] B. Franchi, R. Serapioni and F. Serra Cassano On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. Anal. 13 (2003) 421–466. [31] N. Garofalo, Some properties of sub-Laplacians. Electron. J. Diff. Equ. 25 (2018) 103–131. [32] N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49 (1996) 1081–1144. [33] N. Garofalo and G. Tralli, A Bourgain-Brezis-Mironescu-Dávila theorem in Carnot groups of step two. Preprint available at (2020). [34] P. Hajłasz and P. Koskela, Sobolev met Poincaré. Mem. Am. Math. Soc. 145 (2000) 101. [35] E. Le Donne A primer on Carnot groups: homogenous groups, Carnot-Carathéodory spaces, and regularity of their isometries. Anal. Geom. Metr. Spaces 5 (2017) 116–137. [36] E. Le Donne and T. Moisala, Semigenerated step-3 Carnot algebras and applications to sub-Riemannian perimeter. Preprint available at (2020). [37] L. Lombardini, Fractional perimeters from a fractal perspective. Adv. Nonlinear Stud. 19 (2019) 165–196. [38] A. Maalaoui and A. Pinamonti, Interpolations and fractional Sobolev spaces in Carnot groups. Nonlinear Anal. 179 (2019) 91–104. [39] A. Maione, A. Pinamonti and F. Serra Cassano, Γ-convergence for functionals depending on vector fields. II. Convergence of minimizers. Forthcoming. [40] A. Maione, A. Pinamonti and F. Serra Cassano Γ-convergence for functionals depending on vector fields. I. Integral representation and compactness. J. Math. Pure Appl. 139 (2020) 109–142. [41] M. Marchi, Regularity of sets with constant intrinsic normal in a class of Carnot groups. Ann. Inst. Fourier (Grenoble) 64 (2014) 429–455. [42] J. M. Mazón, J. D. Rossi and J. Toledo, Nonlocal perimeter, curvature and minimal surfaces for measurable sets. J. Anal. Math. 138 (2019) 235–279. [43] J. M. Mazón, J. D. Rossi and J. J. Toledo, Nonlocal Perimeter Curvature and Minimal Surfaces for Measurable sets. Frontiers in Mathematics. Birkhäuser/Springer, Cham (2019). [44] J. Mitchell, On Carnot-Carathéodory metrics. J. Diff. Geom. 21 (1985) 35–45. [45] R. Monti, Distances, Boundaries and Surface Measures in Carnot-Carathéodory Spaces. Ph.D.thesis (2001). Available at cvgmt.sns.it/paper/3706/. [46] V. Pagliari, Halfspaces minimise nonlocal perimeter: a proof via calibrations. Ann. Mater. Pure Appl. 199 (2020) 1685–1696. [47] A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV-functions and the Bourgain-Brezis-Mironescu formula. Adv. Calc. Var. 12 (2019) 225–252. [48] A. Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78 (1984) 143–160. [49] O. Savin and E. Valdinoci, Γ-convergence for nonlocal phase transitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012) 479–500. [50] E. M. Stein and R. Shakarchi, Complex Analysis, Vol. 2 of Princeton Lectures in Analysis. Princeton University Press, Princeton, NJ (2003). [51] E. Valdinoci, A fractional framework for perimeters and phase transitions. Milan J. Math. 81 (2013) 1–23. [52] V. S. Varadarajan, Lie groups, Lie Algebras, and their Representations. Reprint of the 1974 edition. Vol. 102 of Graduate Texts in Mathematics. Springer-Verlag, New York (1984). [53] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Vol. 100 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1992). [54] A. Visintin, Nonconvex functionals related to multiphase systems. SIAM J. Math. Anal. 21 (1990) 1281–1304. [55] A. Visintin, Generalized coarea formula and fractal sets. Jpn. J. Ind. Appl. Math. 8 (1991) 175–201. COCV_2021__27_S1_A13_07b2cd567-12bd-4f2b-a89f-bb016a67973b cocv200015 10.1051/cocv/202005610.1051/cocv/2020056 Social optima in leader-follower mean field linear quadratic control* Huang Jianhui 1 Wang Bing-Chang 2 0000-0002-6987-166X Xie Tinghan 1** 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, PR China. 2 School of Control Science and Engineering, Shandong University, Jinan, PR China. **Corresponding author: [e-mail] 01 03 2021 01 03 2021 2021 cocv/2021/01 Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science OuverteSupplementS12 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF) Full (DJVU)

This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and a large number of weakly-coupled interactive followers. The leader and the followers cooperate to optimize the social cost. Specifically, for any strategy provided first by the leader, the followers would like to choose a strategy to minimize social cost functional. Using variational analysis and person-by-person optimality, we construct two auxiliary control problems. By solving sequentially, the auxiliary control problems with consistent mean field approximations, we can obtain a set of decentralized social optimality strategy with help of a class of forward-backward consistency systems. The relevant Stackelberg equilibrium is further proved under some proper conditions.

Leader-follower problem weakly-coupled stochastic system linear quadratic control social optimality forward-backward stochastic differential equation 91A12 91A23 91A25 93E03 93E20 Research Grants Council, University Grants Committee http://dx.doi.org/10.13039/501100002920 153005/14P Research Grants Council, University Grants Committee http://dx.doi.org/10.13039/501100002920 153275/16P National Natural Science Foundation of China http://dx.doi.org/10.13039/501100001809 61773241 Research Grants Council, University Grants Committee http://dx.doi.org/10.13039/501100002920 P0030808 Research Grants Council, University Grants Committee http://dx.doi.org/10.13039/501100002920 P0008686 Research Grants Council, University Grants Committee http://dx.doi.org/10.13039/501100002920 P0031044 idline ESAIM: COCV 27 (2021) S12 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S12 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020056 www.esaim-cocv.org SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL Jianhui Huang1 , Bing-Chang Wang2 and Tinghan Xie1, Abstract. This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and a large number of weakly-coupled interactive followers. The leader and the followers cooperate to optimize the social cost. Specifically, for any strategy provided first by the leader, the followers would like to choose a strategy to minimize social cost functional. Using variational analysis and person-by-person optimality, we construct two auxiliary control problems. By solving sequentially, the auxiliary control problems with consistent mean field approximations, we can obtain a set of decentralized social optimality strategy with help of a class of forward-backward consistency systems. The relevant Stackelberg equilibrium is further proved under some proper conditions. Mathematics Subject Classification. 91A12, 91A23, 91A25, 93E03, 93E20. Received January 23, 2020. Accepted August 18, 2020. 1. Introduction Mean field games have been studied by researchers from various aspects [7, 8, 10, 25]. They involve a large number of population and the interaction between each individual is negligible. The mean field game approach has been applied in many fields such as finance [14], economics [41], information technology [21], engineering [13, 24] and medicine [4]. The mean field linear quadratic (LQ) control problem is a special class of control problems, which can model many problems in applications and its solution exhibits elegant properties. For more work about the problem, readers can refer to [3, 9, 18, 19, 22, 26, 36, 37, 39, 44]. For the model with one major player and N minor players, the state of the major player has a significant influence on state equations and cost functionals of other minor individuals, which can be considered as a strong effect of the major player on minor ones. Mean field games (control) with major and minor players have been discussed in some literature, such as [23, 32] for LQ problems, [6, 33] for nonlinear problems, [12, 20] for probabilistic approaches and [11] for finite-state problems. In such types of problem, there is no hierarchical structure of decision making between the major player and the minor players. The first author acknowledges the financial support from: RGC 153005/14P, 153275/16P, P0030808; the second author acknowl- edges the support from: NNSF of China: 61773241; the third author acknowledges the support from: P0008686, P0031044; the authors also acknowledge the support from: the PolyU-SDU Joint Research Centre on Financial Mathematics. Keywords and phrases: Leader-follower problem, weakly-coupled stochastic system, linear quadratic control, social optimality, forward-backward stochastic differential equation. 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, PR China. 2 School of Control Science and Engineering, Shandong University, Jinan, PR China. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 J. HUANG ET AL. In contrast to the model discussed above, leader-follower (Stackelberg) problems contain at least two hier- archies of players. One hierarchy of the players is defined as the leaders with a major position and another hierarchy of the players is defined as the followers with a minor position. The leader has priority to announce a strategy first and then the followers seek their strategies to minimize their cost functionals with response to the given strategy of leader. According to the followers' optimal points, the leader will choose his optimal strategy to minimize its cost functional. Leader-follower problem also has been widely investigated. For example, the two-person leader-follower problem combines with stochastic LQ differential game had been studied by Yong in [43] and the problem of one leader and N followers who play a noncooperative game under LQ stochastic differential game had been studied by Moon and Basar in [31]. For further literature related to Stackelberg games, readers can refer to [1, 2, 27, 29, 34, 35, 38]. Different from noncooperative games, the social optimization (team optimization) problem is a joint decision problem which all the players have the same goal and work cooperatively. The aim of each player is to select an optimal strategy and maximize the total payoff. Team optimization problem has been studied for many years. Marschak [30] first considered team optimization based on game theory. Ho and Chu studied team decision theory in optimal control problems [17]. Groves did the research of viewing the incentive problem as a team problem which the information for decisions is incomplete [16]. The team theory and person-by-person optimization with binary decision was investigated by Bauso and Pesenti [5] and the team problems under stochastic information structure with suboptimal solutions was studied in [15]. In this paper, we investigate social optimality of the leader-follower mean field LQ control problem. Our model contains one leader and N followers. The leader's state appears in both state equation and cost functional of each follower. It shows that the dynamics and cost functionals of the N followers are directly influenced by the behavior of the leader. Unlike the model in [23, 31], our model has a population state average term in all state equations and cost functionals. This implies that such state dynamics and cost functionals are highly interactive and coupled. In reality, it is almost impossible for one player to obtain all the information of other players. Therefore, decentralized control which is based on the individual information set will be used instead of centralized control which is based on full information set and the information structure of each agent is different. Compared with previous works, this paper mainly makes the following contributions: ­ A social optimum problem is studied for mean field models with hierarchical structure. Unlike the problem in [31] where the leader and followers play a noncooperative game and try to minimize their own individual cost functional, all individuals in our models aim to minimize the social cost functional which equals the summation of cost functionals of all players. The N followers are coupled by the population state average term. Since the cost functional presents individual performance in the game problems, the order of magnitude of the perturbation is 1 N which can be ignored. The population state average term may be approximated by a stochastic process directly (see [18]). However, in the team problems, the order of magnitude of the perturbation cannot be ignored after summing up all the cost functionals, which makes the problem very complicated. To overcome such difficulties, we approximate some terms as N goes to infinity and use a duality procedure combined with auxiliary equations to transform the variation of the social cost functional into a standard LQ control form. Then, we construct an auxiliary control problem and a forward-backward consistency system which contains four equations to help us obtain the decentralized form of the optimal controls for the N followers. ­ The decentralized controls of the leader-follower problem are obtained and the solvability of a high- dimensional consistency condition system (CC system) is discussed. Since the leader's state equation and cost functional are fully coupled with the followers' state equations and cost functionals, it is more difficult to solve the leader's problem. Except constructing auxiliary problem by mean field approximation as in the former part, we need to construct six auxiliary equations and use duality relations to obtain the decentralized form of the optimal control for the leader. Unlike the problem for N followers, the final CC system of the leader's problem contains ten equations which becomes a high-dimensional problem. To solve such equations directly is very difficult since they are fully coupled and have high-dimensional characteristics. We transform the high-dimensional CC system to a simple form of linear forward-backward SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 3 stochastic differential equation (FBSDE; see [28, 42]) and discuss the solvability of the FBSDE through Ricatti equation method. ­ The decentralized strategies of leader-follower problem are proved to be Stackelberg equilibrium by per- turbation analysis. Different from [22, 23, 31, 40], we discuss the Stackelberg equilibrium for the team optimization problem. First, we need to prove the decentralized strategies for the followers have asymp- totic social optimality. Because of the Stackelberg problem contains two hierarchies, we consider two coupled cost functionals (the leader and the followers) when using the standard method (see [22]). We prove the asymptotic optimality by decoupling them with two duality procedures and some arguments in error estimates. Second, we need to prove the decentralized strategies for the leader-follower problem is Stackelberg equilibrium. Also, some error estimates are very hard to be given directly since they are fully coupling. We decompose them by applying Ricatti equation method and then estimate them in proper order. In the real world, our model can be used to describe some examples. For an automatic machine, the major part first gives an information to the system and the minor parts will adjust their parameters automatically such that the whole system keeps in the best state. For the economic environment, the small companies may hesitate to make decisions and often follow a monopoly company when facing to the volatility of the market. A monopoly company announces a decision first. Once the small companies try to make decisions according to their own situations, the monopoly company adjusts its decision such that the sum of the social wealth can be maximized. Moreover, the relationship between the employer and the employees or the federal government and the government in each state also can be described by our model. The paper is organized as follows. The problem is formulated in Section 2. In Section 3, we solve the optimal controls for followers based on person-by-person optimality and obtain the CC system of the follower's problem. In Section 4, we seek the social optimal solution of the leader's problem and give the CC system of the leader's problem. Then the CC system is transformed to a simple form of linear FBSDE in Section 5 and its wellposeness is discussed. In Section 6, we give the details of proving the Stackelberg equilibrium. Also, some prior lemmas will be introduced and proved. In Section 7, a numerical example is provided to simulate the efficiency of decentralized control. Section 8 is the conclusion of the paper. Notation: throughout this paper, Rn×m and Sn denote the set of all (n × m) real matrices and the set of all (n×n) symmetric matrices, respectively. k·k is the standard Euclidean norm and h·, ·i is the standard Euclidean inner product. For given symmetric matrix S 0, the quadratic form xT Sx may be defined as kxk2 S, where xT is the transpose of x. C1 ([0, T]; Rn×m ) be the space of all Rn×m -valued continuously differentiable functions on [0, T]. For notation o(1), limn o(1) = 0. By [42], for sake of notation simplicity, we will use K to denote a generic constant in following discussion. The value of K may be different at different places and it only depends on the coefficients and initial values. 2. Problem formulation Let (, F, P) be a complete probability space which contains all P-null sets in F. i Rn are the values of the initial states and Wi(·) are d-dimensional standard Brownian motions, where i = 0, 1, . . . , N. i and Wi(·) are defined on (, F, P). Consider a large-population system which contains one leader and N followers. The state processes of the leader and the ith follower, i = 1, 2, . . . , N, are modeled by the following linear stochastic differential equations (SDE) on a finite time horizon [0, T]: ( dx0(t) = [A0(t)x0(t) + B0(t)u0(t) + C0(t)x(N) (t)]dt + D0(t)dW0(t), x0(0) = 0, dxi(t) = [A(t)xi(t) + B(t)ui(t) + C(t)x(N) (t) + F(t)x0(t)]dt + D(t)dWi(t), xi(0) = i, (2.1) where x(N) (t) := 1 N PN i=1 xi(t) is the state average of the followers. Let -algebra Fi t = (Wi(s), 0 s t) and Gi t = Fi t W {i, 0, W0(s), 0 s t}, where 0 i N. Ft = (Wi(s), 0 s t, 0 i N) and Gt = Ft W {i, 0 i N}. Fi = {Fi t }0tT is the natural filtration generated by Wi(·) and Gi = {Gi t}0tT , where 4 J. HUANG ET AL. 0 i N. Correspondingly, we denote F = {Ft}0tT , G = {Gt}0tT . Next we introduce the following spaces: L (0, T; Rn×m ) = : [0, T] Rn×m (·) is bounded and measurable , L2 F(; Rm ) = : Rm is F-measurable, Ekk2 < , L2 F(0, T; Rm ) = x : [0, T] × Rm x(·) is F-progressively measurable, kx(t)k2 L2 := E Z T 0 kx(t)k2 dt < , L2 F(; C([0, T]; Rm )) = n x : [0, T] × Rm x(·) is F-progressively measurable, continuous, E sup t[0,T ] kx(t)k2 < o , M[0, T] := L2 F(;C([0, T]; Rn )) × L2 F(; C([0, T]; Rm )) × L2 F(0, T; Rm×d ). The set of admissible controls for the leader is defined as follows: U0 = u0|u0(t) L2 G0 (0, T; Rm ) , and the set of admissible controls for the ith follower is defined as follows: Ui = ui|ui(t) L2 Gi (0, T; Rm ) , 1 i N. These are the decentralized control sets and we let U = U1 × U2 × · · · × UN . For comparison, the centralized control set is given by Uc = n (u0, u1, . . . , uN )|ui(t) L2 G(0, T; Rm ), 0 i N o . Now we introduce the cost functionals of the leader and the ith follower, 1 i N. For the leader, the cost functional is defined as follows: J0(u0(·); u(·)) = E Z T 0 kx0(t) - 0(t)x(N) (t) - 0(t)k2 Q0(t) + ku0(t)k2 R0(t) dt + kx0(T) - 0x(N) (T) - 0k2 G0 , (2.2) where u(·) = (u1(t), . . . , uN (t)) Uc. Q0(·), R0(·) and G0(·) are weight matrices. Q0(·) and 0(·) represent the coupling between the leader and the population state average. This implies that the states of the followers can influence the cost functional of the leader. For the ith follower, the individual cost functional is defined as follows: Ji(u0(·); u(·)) = E Z T 0 kxi(t) - (t)x(N) (t) - 1(t)x0(t) - (t)k2 Q(t) + kui(t)k2 R(t) dt + kxi(T) - x(N) (T) - 1x0(T) - k2 G , (2.3) where Q(·), R(·) and G(·) are weight matrices. Q(·), (·) and 1(·) represent the coupling between the ith follower, the population state average and the leader. This implies that the cost functional of the ith follower SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 5 will be affected by the behavior of both the leader and the other followers. All the individuals in the system, including the leader and followers, aim to minimize the social cost functional, which is denoted by J (N) soc (u0(·); u(·)) = NJ0(u0(·); u(·)) + N X i=1 Ji(u0(·); u(·)), > 0. Similar to [23] and [32], we have a scaling factor N before J0(u0(·); u(·)) such that J0(u0(·); u(·)) and Ji(u0(·); u(·)) have the same order of magnitude. Otherwise, if N = 1, then the performance of the leader will be insensitive when N becomes larger. Now we introduce our assumptions. (A1) The coefficients of (1), (2) and (3) satisfy ( A0(·), C0(·), A(·), C(·), F(·) L (0, T; Rn×n ), B0(·), B(·) L (0, T; Rn×m ), D0(·), D(·) L (0, T; Rn×d ). Q0(·), Q(·) L (0, T; Sn ), R0(·), R(·) L (0, T; Sm ), 0(·), 1(·), (·) L (0, T; Rn×n ), 0(·), (·) L2 (0, T; Rn ), 0, 1, Rn×n , G0, G Sn , 0, Rn . (A2) x0(0) and W0(·) are mutually independent. {xi(0), 1 i N} and {Wi(·), 1 i N} are indepen- dent of each other. Exi(0) = ^ , 1 i N. For some constant K, which is independent of N, such that sup1iN Ekxi(0)k2 K. Furthermore, x0(0), W0(·) and {xi(0), 1 i N}, {Wi(t), 1 i N} are inde- pendent of each other. (A3) Q0(·) 0, R0(·) > I, G0 0 and Q(·) 0, R(·) > I, G 0, for some > 0. From now on, we may suppress the notation of time t if necessary. We introduce our leader-follower problem: Problem 2.1. Under (A1)­(A3), for any u0 U0, to find a mapping M: U0 U, and a control u0 U0 such that J (N) soc (u0; M(u0)) = inf uUc J (N) soc (u0; u), J (N) soc (u0; M(u0)) = inf u0U0 J (N) soc (u0; M(u0)). Note that the M here is a mapping, which is different from the notation M[0, T] we just introduced. 3. The mean field LQ control problem for the N followers 3.1. Person-by-person optimality Fix u0 U0. The leader firstly announces his own open-loop strategy. Let u = {u1, u2, . . . , uN } be the centralized optimal control of the followers and x = {x1, x2, . . . , xN } be the corresponding states. Now we perturb ui and fix other uj, where j 6= i. Then we denote ui = ui -ui, xi = xi -xi, where ui is the control after perturbing and xi is its corresponding state. The Frechet differential J0(ui) = J0(u0; u)-J0(u0; u)+o(kuik) and Ji(ui) = Ji(u0; u) - Ji(u0; u) + o(kuik), where i = 1, . . . , N. Therefore, the variations of the state 6 J. HUANG ET AL. equations for the leader, the ith follower and the jth follower, where j 6= i, are dx0 = (A0x0 + C0x(N) )dt, x0(0) = 0, dxi = (Axi + Bui + Cx(N) + Fx0)dt, xi(0) = 0, dxj = (Axj + Cx(N) + Fx0)dt, xj(0) = 0, j 6= i, and the variations of their corresponding cost functionals are 1 2 J0(ui) = E Z T 0 hQ0(x0 - 0x(N) - 0), x0 - 0x(N) idt + hG0(x0(T) - 0x(N) (T) - 0), x0(T) - 0x(N) (T)i , 1 2 Ji(ui) = E Z T 0 hQ(xi - x(N) - 1x0 - ), xi - x(N) - 1x0i + hRui, uiidt + hG(xi(T) - x(N) (T) - 1x0(T) - ), xi(T) - x(N) (T) - 1x0(T)i , 1 2 Jj(ui) = E Z T 0 hQ(xj - x(N) - 1x0 - ), xj - x(N) - 1x0idt + hG(xj(T) - x(N) (T) - 1x0(T) - ), xj(T) - x(N) (T) - 1x0(T)i , respectively. Consequently, we have the variation of the social cost functional as: 1 2 J (N) soc (ui) = 1 2 NJ0(ui) + X j6=i Jj(ui) + Ji(ui) = E Z T 0 NhQ0(x0 - 0x(N) - 0), x0i - NhT 0 Q0(x0 - 0x(N) - 0), x(N) i + hQ(xi - x(N) - 1x0 - ), xii - hT Q(xi - x(N) - 1x0 - ), x(N) i - hT 1 Q(xi - x(N) - 1x0 - ), x0i + hRui, uii + X j6=i hQ(xj - x(N) - 1x0 - ), xji - X j6=i hT Q(xj - x(N) - 1x0 - ), x(N) i - X j6=i hT 1 Q(xj - x(N) - 1x0 - ), x0idt + NhG0(x0(T) - 0x(N) (T) - 0), x0(T)i - NhT 0 G0(x0(T) - 0x(N) (T) - 0), x(N) (T)i + hG(xi(T) - x(N) (T) - 1x0(T) - ), xi(T)i - hT G(xi(T) - x(N) (T) - 1x0(T) - ), x(N) (T)i - hT 1 G(xi(T) - x(N) (T) - 1x0(T) - ), x0(T)i + X j6=i hG(xj(T) - x(N) (T) - 1x0(T) - ), xj(T)i - X j6=i hT G(xj(T) - x(N) (T) - 1x0(T) - ), x(N) (T)i - X j6=i hT 1 G(xj(T) - x(N) (T) - 1x0(T) - ), x0(T)i . (3.1) SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 7 When N , it follows that 1 2 J (N) soc (ui) = E Z T 0 hQ0(x0 - 0x(N) - 0), Nx0i - hT 0 Q0(x0 - 0x(N) - 0), Nx(N) i + hQ(xi - x(N) - 1x0 - ), xii + hRui, uii + * 1 N X j6=i Q(xj - x(N) - 1x0 - ), Nxj + - * 1 N X j6=i T Q(xj - x(N) - 1x0 - ), Nx(N) + - * 1 N X j6=i T 1 Q(xj - x(N) - 1x0 - ), Nx0 + dt + hG0(x0(T) - 0x(N) (T) - 0), Nx0(T)i - hT 0 G0(x0(T) - 0x(N) (T) - 0), Nx(N) (T)i + hG(xi(T) - x(N) (T) - 1x0(T) - ), xi(T)i + * 1 N X j6=i G(xj(T) - x(N) (T) -1x0(T) - ), Nxj(T) + - * 1 N X j6=i T G(xj(T) - x(N) (T) - 1x0(T) - ), Nx(N) (T) + - * 1 N X j6=i T 1 G(xj(T) - x(N) (T) - 1x0(T) - ), Nx0(T) + + o(1). Note that E sup0tT kx0k2 = O( 1 N2 ), E sup0tT kx(N) k2 = O( 1 N2 ) and hT Q(xi - x(N) - 1x0 - ), x(N) i + hT 1 Q(xi - x(N) - 1x0 - ), x0i + hT G(xi(T) - x(N) (T) - 1x0(T) - ), x(N) (T)i + hT 1 G(xi(T) - x(N) (T) - 1x0(T) - ), x0(T)i = o(1) (the rigorous proof will be shown in Section 6). Let x 0 = lim N+ (Nx0), x = lim N+ (Nxj) = lim N+ X j6=i xj , j 6= i. (3.2) Here Nx0 converges to x 0 such that E R T 0 kNx0 - x 0k2 = O( 1 N2 ). Similarly, P j6=i xj and Nxj converge to x (see Section 6 for the detailed proof). Then one can obtain ( dx 0 = (A0x 0 + C0xi + C0x )dt, x 0(0) = 0, dx = (Ax + Cxi + Cx + Fx 0)dt, x (0) = 0. (3.3) When N , by mean field approximation, we use x to approximate x(N) . Note that x will be affected by u0 which is given by the leader. Moreover, the influence of individual follower on x may be negligible. Hence, by 8 J. HUANG ET AL. straightforward computation, we simplified the social cost functional as follows: 1 2 J (N) soc (ui) = E Z T 0 hQ01 - T 1 Q3, x 0i + hQi 2 - T Q3 - T 0 Q01, xii + hRui, uii + hQ3 - T Q3 - T 0 Q01, x idt + hG04(T) - T 1 G6(T), x 0(T)i + hG6(T) - T G6(T) - T 0 G04(T), x (T)i + hGi 5(T) - T G6(T) - T 0 G04(T), xi(T)i , (3.4) where ( 1(·) := x0 - 0x - 0, i 2(·) := xi - x - 1x0 - , 3(·) := (I - )x - 1x0 - , are related to time t, and ( 4(T) := x0(T) - 0x(T) - 0, i 5(T) := xi(T) - x(T) - 1x0(T) - , 6(T) := (I - )x(T) - 1x0(T) - , are related to time T which are terminal terms. It is very important to formulate an auxiliary control problem to obtain the decentralized optimal control for analyzing the problem of social optimality. Usually, an auxiliary control problem is a standard LQ control problem (see [22, 40]). However, (3.4) contains x 0 and x , which are the terms we do not want them appear in the social cost functional. Therefore, we need to use a duality procedure (see [44, Chapter 3]) to get off the dependence of J (N) soc (ui) on x 0 and x . To this end, we introduce two auxiliary equations ( dk1 = 1dt + 1dW0, k1(T) = G04(T) - T 1 G6(T), dk2 = 2dt + 2dW0, k2(T) = (I - T )G6(T) - T 0 G04(T). (3.5) Using Ito formula, we have the following duality relations EhG04(T) - T 1 G6(T), x 0(T)i = Ehk1(0), x 0(0)i + E Z T 0 hk1, A0x 0 + C0xi + C0x i + h1, x 0idt, Eh(I - T )G6(T) - T 0 G04(T), x (T)i = Ehk2(0), x (0)i + E Z T 0 hk2, Ax + Cxi + Cx + Fx 0i + h2, x idt. (3.6) SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 9 Putting (3.4) and (3.6) together, we obtain 1 2 J (N) soc (ui) = E Z T 0 hRui, uii + hQ01 - T 1 Q3 + 1 + FT k2 + AT 0 k1, x 0i + hQ3 - T Q3 - T 0 Q01 + CT 0 k1 + CT k2 + 2 + AT k2, x i + hQi 2 - T Q3 - T 0 Q01 + CT 0 k1 + CT k2, xiidt + hGi 5(T) - T G6(T) - T 0 G04(T), xi(T)i . Comparing the coefficients, it follows that ( 1 = -(Q01 - T 1 Q3 + FT k2 + AT 0 k1), 2 = -(Q3 - T Q3 - T 0 Q01 + CT 0 k1 + CT k2 + AT k2). Then, according to above discussion, the two auxiliary equations can be rewritten as: dk1 = -(Q01 - T 1 Q3 + FT k2 + AT 0 k1)dt + 1dW0, dk2 = -(Q3 - T Q3 - T 0 Q01 + CT 0 k1 + CT k2 + AT k2)dt + 2dW0, k1(T) = G04(T) - T 1 G6(T), k2(T) = (I - T )G6(T) - T 0 G04(T), and the variation of social cost functional is equivalent to 1 2 J (N) soc (ui) = E Z T 0 hQxi, xii + hRui, uii + h-Q(x + 1x0 + ) - T Q3 - T 0 Q01 + CT 0 k1 + CT k2, xiidt + hGxi(T), xi(T)i + h-G(x(T) + 1x0(T) + ) - T G6(T) - T 0 G04(T), xi(T)i . (3.7) 3.2. Decentralized strategy design for followers As discussed in previous subsection, when N is sufficiently large, a stochastic process x can be used to approximate x(N) . Now, we can introduce the following auxiliary control problem for the ith follower. Problem 3.1. (P2) Minimize Ji((u0, x); ui) over ui Ui, where dxi = [Axi + Bui + Cx + Fx0(u0)]dt + DdWi, xi(0) = i, i = 1, 2, . . . , N, (3.8) Ji((u0, x); ui) = E Z T 0 kxik2 Q + kuik2 R + 2h1, xiidt + kxi(T)k2 G + 2h2, xi(T)i , (3.9) with ( 1 = -Q(x + 1x0(u0) + ) - T Q3 - T 0 Q01 + CT 0 k1 + CT k2, 2 = -G(x(T) + 1x0(u0)(T) + ) - T G6(T) - T 0 G04(T). 10 J. HUANG ET AL. Here, x0(u0) means x0 is related to u0. x0, x, k1 and k2 are determined by dx0 = [A0x0 + B0u0 + C0x]dt + D0dW0, x0(0) = 0, dx = [Ax + Bu + Cx + Fx0(u0)]dt, x(0) = ^ , dk1 = -(Q01 - T 1 Q3 + FT k2 + AT 0 k1)dt + 1dW0, dk2 = -(Q3 - T Q3 - T 0 Q01 + CT 0 k1 + CT k2 + AT k2)dt + 2dW0, k1(T) = G04(T) - T 1 G6(T), k2(T) = (I - T )G6(T) - T 0 G04(T), (3.10) where x and u are the approximations of x(N) and 1 N PN i=1 ui, respectively. In what follows, we let u = M(u0) = {u1, u2, . . . , uN } U. Note that u here represents the decentralized optimal control, which is different from the same notation in the beginning of Section 3. Proposition 3.2. Assume that (A1)­(A3) hold. For given u0 U0, (P2) has a unique optimal control ui = -R-1 BT pi, where pi is an adaptive solution to the following backward stochastic differential equation (BSDE) dpi = -(AT pi + Qxi + 1)dt + 0dW0 + idWi, pi(T) = Gxi(T) + 2. Proof. The variation of the state equation in (3.8) is dxi = (Axi + Bui)dt, xi(0) = 0, i = 1, 2, . . . , N, and the variation of the corresponding cost functional is 1 2 Ji((u0, x); ui) = E Z T 0 hQxi, xii + hRui, uii + h1, xiidt + hGxi(T), xi(T)i + h2, xi(T)i . (3.11) Using a similar argument from (3.5) to (3.7), we construct the following auxiliary equation dpi = -(AT pi + Qxi + 1)dt + 0dW0 + idWi, pi(T) = Gxi(T) + 2, (3.12) and have the following duality relation between pi and xi by using Ito formula Ehpi(T), xi(T)i = Ehpi(0), xi(0)i + E Z T 0 hAT pi - (AT pi + Qxi + 1), xii + hBT pi, uiidt. For given u0 U0, since Q 0 and R > I, for some > 0, (3.9) is uniformly convex and it has a unique optimal control. Combining above equation with (3.11), we have 1 2 Ji((u0, x); ui) = E Z T 0 hRui + BT pi, uiidt. (3.11) equal zero is equivalent to Rui + BT pi = 0. SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 11 Thus, we have ui = -R-1 BT pi. (3.13) The proposition follows. Substituting (3.13) into (3.8) and combining (3.12), we have the following FBSDE ( dxi = [Axi - BR-1 BT pi + Cx + Fx0]dt + DdWi, xi(0) = i, i = 1, 2, . . . , N, dpi = -(AT pi + Qxi + 1)dt + 0dW0 + idWi, pi(T) = Gxi(T) + 2. (3.14) By taking limits, the above FBSDE can be rewritten as: ( dx = [(A + C)x + Fx0 - BR-1 BT p]dt, x(0) = ^ , dp = -(AT p + Qx + 1)dt + 0dW0, p(T) = Gx(T) + 2. (3.15) 3.3. The consistency condition of the follower problem Let 1 := (I - T )Q(I - ) + T 0 Q00, G 1 := (I - T )G(I - ) + T 0 G00, 2 := (I - T )Q1 + T 0 Q0, G 2 := (I - T )G1 + T 0 G0, 3 := (I - T )Q - T 0 Q00, G 3 := (I - T )G - T 0 G00, 4 := T 1 Q1 + Q0, G 4 := T 1 G1 + G0, 5 := T 1 Q - Q00, G 5 := T 1 G - G00. Combining (3.10) and (3.15), we can obtain the CC system dx = [(A + C)x + Fx0 - BR-1 BT k2]dt, x(0) = ^ , dx0 = [A0x0 + B0u0 + C0x]dt + D0dW0, x0(0) = 0, dk1 = -[4x0 - T 2 x + AT 0 k1 + FT k2 + 5]dt + 1dW0, dk2 = -[1x - 2x0 + CT 0 k1 + (A + C)T k2 - 3]dt + 2dW0, k1(T) = G 4 x0(T) - (G 2 )T x(T) + G 5 , k2(T) = G 1 x(T) - G 2 x0(T) - G 3 , (3.16) where p = k2 can be easily verified. 4. The optimal control problem for the leader Now, let (P2) have a unique solution. Then, for u0 U0 given by leader, the followers choose their optimal control u = M(u0) = {u1, u2, . . . , uN } U, where ui is shown in (3.13). Now we consider the optimal control of the leader to further minimize the social cost functional. In the infinite population system, x(N) may be approximated by x. Hence, we can construct the following auxiliary optimal control problem for the leader. 12 J. HUANG ET AL. Problem 4.1. (P3) Minimize ^ J (N) soc (u0; u) over u0 U0, where dx0 = [A0x0 + B0u0 + C0x]dt + D0dW0, x0(0) = 0, ^ J (N) soc (u0; u) = N ^ J0(u0; u) + N X i=1 ^ Ji(u0; u). (4.1) (P3) is based on (P2). Therefore, combining (3.13), the equations below (3.4) and the equations (2), (3) with mean field approximations, the cost functionals of the leader and the ith follower are ^ J0(u0; u) = E Z T 0 hQ01, 1i + hR0u0, u0idt + hG04, 4i , ^ Ji(u0; u) = E Z T 0 hQi 2, i 2i + hBT pi, R-1 BT piidt + hGi 5, i 5i , where x, x0, k1, k2, xi, pi are determined by (3.16) and (3.14). Using a similar argument in Section 3, we let u0 be the optimal strategy of the leader and perturb u0 in (4.1), where u0 = u0 - u0. Since x0, x, xi and pi are determined by u0, we denote their corresponding perturbations as: x0 = x0(u0) - x0(u0), x = x(u0) - x(u0), xi = xi(u0) - xi(u0) and pi = pi(u0) - pi(u0). For sake of notation simplicity, we drop (u0) in the following x0(u0), x(u0), xi(u0) and pi(u0), etc. Then, one can obtain dx0 = [A0x0 + B0u0 + C0x]dt, x0(0) = 0, and the variations of corresponding cost functionals 1 2 ^ J0(u0) = E Z T 0 hQ01, x0 - 0xi + hR0u0, u0idt + hG04, x0(T) - 0x(T)i , 1 2 N X i=1 ^ Ji(u0) = N X i=1 E Z T 0 hQi 2, xi - x - 1x0i + hR-1 BT pi, BT piidt + hGi 5, xi(T) - x(T) - 1x0(T)i . Here 1, i 2, 4(T), i 5(T), are related to u0. In what follows, 1, i 2, 3, 4(T), i 5(T), 6(T) will be related to u0. Therefore, the variation of the social cost functional is 1 2 ^ J (N) soc (u0) = NE Z T 0 hQ01, x0i - hT 0 Q01, xi + hR0u0, u0idt + N X i=1 E Z T 0 hQi 2, xii - hT Qi 2, xi - hT 1 Qi 2, x0i + hBR-1 BT pi, piidt + NhG04(T), x0(T)i - NhT 0 G04(T), x(T)i - N X i=1 hT 1 Gi 5(T), x0(T)i - N X i=1 hT Gi 5(T), x(T)i + N X i=1 hGi 5(T), xi(T)i. (4.2) SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 13 Similarly, the variations of those equations in (3.14) and (3.16) are given by dxi = [Axi - BR-1 BT pi + Cx + Fx0]dt, xi(0) = 0, i = 1, 2, . . . , N, dpi = -(AT pi + Qxi + [1 - Q]x - 2x0 + CT 0 k1 + CT k2)dt + 0dW0 + idWi, pi(T) = Gxi(T) + [G 1 - G]x(T) - G 2 x0(T), and dx = [(A + C)x + Fx0 - BR-1 BT k2]dt, x(0) = 0, dk1 = -[4x0 - T 2 x + AT 0 k1 + FT k2]dt + 1dW0, dk2 = -[1x - 2x0 + CT 0 k1 + (A + C)T k2]dt + 2dW0, k1(T) = G 4 x0(T) - (G 2 )T x(T), k2(T) = G 1 x(T) - G 2 x0(T). Since (4.2) contains many terms that we do not want them appear in the social cost functional, we will use a similar argument in Section 3 to get off the dependence of ^ J (N) soc (u0) on those terms. Therefore, we need to construct six auxiliary equations to help us obtain the optimal control of the leader. We introduce the first three auxiliary equations: dqi = midt + n0 i dW0 + nidWi, qi(0) = 0, i = 1, 2, . . . , N, dl1 = s1dt + r1dW0, l1(0) = 0, dl2 = s2dt + r2dW0, l2(0) = 0. where mi = -(BR-1 BT pi - BR-1 BT yi - Aqi), s1 = C0l2 + A0l1 - C0qi, s2 = (A + C)l2 - BR-1 BT yi + Fl1 - Cqi, ni = 0, n0 i = 0, r1 = 0, r2 = 0. Here qi, l1 and l2 are used to free ^ J (N) soc (u0) from the dependence on pi, k1 and k2, respectively. By a similar argument from (3.5) to (3.7), we can rewrite the variation of the social cost functional as follows: 1 2 ^ J (N) soc (u0) = E Z T 0 N X i=1 hQ01 - T 1 Qi 2, x0i + N X i=1 h-T 0 Q01 - T Qi 2, xi + N X i=1 hQi 2, xii +NhR0u0, u0i + N X i=1 hBR-1 BT pi, pii + N X i=1 hl1, -(AT 0 k1 + FT k2 + 4x0 - T 2 x)i + N X i=1 hs1, k1i + N X i=1 hs2, k2i + N X i=1 hl2, -((A + C)T k2 + CT 0 k1 + 1x - 2x0)i + N X i=1 h-(AT pi + Qxi + [1 - Q]x - 2x0 + CT 0 k1 + CT k2), qii + N X i=1 hpi, mii + N X i=1 [h1, r1i + h2, r2i] + N X i=1 [h0, n0 i i + hi, nii]dt + N X i=1 hGi 5(T) - Gqi(T), xi(T)i 14 J. HUANG ET AL. + N X i=1 hG04(T) - T 1 Gi 5(T) - (G 4 )T l1(T) + (G 2 )T l2(T) + (G 2 )T qi(T), x0(T)i - N X i=1 hT 0 G04(T) + T Gi 5(T) - (G 2 )T l1(T) + (G 1 )T l2(T) + (G 1 - G)T qi(T), x(T)i. Next, we introduce another three auxiliary equations: dyi = dt + dW0 + N X i=1 idWi, yi (T) = T 0 G04(T) + T Gi 5(T) - (G 2 )T l1(T) + (G 1 )T l2(T) + (G 1 - G)T qi(T), dyi 0 = 0dt + 0dW0 + N X i=1 0 i dWi, yi 0(T) = G04(T) - T 1 Gi 5(T) - (G 4 )T l1(T) + (G 2 )T l2(T) + (G 2 )T qi(T), dyi = idt + 0dW0 + idWi, yi(T) = Gi 5(T) - Gqi(T), i = 1, 2, . . . , N. where 0 = -(Q01 - T 1 Qi 2 + FT yi - FT yi + AT 0 yi 0 - T 4 l1 + T 2 l2 + T 2 qi), = -T 0 Q01 - T Qi 2 + CT yi - (A + C)T yi + CT 0 yi 0 + 2l1 - T 1 l2 - (1 - Q)T qi, i = -(Qi 2 + AT yi - QT qi). Here yi , yi 0 and yi are used to free ^ J (N) soc (u0) from the dependence on x, x0 and xi, respectively. Similarly, by Ito formula and the duality relations, the variation of the social cost functional can be further rewritten as follows: 1 2 ^ J (N) soc (u0) = E Z T 0 N X i=1 hQ01 - T 1 Qi 2 + FT yi - FT yi + AT 0 yi 0 + 0 - T 4 l1 + T 2 l2 + T 2 qi, x0i + N X i=1 h-T 0 Q01 - T Qi 2 + CT yi - (A + C)T yi + CT 0 yi 0 - + 2l1 - T 1 l2 - (1 - Q)T qi, xi + N X i=1 hQi 2 + AT yi + i - Qqi, xii + N X i=1 hBR-1 BT pi - BR-1 BT yi + mi - Aqi, pii + N X i=1 [hni, ii + hn0 i , 0i] + N X i=1 hs1 - C0l2 - A0l1 + C0qi, k1i + N X i=1 hs2 - (A + C)l2 + BR-1 BT yi - Fl1 + Cqi, k2i + N X i=1 [hr1, 1i + hr2, 2i] + * NR0u0 + N X i=1 BT 0 yi 0, u0 + dt, SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 15 which implies, 1 2 ^ J (N) soc (u0) = E Z T 0 * NR0u0 + N X i=1 BT 0 yi 0, u0 + dt. Thus, 1 2 ^ J (N) soc (u0) = 0 is equivalent to NR0u0 + N X i=1 BT 0 yi 0 = 0. Then, we have the centralized form of the optimal control for the leader u0 = - -1 N R-1 0 BT 0 N X i=1 yi 0 := u (N) 0 , (4.3) where u0 relies on N and the following FBSDE dyi = -(AT yi - QT qi + Qi 2)dt + 0dW0 + idWi, yi(T) = Gi 5(T) - Gqi(T), dqi = (BR-1 BT yi + Aqi - BR-1 BT pi)dt, qi(0) = 0, i = 1, 2, . . . , N, dyi = (-T 0 Q01 - T Qi 2 + CT yi - (A + C)T yi + CT 0 yi 0 + 2l1 - T 1 l2 - (1 - Q)T qi)dt + dW0 + N X i=1 idWi, yi (T) = T 0 G04(T) + T Gi 5(T) - (G 2 )T l1(T) + (G 1 )T l2(T) + (G 1 - G)T qi(T), dyi 0 = -(Q01 - T 1 Qi 2 + FT yi - FT yi + AT 0 yi 0 - T 4 l1 + T 2 l2 + T 2 qi)dt + 0dW0 + N X i=1 0 i dWi, yi 0(T) = G04(T) - T 1 Gi 5(T) - (G 4 )T l1(T) + (G 2 )T l2(T) + (G 2 )T qi(T), dl1 = (A0l1 + C0l2 - C0qi)dt, l1(0) = 0, dl2 = [Fl1 + (A + C)l2 - BR-1 BT yi - Cqi]dt, l2(0) = 0. (4.4) Denote y = lim N+ 1 N N X i=1 yi, y = lim N+ 1 N N X i=1 yi , y 0 = lim N+ 1 N N X i=1 yi 0, q = lim N+ 1 N N X i=1 qi, l 1 = lim N+ 1 N N X i=1 l1, l 2 = lim N+ 1 N N X i=1 l2. Here, using a similar argument of (3.2), we can easily prove that 1 N PN i=1 yi, 1 N PN i=1 yi , 1 N PN i=1 yi 0, 1 N PN i=1 qi, 1 N PN i=1 l1 and 1 N PN i=1 l2 converge to y , y , y 0, q , l 1 and l 2, respectively. Thus, combining (3.16) and (4.4), 16 J. HUANG ET AL. when N , we can obtain the CC system for the leader-follower problem dx = [(A + C)x + Fx0 - BR-1 BT k2]dt, x(0) = ^ , dx0 = [A0x0 + C0x - B0(R0)-1 BT 0 y 0]dt + D0dW0, x0(0) = 0, dk1 = -[4x0 - T 2 x + AT 0 k1 + FT k2 + 5]dt + 1dW0, dk2 = -[1x - 2x0 + CT 0 k1 + (A + C)T k2 - 3]dt + 2dW0, k1(T) = G 4 x0(T) - (G 2 )T x(T) + G 5 , k2(T) = G 1 x(T) - G 2 x0(T) - G 3 , dy = -(AT y - QT q + Q3)dt + dW0, y (T) = G6(T) - Gq (T), dq = (BR-1 BT y + Aq - BR-1 BT k2)dt, q (0) = 0, dy = [-T 0 Q01 - T Q3 + CT y - (A + C)T y + CT 0 y 0 + 2l 1 - T 1 l 2 - (1 - Q)T q ]dt + dW0, y (T) = T 0 G04(T) + T G6(T) - (G 2 )T l 1(T) + (G 1 )T l 2(T) + (G 1 - G)T q (T), dy 0 = -(Q01 - T 1 Q3 + FT y - FT y + AT 0 y 0 - T 4 l 1 + T 2 l 2 + T 2 q )dt + 0 dW0, y 0(T) = G04(T) - T 1 G6(T) - (G 4 )T l 1(T) + (G 2 )T l 2(T) + (G 2 )T q (T), dl 1 = (A0l 1 + C0l 2 - C0q )dt, l 1(0) = 0, dl 2 = [Fl 1 + (A + C)l 2 - BR-1 BT y - Cq ]dt, l 2(0) = 0. (4.5) and the decentralized optimal control for the leader u 0 = -(R0)-1 BT 0 y 0. (4.6) The final CC system is highly coupled with five forward equations and five backward equations. The existence and uniqueness of (4.5) is very important for obtaining the optimal control, however it is very difficult to solve such high-dimensional system. We need to simplify the CC system to a FBSDE using block matrices and these will be discussed in next section. 5. Well-posedness of the CC system Note that in (4.5), the equations of (x, x0, k1, k2) form a coupled FBSDE and (y , q , y , y 0, l 1, l 2) form another coupled FBSDE. The two FBSDEs are also fully coupled with each other. Therefore, we try to look at the above FBSDEs in a different way. To this end, we set X = x x0 q l 1 l 2 , Y = y y y 0 k1 k2 , X(0) = ^ 0 0 0 0 , Y(T) = G6 - Gq (T) T 0 G04 - T G6 - (G 2 )T l 1(T) + (G 1 )T l 2(T) + (G 1 - G)T q (T) G04 - T 1 G6 - (G 4 )T l 1(T) + (G 2 )T l 2(T) + (G 2 )T q (T) G 4 x0(T) - (G 2 )T x(T) + G 5 G 1 x(T) - G 2 x0(T) - G 3 . Then (4.5) is equivalent to ( dX = [AX + BY + b]dt + DdW0, X(0) = (^ T T 0 0 0 0)T , dY = [AX + BY + b]dt + DdW0, Y(T) = GX(T) + g, (5.1) SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 17 with A = A + C F 0 0 0 C0 A0 0 0 0 0 0 A 0 0 0 0 -C0 A0 C0 0 0 -C F (A + C) , B = 0 0 0 0 -BR-1 BT 0 0 -B0(R0)-1 BT 0 0 0 BR-1 BT 0 0 0 -BR-1 BT 0 0 0 0 0 0 -BR-1 BT 0 0 0 , b = 0 0 0 0 0 , D = 0 D0 0 0 0 , A = -Q(I - ) Q1 QT 0 0 1 - Q(I - ) -2 + Q1 -(1 - Q)T 2 -T 1 T 2 -4 -T 2 T 4 -T 2 T 2 -4 0 0 0 -1 2 0 0 0 , b = Q -3 + Q -5 -5 3 , D = 0 1 2 , B = -AT 0 0 0 0 CT -(A + C)T CT 0 0 0 -FT FT -AT 0 0 0 0 0 0 -AT 0 -FT 0 0 0 -CT 0 -(A + C)T , g = -G G 3 - G G 5 G 5 -G 3 , G = G(I - ) -G1 -G 0 0 -G 1 + G(I - ) G 2 - G1 (G 1 - G)T -(G 2 )T (G 1 )T -(G 2 )T G 4 (G 2 )T -(G 4 )T (G 2 )T -(G 2 )T G 4 0 0 0 G 1 -G 2 0 0 0 . Denote A = A + BG B A - GA + BG - GBG B - GB , b = b b - Gb , D = D D - GD , Y = Y - GX. Then (5.1) can be rewritten as: d X Y = A X Y + b dt + DdW0, X(0) = (^ T T 0 0 0 0)T , Y(T) = g. (5.2) This is a fully coupled FBSDE. From the Theorem 3.7 of Chapter 2 in [28], the FBSDE (5.2) is solvable for all g L2 F(; R5n ) if and only if the following condition holds: det (0, I)eAt 0 I > 0, t [0, T]. (5.3) In the case, (5.1) admits a unique solution for any given g L2 F(; R5n ). Under the condition (5.3), we may decouple the FBSDE (5.2) by Y = KX + , t [0, T], 18 J. HUANG ET AL. where K C1 ([0, T]; S5n ) is a solution of the following Ricatti equation K + K(A + BG) + KBK - (B - GB)K - (A - GA + BG - GBG) = 0, t [0, T], K(T) = 0, and C1 ([0, T]; R5n ) satisfies + (KB - (B - GB)) + Kb - (b - Gb) = 0, t [0, T], (T) = g. (5.4) By the Theorems 3.7 and 4.3 of Chapter 2 in [28], if (5.3) hold, then the Ricatti equation admits a unique solution K(·) which has the following representation: K = - (0, I)eA(T -t) 0 I -1 (0, I)eA(T -t) I 0 , t [0, T]. (5.5) Example 5.1. Consider the system (5.2) with parameters A0 = 0.1, B0 = 1, C0 = 0.01, D0 = 1, A = 0.05, B = 1, C = 0.05, D = 1, F = 0.3, 0 = 1, Q0 = 1, R0 = 10, G0 = 0, = 0.1, 1 = 1, Q = 0.9, R = 15, G = 0, = 1.02, T = 12, 0 = = 0. Then, we have A = 0.10 0.30 0 0 0 0.01 0.10 0 0 0 0 0 0.05 0 0 0 0 -0.01 0.10 0.01 0 0 -0.05 0.30 0.10 , B = 0 0 0 0 -0.0667 0 0 -0.0980 0 0 0.0667 0 0 0 -0.0667 0 0 0 0 0 0 -0.0667 0 0 0 , A = -0.81 0.90 0.90 0 0 0.939 -0.93 -2.649 1.83 -1.749 1.83 -1.92 -1.83 1.92 -1.83 1.83 -1.92 0 0 0 -1.749 1.83 0 0 0 , B = -0.05 0 0 0 0 0.05 -0.10 0.01 0 0 -0.30 0.30 -0.10 0 0 0 0 0 -0.10 -0.30 0 0 0 -0.01 -0.10 . Hence, according to the simulation through Matlab software, for any t [0, T], we obtain A = A + BG B A - GA + BG - GBG B - GB , det (0, I)eAt 0 I > 0, (e.g. for t = 6, det (0, I)eAt 0 I = 12.7053 > 0). By the argument above, FBSDE (5.1) is solvable. For further analysis, we make the following assumption: (A4) The equation (5.2) has a unique solution and the solution (X, Y, D) belongs to M[0, T]. For the following equation ( dxi = [Axi - BR-1 BT pi + Cx + Fx0]dt + DdWi, xi(0) = i, i = 1, 2, . . . , N, dpi = -[AT pi + Qxi + 1]dt + 0dW0 + idWi, pi(T) = Gxi(T) + 2, (5.6) where 1 and 2 are related to u0. We let pi = Pxi + , t [0, T], where P C1 ([0, T]; Sn ) is a solution of the following Ricatti equation and C1 ([0, T]; Rn ) satisfies ( P + PA - PBR-1 BT P + AT P + Q = 0, t [0, T], P(T) = G, + (AT - PBR-1 BT ) + 1 + PCx + PFx0 = 0, t [0, T], (T) = 2. SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 19 Since the Ricatti equation is standard, it has a unique solution. Hence, the FBSDE (5.6) is uniquely solvable and the solution belongs to M[0, T]. 6. Asymptotically social optimality In this section, we discuss that if the leader announces u 0 obtained in (4.6) to the N followers, then the set of the optimal decentralized controls for the leader and the followers will constitutes an approximated Stackelberg equilibrium. First, for the open-loop decentralized strategy (u 0, u ) in (4.6) and (3.13), we have the realized decentralized state x 0 and x i , satisfies dx 0(t) = [A0x 0(t) - B0(R0)-1 BT 0 y 0(t) + C0(x )(N) (t)]dt + D0dW0(t), dx i (t) = [Ax i (t) - BR-1 BT pi(t) + C(x )(N) (t) + Fx 0(t)]dt + DdWi(t), x 0(0) = 0, x i (0) = i, i = 1, 2, . . . , N, (6.1) where y 0, pi satisfy (4.5) and (5.6), respectively. Then, by [2] and [31], we give the definition of the Stackelberg equilibrium. Definition 6.1. A set of control laws M(u0) U has asymptotic social optimality if 1 N J (N) soc (u0; M(u0)) - 1 N inf (u0,u)Uc J (N) soc (u0; u) = O 1 N , where M is a mapping and M : U0 U. Uc is defined in Section 3 as a set of centralized information-based control. Definition 6.2. A set of control laws (u 0, u ) U0 × U, where u = M(u 0), is an Stackelberg equilibrium with respect to J (N) soc (u0, u) if the following two properties hold: 1. M(u0) has asymptotic social optimality under u0. 2. The following equation is satisfied 1 N J (N) soc (u 0; M(u 0)) - 1 N inf u0Uc J (N) soc (u0; M(u0)) = O 1 N . We first need to introduce some lemmas before proving the Stackelberg equilibrium. Lemma 6.3. Assume that (A1)­(A4) hold. Then E Z T 0 k(x )(N) - xk2 dt + E Z T 0 kp(N) - pk2 dt + E Z T 0 kx 0 - x0k2 dt = O 1 N . Proof. See Appendix A. Lemma 6.4. Assume that (A1)­(A4) hold. There exists a constant K, which is independent of N, such that J (N) soc (u 0; u ) NK. Proof. See Appendix B. 20 J. HUANG ET AL. Proposition 6.5. Assume that (A1)­(A4) hold. For all (u0; u) Uc, there exists a constant K, which is independent of N, such that Nku0k2 L2 + kuk2 L2 NK. Proof. By Lemma 6.4, we have E Z T 0 Nku0k2 + kuk2 dt inf (u0;u) J (N) soc (u0; u) J (N) soc (u 0; u ) NK, ( > 0). Therefore, Nku0k2 L2 + kuk2 L2 NK, where K is independent of N. The proposition follows. The following two propositions will give the rigorous proofs for the approximations in Section 4. Proposition 6.6. Assume that (A1)­(A4) hold. Then, for (3.1), E sup0tT kx0k2 = O( 1 N2 ), E sup0tT kx(N) k2 = O( 1 N2 ) and hT Q(xi -x(N) -1x0 -), x(N) i+hT 1 Q(xi -x(N) -1x0 -), x0i+ hT G(xi(T) - x(N) (T) - 1x0(T) - ), x(N) (T)i + hT 1 G(xi(T) - x(N) (T) - 1x0(T) - ), x0(T)i = o(1). Proof. See Appendix C. Proposition 6.7. Assume that (A1)­(A4) hold. Then, Nxj, Nx0, Nxj converge to P j6=i xj, x 0, x such that E Z T 0 kNxj - X j6=i xjk2 = O 1 N2 , E Z T 0 kNx0 - x 0k2 = O 1 N2 , E Z T 0 kNxj - x k2 = O 1 N2 . Proof. See Appendix C. By the lemmas and propositions, we discussed above, we give the main result. Theorem 6.8. Assume that (A1)­(A4) hold. Then (u 0, u ) given in (4.6) and (3.13) is a Stackelberg equilibrium with respect to the social cost functional. Proof. For (u0; u) Uc, let 1 N J (N) soc (u 0; u ) - 1 N J (N) soc (u0; u) = 1 N J (N) soc (u 0; M(u 0)) - 1 N J (N) soc (u0; M(u0)) + 1 N J (N) soc (u0; M(u0)) - 1 N J (N) soc (u0; u) := 1 + 2, where 1 = 1 N J (N) soc (u 0; M(u 0)) - 1 N J (N) soc (u0; M(u0)), 2 = 1 N J (N) soc (u0; M(u0)) - 1 N J (N) soc (u0; u). Since u0 is fixed, by following the standard method in Huang et al [22], we obtain k2k2 c(ku0k2 L2 ) 1 N . Specifically, we denote xi as the state of the ith follower when its control is Mi(u0), thus xi is equivalent to xi in Section 4. Let ( u0 = u0 - u0 = 0, u = u - M(u0), ui = ui - Mi(u0), x0 = x0 - x0, xi = xi - xi. SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 21 Then we have J (N) soc (u0; u) = NJ0(u0; u) + N X i=1 Ji(u0; u) = NJ0(u0; M(u0)) + NH0 + NI0 + N X i=1 Ji(u0; M(u0)) + N X i=1 Hi + N X i=1 Ii, where J0(u0; M(u0)) = E Z T 0 kx0 - 0x(N) - 0k2 Q0 + ku0k2 R0 dt + kx0(T) - 0x(N) (T) - 0k2 G0 , H0 = E Z T 0 kx0 - 0x(N) k2 Q0 dt + kx0(T) - 0x(N) (T)k2 G0 , Ji(u0; M(u0)) = E Z T 0 kxi - x(N) - 1x0 - k2 Q + kMi(u0)k2 Rdt + kxi(T) - x(N) (T) - 1x0(T) - k2 G , Hi = E Z T 0 kxi - x(N) - 1x0k2 Q + kuik2 Rdt + kxi(T) - x(N) (T) - 1x0(T)k2 G , I0 = E Z T 0 (x0 - 0x(N) - 0)T Q0(x0 - 0x(N) )dt + (x0(T) - 0x(N) (T) - 0)T G0(x0(T) - 0x(N) (T)) , Ii = E Z T 0 (xi - x(N) - 1x0 - )T Q(xi - x(N) - 1x0) + MT i (u)Ruidt + (xi(T) - x(N) (T) - 1x0(T) - )T G(xi(T) - x(N) (T) - 1x0(T)) . By straightforward computation NI0 = E Z T 0 N[T 1 Q0 - (01)T Q0]x0 - [T 1 Q00 - (01)T Q00] N X i=1 xidt + N[4(T)T G0 - (01(T))T G0]x0(T) - [4(T)T G00 - (01(T))T G00] N X i=1 xi(T) , (6.2) N X i=1 Ii = E Z T 0 N X i=1 (i 2)T Qxi - [(1)T Q + T 3 Q - ((I - )1)T Q] N X i=1 xi - N[T 3 Q1 - [(I - )1]T Q1]x0 + MT i (u)Ruidt + N X i=1 (i 5(T))T Gxi(T) - [(1(T))T G + 6(T)T G - ((I - )1(T))T G] N X i=1 xi(T) - N[6(T)T G1 - [(I - )1(T)]T G1]x0(T) . (6.3) 22 J. HUANG ET AL. where 1 = x(N) - x. By (4.5), (5.6) and Ito formula, we obtain following relations: Nhk1(T), x0(T)i = hNG04, x0(T)i - hNT 1 G6, x0(T)i = E Z T 0 -hNQ01, x0i + hNT 1 Q3, x0i - hk2, NFx0i + * CT 0 k1, N X i=1 xi + dt, (6.4) and N X i=1 hpi(T), xi(T)i = E Z T 0 * T Q3, N X i=1 xi + - * Qi 2, N X i=1 xi + + * T 0 Q01, N X i=1 xi + - * CT 0 k1, N X i=1 xi + - * p(N) - k2, C N X i=1 xi + + N X i=1 hpi, Buii + D p(N) , NFx0 E dt. (6.5) Meanwhile, by (3.13), we have N X i=1 hMi(u), Ruii + N X i=1 hpi, Buii = N X i=1 hRMi(u) + BT pi, uii = N X i=1 hR(-R-1 BT pi) + BT pi, uii = 0. (6.6) Combining (6.2)­(6.6), Lemmas 6.3 and 6.4, it follows that 1 N NI0 + N X i=1 Ii ! = O 1 N . Moreover, 1 N NH0 + PN i=1 Hi 0. Thus, we have 2 = 1 N J (N) soc (u0; M(u0)) - 1 N J (N) soc (u0; u) c(ku0kL2 ) 1 N . (6.7) For 1, we decompose it as follows: 1 = 1 N J (N) soc (u 0; M(u 0)) - 1 N J (N) soc (u0; M(u0)) = 1 N J (N) soc (u 0; M(u 0)) - 1 N J (N) soc u (N) 0 ; M u (N) 0 + 1 N J (N) soc u (N) 0 ; M u (N) 0 - 1 N J (N) soc (u0; M(u0)). Note that u (N) 0 is the centralized social optimal control in (4.3), thus one can easily obtain that 1 N J (N) soc u (N) 0 ; M u (N) 0 1 N J (N) soc (u0; M(u0)). (6.8) We know that J (N) soc (u0; M(u0)) continuously depends on u0. Since M(u0) is the solution of FBSDE (5.6) which continuously depends on parameters, we have M(u0) is continuous in u0. Note that J (N) soc (u0; M(u0)) is a quadratic functional and u 0 is fixed. Let x (N) 0 and x (N) i be the state of the leader and the ith follower when SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 23 the control of the leader is u (N) 0 . Denote u0 = u (N) 0 - u 0, M(u0) = M u (N) 0 - M(u 0), Mi(u0) = Mi u (N) 0 - Mi(u 0), x0 = x (N) 0 - x 0, xi = x (N) i - x i . Then we have J (N) soc u (N) 0 ; M u (N) 0 - J (N) soc (u 0; M(u 0)) = J (N) soc u (N) 0 - u 0 + u 0; M(u (N) 0 ) - M(u 0) + M(u 0) - J (N) soc (u 0; M(u 0)) , and J (N) soc u (N) 0 ; M(u (N) 0 ) = N[J0(u 0; M(u 0)) + H0 0 + I0 0] + N X i=1 [Ji(u 0; M(u 0)) + H0 i + I0 i], where J0(u 0; M(u 0)) = E Z T 0 kx 0 - 0(x )(N) - 0k2 Q0 + ku 0k2 R0 dt + kx 0(T) - 0(x )(N) (T) - 0k2 G0 , H0 0 = E Z T 0 kx0 - 0x(N) k2 Q0 + ku0k2 R0 dt + kx0(T) - 0x(N) (T)k2 G0 , Ji(u 0; M(u 0)) = E Z T 0 kx i - (x )(N) - 1x 0 - k2 Q + kMi(u 0)k2 Rdt + kx i (T) - (x )(N) (T) - 1x 0(T) - k2 G , H0 i = E Z T 0 kxi - x(N) - 1x0k2 Q + kMi(u0)k2 Rdt + kxi(T) - x(N) (T) - 1x0(T)k2 G , I0 0 = E Z T 0 (x 0 - 0(x )(N) - 0)T Q0(x0 - 0x(N) )dt + (x 0(T) - 0(x )(N) (T) - 0)T G0(x0(T) - 0x(N) (T)) , I0 i = E Z T 0 (x i - (x )(N) - 1x 0 - )T Q(xi - x(N) - 1x0) + MT i (u 0)RMi(u0)dt + (x i (T) - (x )(N) (T) - 1x 0(T) - )T G(xi(T) - x(N) (T) - 1x0(T)) . By using similar arguments in Lemma A.1 to Lemma A.2 and k2k2 c(ku0k2 L2 ) 1 N , we obtain 1 N H0 0 + 1 N H0 i + I0 0 + 1 N N X i=1 I0 i = O 1 N . 24 J. HUANG ET AL. Hence, we have - 1 N J (N) soc u (N) 0 ; M u (N) 0 + 1 N J (N) soc (u 0; M(u 0)) K 1 N = O 1 N , (6.9) where K is independent of N. By (6.9) and (6.8), it follows that 1 N J (N) soc (u 0; M(u 0)) - 1 N J (N) soc u (N) 0 ; M u (N) 0 = O 1 N , and 1 N J (N) soc u (N) 0 ; M u (N) 0 - 1 N J (N) soc (u0; M(u0)) 0, respectively. Thus, we have 1 = 1 N J (N) soc (u 0; M(u 0)) - 1 N J (N) soc (u0; M(u0)) O 1 N . (6.10) By Proposition 6.5, there exists K independent of N such that ku0kL2 K. Then, combining (6.7), (6.10), we can obtain: 1 + 2 O 1 N + c (ku0kL2 ) 1 N K · O 1 N = O 1 N , where K is independent of N. The theorem follows. 7. Numerical examples We now give a numerical example for Lemma 6.3. By (5.5) and (5.4), K and can be easily computed. Consider Y = KX + , we can obtain that dX = [(A + BK)X + B + b]dt + DdW0, Y = KX + , where X = ((x)T (x0)T (q )T (l 1)T (l 2)T )T , Y = ((y )T (y )T (y 0)T (k1)T (k2)T )T . Since pi = Pxi + , by the following equations below (5.6), we have dxi = [(A - BR-1 BT P)xi - BR-1 BT + Cx + Fx0]dt + DdWi. The realized decentralized state x 0 and (x )(N) , can be derived by (6.1). Combining them with (4.5), one can obtain d x 0 - x0 (x )(N) - x = A0 C0 F A + C x 0 - x0 (x )(N) - x - 0 BR-1 BT (p(N) - p) dt + 1 N 0 PN 1 D dWi, x 0 - x0 (x )(N) - x (0) = 0 1 N PN 1 i - ^ , where p = k2. We continuously use the parameters in Example 5.1. The population N = 100 and the time interval is [0, 12]. By Matlab computation, the trajectories of the realized state x i are shown in Figure 1(a). SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 25 Figure 1. (a) is the trajectories of x i , i = 1, . . . , 100 and (b) is the curves of 2 i , i = 1, 2, 3 when time interval is [0, 12]. We defined 2 1 = E R 12 0 k(x)(N) -xk2 dt, 2 2 = E R 12 0 kx 0 -x0k2 dt, 2 3 = E R 12 0 kp(N) -pk2 dt. When N increases from 1 to 100, the curves of 2 1, 2 2 and 2 3 are shown in Figure 1b. The X axis indicates N and the Y axis indicates 2 i , i = 1, 2, 3. It can be seen that they are approaching to zero when N is growing larger and larger. 8. Conclusion This paper has analyzed the social optima in a class of LQ mean field control problem. We obtain the decentralized form of the optimal controls for the leader and N followers. By Ricatti equation method, we discuss the solvability of the FBSDE. Finally, a Stackelberg equilibrium theorem is established. For future work, one can extend the results of this paper to the hierarchical control with many leaders case. Appendix A. Proof of lemma 6.3 By (5.6) and (6.1), we have d(x )(N) = [A(x )(N) - BR-1 BT p(N) + C(x )(N) + Fx 0]dt + 1 N N X i=1 DdWi, (x )(N) (0) = 1 N N X i=1 i, dx(N) = [Ax(N) - BR-1 BT p(N) + Cx + Fx0]dt + 1 N N X i=1 DdWi, x(N) (0) = 1 N N X i=1 i, dp(N) = -[AT p(N) + 1]dt + 0dW0 + 1 N N X i=1 idWi, p(N) (T) = Gx(N) (T) + 2. (A.1) To prove Lemma 6.3, we need the following two lemmas. Lemma A.1. Assume that (A1)­(A4) hold. Let x(N) = 1 N PN i=1 xi and p(N) = 1 N PN i=1 pi. Then sup 0tT Ekx(N) - xk2 = O 1 N , sup 0tT Ekp(N) - pk2 = O 1 N . 26 J. HUANG ET AL. Proof. Combining (A.1) and (3.15), we can obtain dµ1 = [Aµ1 - BR-1 BT µ2]dt + 1 N N X i=1 DdWi, µ1(0) = 1 N N X i=1 i - ^ , dµ2 = -[AT µ2 + Qµ1 + 1 N N X i=1 idWi, µ2(T) = Gµ1, where µ1 = x(N) - x and µ2 = p(N) - p. Denote µ2 = Pµ1 + , t [0, T], where P C1 ([0, T]; Sn ) is the solution of the following Ricatti equation and C1 ([0, T]; Rn ) satisfies P + PA - PBR-1 BT P + AT P + Q = 0, t [0, T], P(T) = G, d = -(A - BR-1 BT P)T dt + 1 N N X i=1 (PD - i)dWi, t [0, T], (T) = 0. This is a standard Ricatti equation and the latter BSDE has a unique solution = 0, t [0, T]. Thus µ2 = Pµ1 and dµ1 = [A - BR-1 BT P]µ1dt + 1 N N X i=1 DdWi. By Cauchy-Schwarz inequality and Burkholder-Davis-Gundy's inequality, we have sup 0tT Ekµ1k2 = sup 0tT E Z t 0 (A - BR-1 BT P)µ1ds + Z t 0 1 N N X i=1 DdWi 2 2 sup 0tT E Z t 0 (A - BR-1 BT P)µ1ds 2 + 2 sup 0tT E Z t 0 1 N N X i=1 DdWi 2 2K sup 0tT E Z t 0 kµ1k2 ds + 1 N2 N X i=1 E Z T 0 kDk2 ds = 2K sup 0tT E Z t 0 kµ1k2 ds + O 1 N , where constant K is independent of N. Then, by Gronwall's inequality and µ2 = Pµ1, we obtain sup 0tT Ekµ1k2 = O 1 N , sup 0tT Ekµ2k2 = O 1 N . The lemma follows. Lemma A.2. Assume that (A1)­(A4) hold. Let (x )(N) = 1 N PN i=1 x i . Then sup 0tT Ekx 0 - x0k2 = O 1 N , sup 0tT Ek(x )(N) - x(N) k2 = O 1 N . Proof. Denote µ3 = x 0 - x0 and µ4 = (x )(N) - x(N) . By (A.1), we can obtain d µ3 µ4 = A0 C0 F A + C µ3 µ4 + C0 C µ1 dt, µ3 µ4 (0) = 0 0 . SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 27 For some constant K which is independent of N such that sup 0tT E µ3 µ4 2 = sup 0tT E Z t 0 A0 C0 F A + C µ3 µ4 + C0 C µ1 ds 2 2K sup 0tT E Z t 0 µ3 µ4 2 ds + sup 0tT E Z t 0 kµ1k2 ds = 2K sup 0tT E Z t 0 µ3 µ4 2 ds + O 1 N . By Gronwall's inequality, one can obtain sup 0tT E µ3 µ4 2 = O 1 N . Thus, the lemma follows. Proof of Lemma 6.3. Since k(x )(N) - xk2 = k(x )(N) - x(N) + x(N) - xk2 2k(x )(N) - x(N) k2 + 2kx(N) - xk2 . Combining Lemma A.1 and Lemma A.2, it leads to E Z T 0 k(x )(N) - xk2 dt + E Z T 0 kp(N) - pk2 dt + E Z T 0 kx 0 - x0k2 dt T · O 1 N = O 1 N . The lemma follows. Appendix B. Proof of lemma 6.4 Proof. By (4.5), (A.1), (6.1) and using a similar argument in Lemma A.2, one obtain that for some constant K which is not dependent on N such that sup 0tT Ekx 0k2 K, sup 0tT Ek(x )(N) k2 K. By Cauchy-Schwarz inequality and Burkholder-Davis-Gundy's inequality, we obtain sup 0tT Ekx i k2 = sup 0tT E Z t 0 (Ax i - BR-1 BT pi + C(x )(N) + Fx 0)ds + DdWi 2 sup 0tT E Z t 0 2kAx i k2 + 2kBR-1 BT pik2 + 2kC(x )(N) k2 + 2kFx 0k2 ds + sup 0tT Z t 0 2kDdWik2 2E Z T 0 kAx i k2 + kBR-1 BT pik2 + kC(x )(N) k2 + kFx 0k2 ds + Z T 0 kDk2 ds 2A2 sup 0tT E Z T 0 kx i k2 ds + K, where constant K is independent of N. By Gronwall's inequality, we have sup 1iN sup 0tT Ekx i k2 K, 28 J. HUANG ET AL. where K is not dependent on N. Then, according to Cauchy-Schwarz inequality, Burkholder-Davis-Gundy's inequality and the above discussion, we have J (N) soc (u 0; u ) = NJ0(u 0; u ) + N X i=1 Ji(u 0; u ) = NE Z T 0 kx 0 - 0(x )(N) - 0k2 Q0 + k - (R0)-1 BT 0 y 0k2 R0 dt + kx 0(T) - 0(x )(N) (T) - 0k2 G0 + N X i=1 E Z T 0 kx i - (x )(N) (t) - 1x 0 - k2 Q + k - R-1 BT pik2 R dt + kx i (T) - (x )(N) (T) - 1x 0(T) - k2 G NK, where K is independent of N. The lemma follows. Appendix C. Proof of propositions 6.6 and 6.7 Proof of Proposition 6.6. Since dx0 = (A0x0 + C0x(N) )dt, x0(0) = 0, dx(N) = [(A + C)x(N) + B N ui]dt, x(N) (0) = 0, and by Proposition 6.5, we have kuik2 L2 K, K is independent of N. Using Cauchy-Schwarz inequality, it follows that E sup 0st kx(N) k2 = E sup 0st Z s 0 [(A + C)x(N) + B N ui]dr 2 KE Z t 0 kx(N) k2 dr + 1 N2 kKk2 dr KE Z t 0 kx(N) k2 dr + O 1 N2 , where K is independent of N. By Gronwall's inequality E sup 0tT kx(N) k2 = O 1 N2 . For x0, we have E sup 0st kx0k2 =E sup 0st Z s 0 [A0x0 + C0x(N) ]dr 2 KE Z t 0 kx0k2 dr + O 1 N2 , where K is independent of N. By Gronwall's inequality E sup 0tT kx0k2 = O 1 N2 . SOCIAL OPTIMA IN LEADER-FOLLOWER MEAN FIELD LINEAR QUADRATIC CONTROL 29 Moreover, T Q(xi - x(N) - 1x0 - ) 1 N inf (u0;u) J(N) soc (u0; u) K. Similarly, T 1 Q(xi - x(N) - 1x0 - ), T G(xi(T) - x(N) (T) - 1x0(T) - ), T 1 G(xi(T) - x(N) (T) - 1x0(T) - ) are bounded. Thus, hT Q(xi - x(N) - 1x0 - ), x(N) i + hT 1 Q(xi - x(N) - 1x0 - ), x0i + hT G(xi(T) - x(N) (T) - 1x0(T) - ), x(N) (T)i + hT 1 G(xi(T) - x(N) (T) - 1x0(T) - ), x0(T)i = o(1). The proposition follows. Proof of Proposition 6.7. Since d X j6=i xj = A X j6=i xj + C N - 1 N x(N) + F(N - 1)x0 dt, X j6=i xj (0) = 0, d(Nx0) = [A0(Nx0) + C0(Nx(N) )]dt, (Nx0)(0) = 0, d(Nxj) = [A(Nxj) + C(Nx(N) ) + F(Nx0)]dt, (Nxj)(0) = 0. According to equations in (3.3), one can obtain d Nxj - X j6=i xj = A Nxj - X j6=i xj + C N x(N) + Fx0 dt, Nxj - X j6=i xj (0) = 0, d(Nx0 - x 0) = A0 Nx0 - x 0 + C0 Nxj - X j6=i xj + C0 Nxj - dx dt, Nx0 - x 0 (0) = 0, d(Nxj - x ) = [(A + C)(Nxj - x ) + C Nxj - X j6=i xj + F(Nx0 - x 0)]dt, (Nxj - x )(0) = 0. Combing with the results in Proposition 6.6, we have E sup 0st Nxj - X j6=i xj 2 = E sup 0st Z s 0 A Nxj - X j6=i xj + C N x(N) + Fx0 dr 2 KE Z t 0 Nxj - X j6=i xj 2 dr + O 1 N2 , where constant K is independent of N. By Gronwall's inequality E sup 0tT Nxj - X j6=i xj 2 = O 1 N2 . 30 J. HUANG ET AL. Since Nx0 - x 0 and Nxj - x are coupled, we have E sup 0st Nx0 - x 0 Nxj - dx 2 =E sup 0st Z s 0 " A0 C0 F A + C Nx0 - x 0 Nxj - dx + C0 C (Nxj - X j6=i xj) # dr 2 KE Z t 0 Nx0 - x 0 Nxj - dx 2 dr + O 1 N2 , where real-valued matrix K is independent of N. By Gronwall's inequality E sup 0tT Nx0 - x 0 Nxj - dx 2 = O 1 N2 . The proposition follows. References [1] T. Basar, A. 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Zhang, Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters. SIAM J. Control Optim. 55 (2017) 429­456. [40] B. Wang and J. Huang, Social optima in robust mean field LQG control, in Proceedings of the 11th Asian Control Conference. Gold Coast, QLD (2017) 2089­2094. [41] G.Y. Weintraub, C.L. Benkard and B.V. Roy, Markov perfect industry dynamics with many firms. Econometrica 76 (2008) 1375­1411. [42] J. Yong, Linear forward­backward stochastic differential equations. Appl. Math. Optim. 39 (1999) 93­119. [43] J. Yong, A leader-follower stochastic linear quadratic differential game. SIAM J. Control Optim. 41 (2002) 1015­1041. [44] J. Yong and X.Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). [1] T. Başar, A. Bensoussan and S. P. Sethi, Differential games with mixed leadership: the open-loop solution. Appl. Math. Comput. 217 (2010) 972–979. [2] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory. SIAM Philadelphia (1999). [3] M. Bardi and F. S. Priuli, Linear-quadratic N-person and mean-field games with ergodic cost. SIAM J. Control Optim. 52 (2014) 3022–3052. [4] C. T. Bauch and D. J. D. Earn, Vaccination and the theory of games. Proc. Natl. Acad. Sci. U.S.A. 101 (2004) 13391–13394. [5] D. Bauso and R. Pesenti, Team theory and person-by-person optimization with binary decisions. SIAM J. Control Optim. 50 (2012) 3011–3028. [6] A. Bensoussan, M. H. M. Chau and S. C. P. Yam, Mean field games with a dominating player. Appl. Math. Optim. 74 (2016) 91–128. [7] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013). [8] P. E. Caines, Mean field games, in Encyclopedia of Systems and Control, edited by T. Samad and J. Baillieul. Springer-Verlag, Berlin (2014). [9] R. Carmona and F. Delarue, Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51 (2013) 2705–2734. [10] R. Carmona, F. Delarue and D. Lacker, Mean field games with common noise. Ann. Probab. 44 (2016) 3740–3803. [11] R. Carmona and P. Wang, Finite state mean field games with major and minor players. Preprint (2016). [12] R. Carmona and X. Zhu, A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26 (2016) 1535–1580. [13] R. Couillet, S. M. Perlaza, H. Tembine and M. Debbah, Electrical vehicles in the smart grid: a mean field game analysis. IEEE. J. Sel. Area. Commun. 30 (2012) 1086–1096. [14] D. Firoozi and P. E. Caines, Mean field game ε-Nash equilibria for partially observed optimal execution problems in finance, in Proceedings of the IEEE 55th Conference on Decision and Control. Las Vegas (2016) 268–275. [15] G. Gnecco, M. Sanguineti and M. Gaggero, Suboptimal solutions to team optimization problems with stochastic information structure. SIAM J. Optimiz. 22 (2012) 212–243. [16] T. Groves, Incentives in teams. Econometrica 41 (1973) 617–631. [17] Y. C. Ho and K. C. Chu, Team decision theory and information structures in optimal control. Part I. IEEE Trans. Automat. Contr. 17 (1972) 15–22. [18] J. Huang and M. Huang, Robust mean field linear-quadratic-Gaussian games with unknown L² -disturbance. SIAM J. Control Optim. 55 (2017) 2811–2840. [19] J. Huang and N. Li, Linear quadratic mean-field game for stochastic delayed systems. IEEE Trans. Automat. Contr. 63 (2018) 2722–2729. [20] J. Huang, S. Wang and Z. Wu, Backward–forward linear–quadratic mean-field games with major and minor agents. Probab. Uncertain. Quant. Risk 1 (2016) 1–27. [21] M. Huang, P. E. Caines and R. P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions, in Proceedings of 42nd IEEE International Conference on Decision and Control. Maui (2003) 98–103. [22] M. Huang, P. E. Caines and R. P. Malhamé, Social optima in mean-field LQG control: centralized and decentralized strategies. IEEE Trans. Automat. Contr. 57 (2012) 1736–1751. [23] M. Huang and S. L. Nguyen, Linear-quadratic mean field teams with a major agent, in Proceedings of IEEE 55th Conference on Decision and Control. Las Vegas (2016) 6958–6963. [24] A. C. Kizilkale and R. P. Malhame, Collective target tracking mean field control for Markovian jump-driven models of electric water heating loads, in Proceedings of the 19th IFAC World Congress. Cape Town, South Africa (2014) 1867–1972. [25] J. Lasry and P. Lions, Mean field games. Jpn. J. Math. 2 (2007) 229–260. [26] T. Li and J. Zhang, Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans. Automat. Contr. 53 (2008) 1643–1660. [27] Y. N. Lin, X. S. Jiang and W. H. Zhang, An open-loop Stackelberg strategy for the linear quadratic mean-field stochastic differential game. IEEE Trans. Automat. Contr. 64 (2019) 97–110. [28] J. Ma and J. Yong, Forward–Backward Stochastic Differential Equations and their Applications. Springer-Verlag, Berlin (1999). [29] S. Maharjan, Q. Zhu, Y. Zhang, S. Gjessing and T. Basar, Dependable demand response management in the smart grid: a Stackelberg game approach. IEEE Trans. Smart Grid. 4 (2013) 120–132. [30] J. Marschak, Elements for a theory of teams. Manage. Sci. 1 (1955) 127–137. [31] J. Moon and T. Başar, Linear quadratic mean field Stackelberg differential games. Automatica 97 (2018) 200–213. [32] S. L. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players. SIAM J. Control Optim. 50 (2012) 2907–2937. [33] M. Nourian and P. E. Caines, ϵ-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim. 51 (2013) 3302–3331. [34] J. Shi, G. Wang and J. Xiong, Leader-follower stochastic differential game with asymmetric information and applications. Automatica 63 (2016) 60–73. [35] M. Simaan and J. Cruz, A Stackelberg solution for games with many players. IEEE Trans. Automat. Contr. 18 (1973) 322–324. [36] H. Tembine, Q. Zhu and T. Başar, Risk-sensitive mean-field games. IEEE Trans. Automat. Contr. 59 (2014) 835–850. [37] B. Wang and J. Zhang, Mean field games for large population multiagent systems with Markov jump parameters. SIAM J. Control Optim. 50 (2012) 2308–2334. [38] B. Wang and J. Zhang, Hierarchical mean field games for multiagent systems with tracking-type costs: distributed ε-Stackelberg equilibria. IEEE Trans. Automat. Contr. 59 (2014) 2241–2247. [39] B. Wang and J. Zhang, Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters. SIAM J. Control Optim. 55 (2017) 429–456. [40] B. Wang and J. Huang, Social optima in robust mean field LQG control, in Proceedings of the 11th Asian Control Conference. Gold Coast, QLD (2017) 2089–2094. [41] G. Y. Weintraub, C. L. Benkard and B. V. Roy, Markov perfect industry dynamics with many firms. Econometrica 76 (2008) 1375–1411. [42] J. Yong, Linear forward–backward stochastic differential equations. Appl. Math. Optim. 39 (1999) 93–119. [43] J. Yong, A leader-follower stochastic linear quadratic differential game. SIAM J. Control Optim. 41 (2002) 1015–1041. [44] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). COCV_2021__27_S1_A14_096e4372b-0f6d-407a-a3bc-0ce966218491cocv20002010.1051/cocv/202006010.1051/cocv/2020060 Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation* Christof Constantin 1** Müller Georg 2 1 Technische Universität München, Chair of Optimal Control, Center for Mathematical Sciences, M17, Boltzmannstraße 3, 85748 Garching, Germany. 2 Universität Konstanz, Department of Mathematics and Statistics, WG Numerical Optimization, Universitätsstraße 10, 78457 Konstanz, Germany. **Corresponding author: [e-mail] SupplementS13 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF)Full (DJVU)

This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are – in a certain sense – exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.

Multiobjective optimal control non-smooth optimization first-order necessary optimality condition strong stationarity semilinear partial differential equation Pareto front 35J20 49J52 49K20 58E17 90C29 idline ESAIM: COCV 27 (2021) S13 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S13 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020060 www.esaim-cocv.org MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH SEMILINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION Constantin Christof1, and Georg Muller2 Abstract. This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are ­ in a certain sense ­ exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures. Mathematics Subject Classification. 35J20, 49J52, 49K20, 58E17, 90C29. Received January 27, 2020. Accepted September 19, 2020. This research has been partially funded by the German Research Foundation (DFG) through the Priority Programme SPP 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimiza- tion", Project P02 "Multiobjective Optimization of Non-smooth PDE-constrained Problems ­ Switches, State Constraints, and Model Order Reduction". The first author gratefully acknowledges the support by the International Research Training Group IGDK 1754, funded by the German Research Foundation (DFG) and the Austrian Science Fund (FWF) under project number 188264188/GRK1754. Keywords and phrases: Multiobjective optimal control, non-smooth optimization, first-order necessary optimality condition, strong stationarity, semilinear partial differential equation, Pareto front. 1 Technische Universitat Munchen, Chair of Optimal Control, Center for Mathematical Sciences, M17, Boltzmannstraße 3, 85748 Garching, Germany. 2 Universitat Konstanz, Department of Mathematics and Statistics, WG Numerical Optimization, Universitatsstraße 10, 78457 Konstanz, Germany. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 C. CHRISTOF AND G. MULLER 1. Introduction The aim of this paper is to study first-order necessary optimality conditions for multiobjective optimal control problems of the form Minimize J1(y, u) := j1(y) + 1 2 kuk2 L2 . . . JN (y, u) := jN (y) + N 2 kuk2 L2 w.r.t. u L2 (), y H1 0 () H2 (), s.t. - y + max(0, y) = u a.e. in . (P) Here, Rd , d 1, is a bounded domain with a sufficiently regular boundary, n, n = 1, . . . , N, are non- negative Tikhonov parameters with N > 0, and jn, n = 1, . . . , N, are given objective functions with suitable mapping and smoothness properties. For the precise assumptions on the quantities in (P), we refer to Section 2. Problems of the type (P) ­ i.e., multicriteria optimization problems governed by semilinear partial differential equations involving non-smooth Nemytskii operators - arise, for instance, in mechanics, plasma physics, and the context of certain combustion processes when the state of the system is supposed to meet several, potentially conflicting design goals. See, e.g., [33, 47, 53, 55] for some examples of possible application areas. From the mathematical point of view, problems of the type (P) are challenging for a number of reasons. The first and probably most obvious one is the presence of the non-smooth Nemytskii operator in the governing partial differential equation. Because of this term, the control-to-state mapping S : u 7 y of (P) does not possess a Gateaux derivative, but is only directionally differentiable, and it is not possible to apply standard results to derive, for instance, first-order necessary optimality conditions or to devise efficient numerical solution algorithms. The strategy that is most commonly used in the literature to handle such a lack of smoothness is to work with elements of appropriately defined subdifferentials (e.g., those of Clarke, Dini, Mordukhovich, or Frechet) instead of the non-existing gradients of the involved (reduced) objective functions. See, for example, [17, 23, 43, 50, 52] for an overview of these concepts and references on the use of subgradients in the single- objective context, [9, 10, 26, 27, 30, 31, 36, 39, 42, 56] for applications in the multiobjective setting, and [3, 6, 35] for results that additionally also rely on regularization techniques. Unfortunately, for problems of the type (P), a standard subgradient-based analysis turns out to be not very rewarding either. Since the non-smoothness enters (P) only indirectly via the non-differentiable Nemytskii operator in the governing PDE, a full characterization of quantities like the Clarke subdifferentials of the reduced objective functions Jn(S(·), ·), n = 1, . . . , N, of (P) is only available in pathological situations. Optimality conditions involving these generalized differentials are thus rather academic and barely usable in practice. Note that this is a major difference to situations, in which the non-smoothness stems from the "outer" functions Jn and not from the governing PDE, and in which, as a consequence, classical chain rules for subgradients can be applied, cf. [7]. A possible way around this difficulty, that has recently been employed in [14, 48, 49] in the single-objective setting, is to work with notions of generalized derivatives on the level of the solution operator S : u 7 y. Such approaches typically result in necessary optimality conditions of intermediate strength that can indeed be solved with standard solution algorithms. For more details on this topic and further remarks on how the findings of [14, 48, 49] are related to the present paper, we refer to [14], Section 4 and Sections 4 and 5. For results on smooth multiobjective optimization and optimal control problems, see also [4, 8, 22, 29, 44­46]. A second factor that significantly complicates the derivation of necessary optimality conditions for problems of the type (P) and that is completely absent in the single-objective setting is that - even in the smooth case ­ there are several sensible, purely primal optimality and stationarity concepts for multiobjective optimization problems. Compare, for instance, with the notions of weak, ordinary, and proper Pareto optimality and stationarity in Definitions 2.3 and 3.2 in this context. Even worse, the multiobjective aspect of (P) also turns out to add an additional layer of non-smoothness to the problem. To see this, consider, for example, the notion of weak Pareto MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 3 optimality for (P), i.e., the optimality condition given by @u L2 () : Jn(S(u), u) < Jn(S(u), u) n = 1, . . . , N. This condition is clearly equivalent to u being a global optimum of the scalar problem min uL2() max J1(S(u), u) - J1(S(u), u), . . . , JN (S(u), u) - JN (S(u), u) , (1.1) whose objective function not only contains the non-smooth solution map S, but also the non-differentiable maximum-value function on RN . We thus indeed have to deal with two sources of non-smoothness here which are even nested. Note that, in first-order necessary optimality conditions for weak Pareto optima, the presence of the maximum-value function in (1.1) typically manifests itself in the form of additional multipliers. Compare, for instance, with the results for multiobjective problems with smooth objective functions in [39], Theorems 3.1.1, 3.1.5 in this context and also with the stationarity systems established in Theorem 4.5. The main goal of the present paper is to demonstrate that, despite all of the above difficulties, it is indeed possible to derive very rigorous first-order necessary optimality conditions for multiobjective optimal control problems of the type (P). To be more precise, we will show how to prove so-called strong stationarity conditions for this problem class. In the single-objective setting, such conditions are well-known, e.g., for the optimal control of elliptic variational inequalities of the first and the second kind and the optimal control of non-smooth elliptic and parabolic partial differential equations, see [13, 14, 18, 19, 38, 40, 41]. In the multiobjective context, strong stationarity conditions have, at least to the best of the authors' knowledge and with some exceptions for finite-dimensional MPECs in [56], not been considered so far (most likely because of the doubly non-smooth behavior in (1.1) that is generally hard to handle). Recall that the distinguishing feature of a strong stationarity system is its equivalence to the first-order necessary optimality condition in primal form. In the single-objective case, this means that a point is strongly stationary if and only if it is Bouligand stationary in the sense of [20], Definition 5.4, i.e., stationary in the sense that the directional derivative is non-negative in all directions. When considering the problem (P) with its multiple sensible purely primal necessary optimality conditions, one, of course, has to differentiate at this point. We will thus establish not only one but even two strong stationarity systems for (P) ­ one equivalent to weak Pareto stationarity and one equivalent to both ordinary and proper Pareto stationarity, see Definition 3.2 and Theorem 4.5. (Note that this implies in particular that the concepts of ordinary and proper Pareto stationarity are the same for the problem (P).) A main ingredient of our analysis is the - at first glance rather surprising ­ fact that the non-linear and non-differentiable inverse S0 (u; ·)-1 : H1 0 () H2 () L2 (), z 7 -z + 1{S(u)=0} max(0, z) + 1{S(u)>0}z, of the directional derivative S0 (u; ·) of the control-to-state mapping S : u 7 y of (P) is self-adjoint in the sense that u, S0 (u; ·)-1 (z) L2 = S0 (u; ·)-1 (u), z H1 0 H2 (1.2) holds for all u L2 () and all z H1 0 () H2 (), where, on the right-hand side of (1.2), the map S0 (u; ·)-1 is interpreted as a function from L2 () to (H1 0 () H2 ()) . This behavior makes it possible to resolve the difficulties related to the additional layer of non-smoothness in (1.1) that normally prevent the derivation of strong stationarity conditions in the multiobjective context, cf. Lemma 4.2 and Theorem 4.5. Note that the self-adjointness of the operator S0 (u; ·)-1 in (1.2) is also of relevance for scalar Tikhonov regularized optimal control problems governed by non-smooth semilinear partial differential equations as it allows to employ an adjoint calculus similar to that available in the smooth setting in the non-smooth case. Quite interestingly, the self-adjointness property in (1.2) is also directly related to certain non-standard regularization effects that have apparently not been documented so far in the literature ­ neither in the single- nor in the multiobjective context. These effects cause the problem (P) to be Gateaux differentiable when it is reduced to the state y although it is not Gateaux when reduced to the control u. For further details on this topic, see Section 6. We remark that the results of Section 6 particularly imply that the problems considered in [18], 4 C. CHRISTOF AND G. MULLER Section 5, [14], Section 5, and [21], Section 5, Case 1 all admit a Gateaux differentiable reformulation. As we will see in Section 5, they further yield that regularization approaches are exact for the problem (P) in the sense that weak L2 -accumulation points of weakly Pareto stationary points of the regularized multiobjective optimal control problems are weakly Pareto stationary for the non-smooth limit problem. Note that this behavior is again quite surprising as such effects can typically not even be observed in simple, one-dimensional examples. Compare, for instance, with the situation, where the function f(x) := -|x| is approximated by the sequence f(x) := - x2 + , > 0, and where the point x = 0 is Bouligand stationary for all f but not for the limit function f, in this context. Before we begin with our analysis, we would like to point out that we consider the problem (P) as a model problem in this paper. It is easy to check that our arguments can be extended straightforwardly to cases where the governing PDE contains a more general elliptic second-order partial differential operator or a more complicated piecewise C1 -function with properties similar to that of max(0, ·). An extension to the parabolic setting, cf. [38], is also possible by invoking the results in the appendix of [15]. We restrict our attention to the setting in (P) to avoid obscuring the basic ideas of our analysis with unnecessary technicalities. To help the reader navigate the paper, we conclude this introduction with a brief overview of the content of the upcoming sections: Sections 1.1 and 2 deal with preliminaries. Here, we comment on the notation used, state our standing assumptions on jn, n, and (see Asm. 2.1), and recall known results on the properties of the control-to-state mapping S : u 7 y of (P), notions of optimality for multiobjective optimization problems, and the existence of Pareto optimal controls. Section 3 is concerned with first-order necessary optimality conditions in purely primal form (i.e., conditions based on directional derivatives). We remark that, in the finite-dimensional setting, such stationarity concepts have already been discussed, e.g., in [26], Section 4 and [30], Section 3. In Section 4, we prove the already mentioned strong stationarity conditions for the multiobjective optimal control problem (P). This section contains the main result of the paper, Theorem 4.5. It further addresses in detail the self-adjointness property in (1.2) (see Lem. 4.2), the consequences that our findings have for the regularity properties of Pareto stationary points (see Cor. 4.7), and the relationship of our results to the notion of subdifferential studied in [14, 48, 49] and classical scalarization techniques (see Rem. 4.6). Section 5 demonstrates that smoothing methods are indeed exact for problems of the type (P) in the sense that they allow to determine weakly Pareto stationary points when the regularization parameter is driven to zero, see Theorem 5.3. Corollary 5.4 in this section moreover shows that the concepts of strong and C-stationarity are the same for the problem (P) in the single-objective case N = 1. In Section 6, we discuss the non-standard regularization effects that are responsible for the results of the previous sections and that cause the problem (P) to be Gateaux differentiable when reduced to the state y. Here, we further give an alternative interpretation of the strong stationarity conditions in Theorem 4.5 and prove an auxiliary result on non-smooth Nemytskii operators that is also interesting on its own, see Theorem 6.1. Section 7 of the paper finally contains numerical experiments which demonstrate that the multiplier systems in Sections 4 and 5 are amenable to numerical solution procedures and allow to compute approximations of the Pareto front of (P). 1.1. Remarks on the notation In what follows, we use the standard symbols Lq (), Ck, (), Hk 0 (), Hk (), and Wk,q (), 1 q , k N, 0 < 1, to denote the Lebesgue-, Holder-, and Sobolev spaces on a bounded Lipschitz domain Rd , d 1, respectively. For details on these spaces, we refer to [1, 2, 24, 28]. Given two Banach spaces X and Y , we further define X to be the dual space of X and L(X, Y ) to be the space of linear and continuous functions from X to Y . In the special case X = H1 0 (), we also set H-1 () := H1 0 () . As usual, we interpret H1 0 (), L2 (), and H-1 () as a Gelfand triple, i.e., H1 0 () , L2 () = L2 () , H-1 (). The same convention is used for the space H1 0 () H2 (), cf. [32], Section 1.9. Norms, scalar products, and dual pairings are denoted by the symbols k · k, (·, ·), and h·, ·i in this paper, and the modes of weak and strong convergence by the arrows * and MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 5 . If we want to specify the space/topology that we are referring to, then we add suitable sub- or superscripts and write, e.g., k · kL2 . With , , c, and i, i = 1, . . . , d, we denote the (distributional) Laplacian, the (weak) gradient, the convex subdifferential, and the first (weak) partial derivatives of a function, respectively. For higher-order derivatives, we also use the multi-index notation , Nd 0. Gateaux-, Frechet-, and directional derivatives in the functional analytic sense are denoted by a prime in the usual way. Given a function v : R and a measurable set D , we finally define {v 0}, {=, 6=, <, >, , }, to be the set {x | v(x) 0} and 1D to be the indicator function of D (with values in {0, 1}). If v is an element of an Lq -space and D is only defined up to sets of measure zero, then we consider {v 0} to be defined up to sets of measure zero as well and identify 1D with an element of L (). Note that, in the remainder of this paper, new symbols etc. are introduced whenever necessary. For the sake of readability, this additional notation is defined where it first appears in the text. 2. Problem setting and preliminaries As already mentioned in the introduction, the aim of this paper is to study multiobjective optimal control problems of the type Minimize J1(y, u) := j1(y) + 1 2 kuk2 L2 . . . JN (y, u) := jN (y) + N 2 kuk2 L2 w.r.t. u L2 (), y H1 0 () H2 (), s.t. - y + max(0, y) = u a.e. in . (P) Our standing assumptions on the quantities in (P) are as follows: Assumption 2.1 (Standing assumptions for the study of problem (P)). i) Rd , d 1, is a bounded domain that is convex or possesses a C1,1 -boundary (in the sense of [25], Sect. 6.2). ii) jn : H1 0 () H2 () R, n = 1, . . . , N, N N, are functions that are weakly lower semicontinuous, continuously differentiable, and bounded from below. iii) n, n = 1, . . . , N, are given non-negative real numbers and N is positive. Note that the PDE in (P) is uniquely solvable for all u L2 () by the theorem of Browder and Minty, [51], Theorem 3-1.5. To be more precise, we have: Proposition 2.2 (Properties of the PDE in (P)). For every right-hand side u L2 (), there exists a unique solution y H1 0 () H2 () of the partial differential equation - y + max(0, y) = u a.e. in . (2.1) Further, the solution operator S : L2 () H1 0 () H2 (), u 7 y, associated with the PDE (2.1) satisfies: i) S is globally Lipschitz continuous, i.e., there exists an absolute constant C > 0 with kS(u1) - S(u2)kH2 Cku1 - u2kL2 u1, u2 L2 (). ii) S is weakly continuous, i.e., for every u L2 () it holds uk L2 * u = S(uk) H2 * S(u). (2.2) 6 C. CHRISTOF AND G. MULLER iii) S is strongly and weakly Hadamard directionally differentiable in every point u L2 () in every direc- tion v L2 (), i.e., for all u, v L2 (), there exists a unique S0 (u; v) H1 0 () H2 () such that the implications vk L2 * v, tk 0+ = S(u + tkvk) - S(u) tk H2 * S0 (u; v) and vk L2 v, tk 0+ = S(u + tkvk) - S(u) tk H2 S0 (u; v) hold. Moreover, the directional derivative v := S0 (u; v) H1 0 () H2 () in a point u with state y := S(u) in a direction v is uniquely characterized by the partial differential equation - v + 1{y=0} max(0, v) + 1{y>0}v = v a.e. in . (2.3) iv) S is Gateaux differentiable in a point u L2 () with state y := S(u) (i.e., S0 (u; ·) is an element of the space L(L2 (), H1 0 () H2 ())) if and only if the set {y = 0} has measure zero. Proof. All assertions of the proposition have been proved in [14], Proposition 2.1, Theorem 2.2, Corollary 2.3, 3.8. Note that, under our assumptions on , the space (Y, k·kY ) used in [14] is isomorphic to (H1 0 ()H2 (), k·kH2 ) by [25], Theorem 9.15, Lemma 9.17 and [28], Theorem 3.2.1.2. The results of [14] thus indeed yield the asserted mapping properties of the solution operator S. As usual in the context of multiobjective optimization, in the remainder of this paper, we are interested in finding controls u L2 () that yield ­ at least in some sense ­ an optimal compromise between the different objective functions Jn(S(·), ·), n = 1, . . . , N, of (P). The notions of optimality that we are mainly concerned with in our analysis are the following (cf. [30], Defs. 3.1, 3.2 and also [22, 39]): Definition 2.3 (Notions of Pareto optimality). A control u L2 () with associated state y := S(u) is called: i) a local weak Pareto optimum of (P) if there exists an r > 0 such that there is no u L2 () satisfying ku - ukL2 < r, Jn(S(u), u) < Jn(y, u) n = 1, . . . , N. ii) a local Pareto optimum of (P) (in the ordinary sense) if there exists an r > 0 such that there is no u L2 () satisfying ku - ukL2 < r, Jn(S(u), u) Jn(y, u) n = 1, . . . , N, Jn(S(u), u) < Jn(y, u) for at least one n {1, . . . , N}. iii) a local proper Pareto optimum of (P) (in the sense of Geoffrion) if there exist constants r, C > 0 such that, for every control u L2 () satisfying ku - ukL2 < r and Jl(S(u), u) < Jl(y, u) for some l {1, . . . , N}, there exists an index m {1, . . . , N} with Jl(y, u) - Jl(S(u), u) C Jm(S(u), u) - Jm(y, u) . iv) a global weak/ordinary/proper Pareto optimum, respectively, of (P) if the condition in i)/ii)/iii), respectively, holds with r = . MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 7 Note that we trivially have u properly Pareto optimal u Pareto optimal u weakly Pareto optimal, (2.4) and that the concepts of local (respectively, global) weak, ordinary, and proper Pareto optimality coincide with each other and the classical notion of local (respectively, global) optimality in the single-objective case N = 1. Using our standing assumptions and the properties of the map S in Proposition 2.2, it is easy to prove: Theorem 2.4 (Existence of proper Pareto optima). There exists at least one global proper Pareto optimum u L2 () of the problem (P). Proof. We use a scalarization approach, cf. [22], Theorem 3.11 and [39]: Consider the auxiliary problem min uL2() N X n=1 jn(S(u)) + n 2 kuk2 L2 . (2.5) Then, it follows from our assumptions on jn and n, the weak continuity of the map S in Proposition 2.2ii), and the weak lower semicontinuity of convex and continuous functions that the objective of (2.5) is weakly lower semicontinuous, bounded from below, and radially unbounded as a function from L2 () to R. These properties imply, in combination with the direct method of calculus of variations, that (2.5) admits at least one global minimum u L2 (), i.e., at least one u L2 () with associated state y := S(u) such that N X n=1 Jn(y, u) N X n=1 Jn(S(u), u) u L2 (). (2.6) We claim that this u is also a global proper Pareto optimum of the problem (P). To see this, suppose that we are given a u L2 () that satisfies Jl(S(u), u) < Jl(y, u) for some l {1, . . . , N}. Then, (2.6) yields 0 < Jl(y, u) - Jl(S(u), u) X n6=l Jn(S(u), u) - Jn(y, u) N max n=1,...,N Jn(S(u), u) - Jn(y, u) , (2.7) and we may deduce that there exists an m {1, . . . , N} with Jl(y, u) - Jl(S(u), u) N Jm(S(u), u) - Jm(y, u) . This shows that the condition in Definition 2.3iii) holds (with r = and C = N) and completes the proof. 3. First-order necessary optimality conditions in primal form Having discussed the properties of the PDE (2.1) and the solvability of the problem (P), we now turn our attention to first-order necessary optimality conditions. We begin with "purely primal" optimality conditions that rely only on directional derivatives and do not involve additional multipliers. In the finite-dimensional setting, such conditions have already been discussed, for instance, in [26], Section 4 and [30], Section 3. Theorem 3.1 (First-order necessary optimality conditions in primal form). i) If u L2 () is a local weak Pareto optimum of (P) with associated state y := S(u), then there exists no v L2 () satisfying hj0 n(y), S0 (u; v)iH1 0 H2 + n (u, v)L2 < 0 n = 1, . . . , N. (3.1) 8 C. CHRISTOF AND G. MULLER ii) If u L2 () is a local proper Pareto optimum of (P) with state y := S(u) and constants r, C > 0 as in Definition 2.3iii), then, for every direction v L2 () satisfying hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 < 0 for some l {1, . . . , N}, there exists an index m {1, . . . , N} with - hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 C hj0 m(y), S0 (u; v)iH1 0 H2 + m (u, v)L2 . (3.2) Proof. From our assumptions on the functions jn, the differentiability properties of the solution map S in Proposition 2.2, and the chain rule, [11], Proposition 2.47, it follows straightforwardly that lim t0+ Jn(S(u + tv), u + tv) - Jn(S(u), u) t = hj0 n(S(u)), S0 (u; v)iH1 0 H2 + n (u, v)L2 (3.3) holds for all u, v L2 () and all n = 1, . . . , N. Suppose now that we are given a local weak Pareto optimum u L2 () of (P) such that there exists a direction v L2 () with (3.1). Then, (3.3) yields that we can find arbitrarily small numbers t > 0 with Jn(S(u+tv), u+tv)-Jn(S(u), u) < 0 for all n = 1, . . . , N. This contradicts the local weak Pareto optimality of u, shows that every local weak Pareto optimum has to satisfy the condition in i), and proves the first part of the theorem. To establish ii), we can proceed along similar lines: If we are given a local proper Pareto optimum u L2 () with state y := S(u) and constants r, C > 0 as in Definition 2.3iii) and a v L2 () satisfying hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 < 0 for some l {1, . . . , N}, then (3.3) yields that Jl(S(u + tv), u + tv) - Jl(y, u) < 0 holds for all sufficiently small t > 0, and we may use the condition in Definition 2.3iii) to deduce that there exist an m {1, . . . , N} and a sequence {tk} R+ with tk 0+ and 0 < Jl(y, u) - Jl(S(u + tkv), u + tkv) C Jm(S(u + tkv), u + tkv) - Jm(y, u) for all k N. Due to (3.3) and the properties of l, the above implies 0 < - hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 C hj0 m(y), S0 (u; v)iH1 0 H2 + m (u, v)L2 . This establishes (3.2) and completes the proof. Theorem 3.1 motivates the following definition: Definition 3.2 (Notions of stationarity). A control u L2 () with associated state y := S(u) is called: i) a weakly Pareto stationary point of (P) if there is no v L2 () satisfying hj0 n(y), S0 (u; v)iH1 0 H2 + n (u, v)L2 < 0 n = 1, . . . , N. (3.4) ii) a Pareto stationary point of (P) if there is no v L2 () satisfying hj0 n(y), S0 (u; v)iH1 0 H2 + n (u, v)L2 0 n = 1, . . . , N, hj0 n(y), S0 (u; v)iH1 0 H2 + n (u, v)L2 < 0 for at least one n {1, . . . , N}. (3.5) iii) a properly Pareto stationary point of (P) if there exists a C > 0 such that, for every v L2 () satisfying hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 < 0 for some l {1, . . . , N}, there is an index m {1, . . . , N} with - hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 C hj0 m(y), S0 (u; v)iH1 0 H2 + m (u, v)L2 . (3.6) MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 9 Some remarks are in order regarding the concepts in Definition 3.2: Remark 3.3. i) Pareto optima satisfying the condition in Definition 3.2ii) are sometimes also referred to as "efficient in Kuhn and Tucker's sense" or "KT-proper", see [22], Definition 2.49 and [30], Definition 3.3. ii) Analogously to the hierarchy in (2.4), we have u properly stationary u stationary u weakly stationary. iii) The conditions i), ii), and iii) in Definition 3.2 are equivalent to the global weak, global ordinary, and global proper Pareto optimality of the function v = 0 in the first-order approximation of (P) defined by Minimize J1(y, u) + hj0 1(y), S0 (u; v)iH1 0 H2 + 1 (u, v)L2 . . . JN (y, u) + hj0 N (y), S0 (u; v)iH1 0 H2 + N (u, v)L2 w.r.t. v L2 (), respectively. This shows in particular that the concepts in Definition 3.2 extend the notion of Bouligand stationarity, see [20], Definition 5.4, to the multiobjective setting. (It is easy to check that, in the single- objective case N = 1, all of the conditions in Definition 3.2 coincide with each other and with the notion of Bouligand stationarity.) iv) While the conditions i) and iii) in Definition 3.2 are always necessary for weak and proper Pareto optimality, respectively, by Theorem 3.1, the property in Definition 3.2ii) is typically not a necessary condition for ordi- nary Pareto optimality. Compare, e.g., with the simple bi-criterial optimization problem min(f1(x), f2(x)) with f1(x) := -x and f2(x) := x3 in this context, where the point x := 0 is a Pareto optimum but does not satisfy ii). In the finite-dimensional setting, the necessity of ii) can be recovered under a generalized Abadie constraint qualification, see [26], Theorem 4.1. Due to their reliance on the non-linear directional derivative S0 (u; ·) of the solution map S and their for- mulation as variational inequalities, the necessary optimality conditions in Theorem 3.1 and the stationarity concepts in Definition 3.2 are typically not very useful in practical applications. In the following Sections 4 and 5, we will derive multiplier systems that are easier to work with and more suitable as starting points for the development of numerical solution algorithms. 4. Strong stationarity conditions for the problem (P) The aim of this section is to establish so-called strong stationarity conditions for the multiobjective optimal control problem (P), i.e., multiplier systems that are equivalent to the purely primal necessary optimality conditions in Definition 3.2. As already pointed out in the introduction, in the single-objective setting, such stationarity systems are well-known for various problem classes. See, e.g., [13, 14, 18, 19, 38, 40, 41] and the references therein for some examples. In the multiobjective context, the situation is different since the additional layer of non-smoothness in (1.1) and the non-differentiability of the control-to-state mapping S create a nested structure that is generally hard to handle analytically. In the following, we will show that, for the problem (P), the difficulties arising from the doubly non-smooth behavior in (1.1) can be resolved by exploiting the self-adjointness property of the operator S0 (u; ·)-1 in (1.2). The starting point of our investigation is: Lemma 4.1 (Behavior of higher-order weak derivatives on level sets). Suppose that a function w Wk,1 (), k N, is given. Then, for every Nd 0 with 1 || k and every b R, it holds w = 0 a.e. in {w = b}. Proof. We use induction w.r.t. the absolute value l := || of the multi-index to establish the claim: Suppose that an arbitrary but fixed k N and a w Wk,1 () are given. Then, for every i = 1, . . . , d, the classical lemma of Stampacchia, see [2], Proposition 5.8.2, implies that (iw)1{w=b} = 0 holds as an identity in L2 (). This proves the assertion for l = 1. It remains to perform the induction step l 7 l + 1. To this end, let us assume 10 C. CHRISTOF AND G. MULLER that a multi-index Nd 0 with || = l + 1 k, l 1, is given. Then, we can find multi-indices , Nd 0 with || = l, || = 1, and = + . From the induction hypothesis, we obtain that w vanishes a.e. on {w = b}, i.e., it holds ( w)1{w=b} = 0 L2 () and, as a consequence, 1{w=b} = 1{ w=0}1{w=b} L2 (). Using again the classical version of Stampacchia's lemma, we further obtain that (+ w)1{ w=0} = 0 L2 (). Combining the last two identities yields that ( w)1{w=b} = (+ w)1{ w=0}1{w=b} = 0 L2 (). Thus, w = 0 a.e. in {w = b} and the induction step is complete. This proves the claim of the lemma. We remark that, in a less general format, the above result has already been used in [15], proof of Lemma A.1 and [16], proof of Theorem 2.2. By applying Lemma 4.1 to the PDE (2.3), it is straightforward to check that the directional derivative S0 (u; ·) indeed satisfies the identity (1.2). To be more precise, we have: Lemma 4.2 (Properties of S0 (u; ·) and S0 (u; ·)-1 ). Consider an arbitrary but fixed control u L2 () with state y := S(u). Then, the following is true: i) The map S0 (u; ·) : L2 () H1 0 () H2 () is bi-Lipschitz and its inverse is given by S0 (u; ·)-1 : H1 0 () H2 () L2 (), w 7 -w + 1{y=0} max(0, w) + 1{y>0}w. (4.1) ii) The map S0 (u; ·)-1 : H1 0 () H2 () L2 () admits a unique, globally Lipschitz continuous extension S0 (u; ·)-1 : L2 () (H1 0 () H2 ()) , and this extension satisfies S0 (u; ·)-1 (w), z H1 0 H2 = Z w(-z) + 1{y=0} max(0, w)z + 1{y>0}wz dx z H1 0 () H2 () w L2 (). (4.2) iii) For every z H1 0 () H2 (), it holds u, S0 (u; ·)-1 (z) L2 = - u + 1{y>0}u, z H1 0 H2 . (4.3) Here, u (H1 0 () H2 ()) denotes the "very weak" Dirichlet Laplacian of the function u L2 (), i.e., the unique element of (H1 0 () H2 ()) that satisfies the adjoint relation (u, z)L2 = hu, ziH1 0 H2 for all z H1 0 () H2 (). iv) For every z H1 0 () H2 (), it holds u, S0 (u; ·)-1 (z) L2 = S0 (u; ·)-1 (u), z H1 0 H2 . Proof. Part i) of the lemma is a trivial consequence of Proposition 2.2. To prove ii), we note that the right-hand side of (4.2) defines a globally Lipschitz continuous map from L2 () to (H1 0 () H2 ()) that coincides with S0 (u; ·)-1 on H1 0 () H2 () by Green's formula. (Recall that H1 0 () H2 (), L2 (), and (H1 0 () H2 ()) are interpreted as a Gelfand triple, i.e., H1 0 ()H2 () , L2 () , (H1 0 ()H2 ()) .) The function S0 (u; ·)-1 thus admits an extension with the desired properties. Since H1 0 () H2 () is dense in L2 (), we further know that there can only be one L2 -Lipschitz continuous extension of S0 (u; ·). This establishes ii). It remains to prove iii) and iv). To this end, we note that Lemma 4.1 and the definition of S imply Z 1{y=0} max(0, z)u dx = Z 1{y=0} max(0, z)(-y + max(0, y))dx = 0 MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 11 and, analogously, Z 1{y=0}z max(0, u)dx = Z 1{y=0}z max(0, -y + max(0, y))dx = 0 for all z H1 0 () H2 (). Combining the above with (4.1) and (4.2) yields u, S0 (u; ·)-1 (z) L2 = Z u -z + 1{y=0} max(0, z) + 1{y>0}z dx = Z u -z + 1{y>0}z dx = Z u(-z) + 1{y=0} max(0, u)z + 1{y>0}uz dx = S0 (u; ·)-1 (u), z H1 0 H2 for all z H1 0 () H2 (). This establishes iii) and iv) and completes the proof. Remark 4.3. An argument based on Stampacchia's lemma similar to that in the proof of Lemma 4.2 has also been used in [18], Section 5 for the analysis of the solution map of a quasilinear partial differential equation involving a term of the form g(y)y with a piecewise smooth g : R R. For this PDE, the lemma of Stampacchia for first weak derivatives allows to show that all terms that could possibly prevent the expression g0 (y; z)y from being linear in z are negligible, and to establish that the solution map of the considered quasilinear partial differential equation is Gateaux differentiable in spite of the fact that it contains the non-smooth Nemytskii operator g. Compare also with Theorem 6.1 and Remark 6.3 in this context. We point out that, for the PDE (2.1), the situation is different since the maps S, S0 (u; ·), and S0 (u; ·)-1 are not Gateaux differentiable but contain "proper" non-differentiabilities, cf. Proposition 2.2iv) and also the example in Section 6. As we will see below, Lemma 4.2 makes it possible to handle the non-linearity of the directional derivative S0 (u; ·) in the necessary optimality conditions (3.4), (3.5), and (3.6). To deal with the multiobjective aspect in these conditions, we need the following infinite-dimensional version of Tucker's/Motzkin's theorem of the alternative (cf. [22], Thms. 3.22, 3.24 and also [26], Prop. 2.2): Lemma 4.4 (Existence of multipliers in general Hilbert spaces). Suppose that V is a real Hilbert space and that w 1,. . . , w N , N N, are given elements of V . Then, it holds @z V : hw n, ziV < 0 n = 1, . . . , N RN : n 0 n = 1, . . . , N, N X n=1 n = 1, N X n=1 nw n = 0 (4.4) and @z V : hw n, ziV 0 n = 1, . . . , N, hw n, ziV < 0 for at least one n RN : n > 0 n = 1, . . . , N, N X n=1 n = 1, N X n=1 nw n = 0. (4.5) Proof. We begin with (4.4): If we assume that the right-hand side of (4.4) holds and that there exists a z V with hw n, ziV < 0 for all n = 1, . . . , N, then we arrive at the contradiction 0 = * N X n=1 nw n, z + V < 0. 12 C. CHRISTOF AND G. MULLER This proves "". To establish "", we note that the condition on the left-hand side of (4.4) can be recast as max (hw 1, ziV , . . . , hw N , ziV ) 0 z V, or, equivalently, as 0 c(g F)(0), where c denotes the convex subdifferential and where g and F are the functions defined by g : RN R, g(x1, . . . , xN ) := max(x1, . . . , xN ) and F : V RN , F(z) := (hw n, ziV )n=1,...,N . Using the chain rule for the convex subdifferential, see [23], Proposition I-5.7, and the formula for the subdifferential of the vector-maximum ([50], Ex. 8.26), we may now deduce that 0 F cg(F(0)) = F cg(0) = F ( µ RN µn 0 n = 1, . . . , N, N X n=1 µn = 1 ) . This proves the existence of a RN with the properties on the right-hand side of (4.4) and establishes "". It remains to show (4.5). As in the case of (4.4), the implication "" in (4.5) follows immediately by contradiction. To prove "", we note that, by the Riesz representation theorem, we can find elements wn V , n = 1, . . . , N, such that hw n, ·iV = (wn, ·)V holds for all n. Suppose that at least one of these wn is not zero (else the proof is trivial) and denote the subspace spanned by the elements wn with W. Then, W is obviously finite-dimensional and possesses a (·, ·)V -orthonormal basis e1, . . . , eM , 1 M N. Let n m, n = 1, . . . , N, m = 1, . . . , M, be the coordinates of wn w.r.t. the basis {em}. Then, it holds * w n, M X m=1 mem + V = M X l=1 n l el, M X m=1 mem ! V = M X m=1 n mm for all RM , and we may use the left-hand side of (4.5) to conclude that @ RM : A (-, 0]N , A 6= 0, where A RN×M is the matrix defined by A := (n m)n=1,...,N,m=1,...,M . From the classical, finite-dimensional version of Tucker's theorem of the alternative, see [37], Theorem II-4.3, we may now deduce that there exists a RN satisfying AT = 0, PN n=1 n = 1, and n > 0 for all n = 1, . . . , N, and from the definition of A that N X n=1 nwn = M X m=1 N X n=1 nn mem = 0. This establishes the implication "" in (4.5) and completes the proof. MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 13 We are now in the position to prove the main result of this section and the paper as a whole: Theorem 4.5 (Strong stationarity conditions). i) A control u L2 () with state y := S(u) is weakly Pareto stationary for (P) if and only if there exist an adjoint state p and a multiplier such that u, y, p, and satisfy the system u, p L2 (), y H1 0 () H2 (), RN , n 0 n = 1, . . . , N, N X n=1 n = 1, -y + max(0, y) = u, -p + 1{y>0}p = N X n=1 nj0 n(y), p + N X n=1 nnu = 0. (4.6) ii) A control u L2 () with state y := S(u) is Pareto stationary for (P) (in the ordinary sense) if and only if there exist an adjoint state p and a multiplier such that u, y, p, and satisfy the system u, p L2 (), y H1 0 () H2 (), RN , n > 0 n = 1, . . . , N, N X n=1 n = 1, -y + max(0, y) = u, -p + 1{y>0}p = N X n=1 nj0 n(y), p + N X n=1 nnu = 0. (4.7) iii) A control u L2 () is Pareto stationary for (P) (in the ordinary sense) if and only if it is properly Pareto stationary for (P). Proof. We begin with i): From Definition 3.2i), equation (4.3), and Lemma 4.2i), it follows that a control u L2 () with state y := S(u) is weakly Pareto stationary for (P) if and only if @z H1 0 () H2 () : hj0 n(y), ziH1 0 H2 + n u, S0 (u; ·)-1 (z) L2 = j0 n(y) + n -u + 1{y>0}u , z H1 0 H2 < 0 n = 1, . . . , N. (4.8) Here, the Laplacian u is again understood in the very weak sense. Due to Lemma 4.4, we further know that (4.8) is equivalent to the statement RN : n 0 n = 1, . . . , N, N X n=1 n = 1, N X n=1 n j0 n(y) + n -u + 1{y>0}u = 0 H1 0 () H2 () . (4.9) If we now define p := - PN n=1 nnu, then the assertion in i) follows immediately. To establish ii), we can use exactly the same arguments as for i) (with (4.4) replaced by (4.5)). It remains to prove iii). To this end, let us suppose that u is Pareto stationary in the ordinary sense, that p and are as in (4.7), and that we are given a v L2 () with associated directional derivative z := S0 (u; v) H1 0 () H2 () such that hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 < 0 holds for some l {1, . . . , N}. Then, Lemma 4.2iii) yields hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 = j0 l(y) + l -u + 1{y>0}u , z H1 0 H2 , 14 C. CHRISTOF AND G. MULLER and we may use the strong stationarity system (4.7) and the same arguments as in (4.9) to deduce that N X n=1 n j0 n(y) + n -u + 1{y>0}u , z H1 0 H2 = 0. The above implies (completely analogously to (2.7), again by Lemma 4.2iii), and since n > 0 for all n) that 0 < - hj0 l(y), S0 (u; v)iH1 0 H2 + l (u, v)L2 = 1 l X n6=l n j0 n(y) + n -u + 1{y>0}u , z H1 0 H2 1 l max n=1,...,N j0 n(y) + n -u + 1{y>0}u , z H1 0 H2 min n=1,...,N n -1 max n=1,...,N hj0 n(y), S0 (u; v)iH1 0 H2 + n (u, v)L2 . The proper Pareto stationarity of u now follows immediately, see Definition 3.2iii). Since the reverse implication is trivial, this completes the proof. Several things are noteworthy regarding the last result: Remark 4.6. i) The structure of the strong stationarity systems in Theorem 4.5 is completely analogous to that of classi- cal first-order necessary optimality conditions for smooth, finite-dimensional multiobjective optimization problems, see, e.g., [22], Section 3.3, [39], Section 3.1.1, and [27], Section 4. In particular, (4.6) and (4.7) do not contain any kind of evaluation of the subdifferential of the function max(0, ·). Section 6 will give an explanation for this behavior. ii) From [14], Theorem 3.18, it follows straightforwardly that the lines -p + 1{y>0}p = N X n=1 nj0 n(y) and p + N X n=1 nnu = 0 in the strong stationarity conditions (4.6) and (4.7) imply the existence of a generalized derivative G ss B S(u) with N X n=1 n G j0 n(y) + nu = 0. (4.10) Here, ss B S(u) L(L2 (), H1 0 () H2 ()) is the strong-strong Bouligand subdifferential of the solution map S associated with (2.1) in the sense of [14], Definition 3.1, and G L((H1 0 () H2 ()) , L2 ()) denotes the adjoint of G. Due to the chain rule for the Bouligand subdifferential B and the fact that the Bouligand subdifferential is smaller than the subdifferential of Clarke (which we denote by C in the following), (4.10) further yields that 0 N X n=1 n B(jn S)(u) + nu N X n=1 nB Jn(S(·), ·) (u) N X n=1 nC Jn(S(·), ·) (u). MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 15 The strong stationarity conditions (4.6) and (4.7) thus imply that there exist elements of the Clarke sub- differentials CJn(S(·), ·) of the reduced objective functions Jn(S(·), ·), n = 1, . . . , N, of (P) at u such that a linear combination of these elements with the coefficients n vanishes. Note that this is precisely the "standard" first-order necessary optimality condition for non-smooth multiobjective optimization prob- lems, see, e.g., [10, 30, 39]. The systems (4.6) and (4.7) are thus more rigorous than ordinary optimality conditions based on Clarke's generalized differential (and also stronger than conditions which exploit the subdifferential calculus of [14]). This, along with the equivalence to the purely primal stationarity concepts in Definition 3.2, justifies calling (4.6) and (4.7) strong stationarity conditions. iii) Theorem 4.5 is closely related to classical scalarization approaches (as found, e.g., in [22], Sects. 3 and 4) in the sense that the systems (4.6) and (4.7) can be identified with strong stationarity conditions (in the ordinary, single-objective sense) of optimal control problems governed by (2.1) with objective functions of the form u 7 PN n=1 nJn(S(u), u). Compare, e.g., with [14], Theorem 4.12 in this context. The main insight provided by Theorem 4.5 is, of course, that every weak, ordinary, or proper stationary point of (P) is a strongly stationary point of a suitably scalarized auxiliary problem. iv) We would like to point out that the proof of Theorem 4.5 relies heavily on the fact that the non-smooth Nemytskii operator max(0, ·) in the PDE (2.1) interacts in a special way with the Tikhonov regularization terms in the objective functions of (P) and that the control space of (P) is all of L2 (). If more general non-smooth Nemytskii operators or objective functions Jn : H1 0 () H2 () × L2 () R are considered or if the controls are subject to additional constraints, then it is still often possible to establish stationarity systems similar to those in (4.6) and (4.7) by using a scalarization approach and regularization. However, in this general situation, it is typically completely unclear how the resulting optimality conditions are related to the notions of stationarity in Definition 3.2 (or if they are of a sensible strength at all). Compare, for instance, with the optimality conditions derived by regularization for scalar optimal control problems with general objective functions governed by non-smooth semilinear PDEs in [14], Section 4.1 in this context and also with the analysis of Section 5. As a direct consequence of Theorem 4.5, we obtain: Corollary 4.7 (Increased regularity of Pareto stationary points). i) If the maps jn, n = 1, . . . , N, are continuously differentiable as functions from H1 0 () to R, i.e., if there exist continuously differentiable jn : H1 0 () R such that jn = jn EH1 0 holds for all n, where EH1 0 : H1 0 () H2 () H1 0 () denotes the canonical embedding of H1 0 () H2 () into H1 0 (), then every Pareto stationary control u of (P) is an element of H1 0 (). ii) If the maps jn, n = 1, . . . , N, are continuously differentiable as functions from L2 () to R, i.e., if there exist continuously differentiable jn : L2 () R such that jn = jn EL2 holds for all n, where EL2 : H1 0 () H2 () L2 () denotes the canonical embedding of H1 0 () H2 () into L2 (), then every Pareto stationary control u of (P) is an element of H1 0 () H2 (). Proof. From Theorem 4.5, we obtain that, for every Pareto stationary u L2 () with state y := S(u), we can find a multiplier RN and an adjoint state p L2 () with n > 0 n = 1, . . . , N, N X n=1 n = 1, p + N X n=1 nnu = 0, -p + 1{y>0}p = N X n=1 nj0 n(y). (4.11) If we assume that there exist continuously differentiable functions jn : H1 0 () R such that jn = jn EH1 0 holds for all n, then it follows from (4.11), the chain rule, and the definitions of the very weak Laplacian and 16 C. CHRISTOF AND G. MULLER the dual pairings in H1 0 () and H1 0 () H2 () that p satisfies (p, -z)L2 = h-p, ziH1 0 H2 = * -1{y>0}p + N X n=1 nj0 n(y), z + H1 0 H2 = * -1{y>0}p + N X n=1 nj0 n(EH1 0 y), z + H1 0 for all z H1 0 () H2 (). Let us denote the element of H-1 () appearing on the right-hand side of the last equation with f. Then, it follows from classical results on the Poisson problem that there exists a unique solution p H1 0 () of the variational identity (p, z)L2 = hf, ziH1 0 for all z H1 0 (). From Green's formula, we obtain that this p also satisfies (p, -z)L2 = hf, ziH1 0 = (p, -z)L2 for all z H1 0 () H2 () and, as a consequence, that (p - p, -z)L2 = 0 for all z H1 0 () H2 (). Using our assumptions on and classical results on the H2 - regularity of solutions of the Poisson problem, see [25], Theorem 9.15, Lemma 9.17 and [28], Theorem 3.2.1.2, we may further deduce that there exists a unique z H1 0 () H2 () satisfying -z = p - p L2 (). Choosing this z as a test function yields (p - p, -z)L2 = (p - p, p - p)L2 = 0. We thus have p = p H1 0 () and, since the number PN n=1 nn is positive by our assumption N > 0 and due to the properties of , u H1 0 (). This establishes i). The proof of ii) is completely along the same lines. We conclude this section with an existence result that shows that it makes sense to use the strong stationarity conditions in Theorem 4.5 for the calculation of Pareto stationary points: Lemma 4.8 (Solvability of the strong stationarity system). Suppose that a vector RN satisfying n 0 n = 1, . . . , N, N X n=1 n = 1, and N > 0 is given. Then, there exists at least one solution (u, y, p) of the system u, p L2 (), y H1 0 () H2 (), -y + max(0, y) = u, -p + 1{y>0}p = N X n=1 nj0 n(y), p + N X n=1 nnu = 0. Proof. Since N is positive and since N > 0 holds by our standing assumptions, it follows completely analogously to the proof of Theorem 2.4 that there exists at least one global solution u L2 () of the problem min uL2() N X n=1 n jn(S(u)) + n 2 kuk2 L2 , (4.12) and from the first-order necessary optimality condition of (4.12) and the properties of S in Proposition 2.2, we obtain that this u and its state y := S(u) have to satisfy N X n=1 n hj0 n(y), S0 (u; v)iH1 0 H2 + n (u, v)L2 0 v L2 (). (4.13) Using Lemma 4.2 and exactly the same arguments as in the proof of Theorem 4.5, we can rewrite (4.13) as N X n=1 n j0 n(y) + n -u + 1{y>0}u , z H1 0 H2 0 z H1 0 () H2 (), MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 17 and this, in turn, entails N X n=1 n j0 n(y) + n -u + 1{y>0}u = 0 H1 0 () H2 () . If we now define p := - PN n=1 nnu, then the claim follows immediately. Note that Theorem 4.5 and Lemma 4.8 show that, to calculate Pareto stationary points of the problem (P), it is a good strategy to fix vectors RN with n 0 for all n = 1, . . . , N and PN n=1 n = 1, and to subsequently try to solve the remaining equations in the strong stationarity condition (4.6) for (u, y, p). Due to Lemma 4.8, we know that the system that is obtained along these lines has at least one solution for all with N > 0, and from Theorem 4.5 it follows that every point that we calculate in this way is weakly or, in the case n > 0 for all n, properly Pareto stationary. Moreover, Theorem 4.5 yields that we can determine all weak (respectively, proper, respectively, ordinary) Pareto stationary points of (P) by solving systems of the form (4.6) (respectively, (4.7)). We will get back to this topic in Section 7, where we will use a similar approach in the discrete setting for our numerical experiments. 5. Relation to regularization approaches and C-stationarity The aim of this section is to demonstrate that the strong stationarity conditions in Theorem 4.5 are not only interesting for their own sake but also have quite surprising consequences for the analysis of regularization techniques for problems of the type (P). To be more precise, in what follows, we will show that, when a problem of the form (P) satisfying the conditions of Corollary 4.7i) is regularized and the regularization parameter is driven to zero, then all weak L2 -accumulation points of the weakly Pareto stationary points of the regularized multiobjective optimal control problems are weakly Pareto stationary for the unregularized limit problem (P). Note that this observation is indeed remarkable since similar effects cannot be observed even in very simple one-dimensional examples. Compare, e.g., with the situation already mentioned in the introduction where the function f(x) := -|x| is approximated by the family f(x) := - x2 + , > 0, in this context. To regularize the problem (P), we follow the lines of [14] and replace the max-function in (2.1) with a suitably chosen differentiable approximation max : R R. This gives rise to a family of regularized multiobjective optimal control problems of the form Minimize J1(y, u) := j1(y) + 1 2 kuk2 L2 . . . JN (y, u) := jN (y) + N 2 kuk2 L2 w.r.t. u L2 (), y H1 0 () H2 (), s.t. - y + max(y) = u a.e. in . (P) Our standing assumptions on the approximations max are the same as in [14]: Assumption 5.1 (Standing assumptions on the functions max). The functions max : R R, > 0, satisfy the following: i) It holds max C1 (R) for all > 0. ii) There is a constant C > 0 with | max(x) - max(0, x)| C for all x R. iii) For all x R and all > 0, we have 0 max0 (x) 1. iv) For every arbitrary but fixed > 0, the derivatives {max0 }>0 converge uniformly to one in [, ) and uniformly to zero in (-, -] for 0+ . 18 C. CHRISTOF AND G. MULLER Note that, e.g., the family max(x) := 1 2 x2 + 2 + x , > 0, has all of the above properties. Under Assumption 5.1, the following can be established for the PDE in (P), cf. [14], Section 4.1: Proposition 5.2 (Properties of the regularized PDE). For every u L2 () and every > 0, there exists a unique solution y H1 0 () H2 () of the PDE - y + max(y) = u a.e. in . (5.1) Further, the solution operator S : L2 () H1 0 () H2 (), u 7 y, associated with the partial differential equation (5.1) satisfies: i) S is weakly continuous (in the sense of (2.2)). ii) S is Frechet differentiable, and the Frechet derivative S0 (u) L(L2 (), H1 0 () H2 ()) of S in a point u L2 () is precisely the solution map v 7 v of the PDE -v + max0 (y)v = v. iii) There exists a constant C > 0 such that, for all u L2 (), it holds kS(u) - S(u)kH1 0 H2 C > 0. iv) For all {uk} L2 () and {k} (0, ) satisfying uk * u in L2 () for some u L2 () with state y := S(u) and k 0+ , there exist a subsequence {kl} and a function L () such that = 0 a.e. in {y < 0}, = 1 a.e. in {y > 0}, [0, 1] a.e. in {y = 0} holds and such that the directional derivatives S0 kl (ukl )v converge weakly in H1 0 () H2 () to the unique solution v of the partial differential equation -v + v = v for all v L2 (). Proof. The assertions of i), ii), and iii) have been proved in [14], Lemmas 4.2, 4.3. The proof of iv) is completely analogous to that of the second part of [14], Lemma 4.3 with the only difference that, in iv), we have to work with weak L2 -convergence instead of strong L2 -convergence. This, however, is not a problem: From the weak convergence uk * u in L2 (), iii), and the weak continuity of S, we obtain that Sk (uk) = S(uk) + (Sk - S)(uk) * S(u) in H1 0 () H2 (). (5.2) The above implies in particular that Sk (uk) converges to S(u) in L2 () and, at least after the transition to a subsequence (still denoted by the same symbol), that Sk (uk) S(u) = y pointwise a.e. in . Using this pointwise-a.e. convergence, we can argue exactly as in the second part of the proof of [14], Lemma 4.3 to establish the assertion of iv). This completes the proof. We are now in the position to prove that weak L2 -accumulation points of weakly Pareto stationary points of (P) are indeed weakly Pareto stationary for the original problem (P) in the situation of Corollary 4.7i). Theorem 5.3 (Preservation of weak Pareto stationarity in the limit 0+ ). Suppose that the maps jn are continuously differentiable as functions from H1 0 () to R (in the sense of Cor. 4.7i)), and that sequences MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 19 {k} (0, ) and {uk} L2 () are given such that k 0+ holds and such that uk is weakly Pareto stationary for (Pk ) for all k, i.e., such that there is no v L2 () satisfying j0 n(Sk (uk)), S0 k (uk)v H1 0 H2 + n (uk, v)L2 < 0 n = 1, . . . , N. (5.3) Then, every weak L2 -accumulation point of {uk} is weakly Pareto stationary for (P). Proof. Suppose that u L2 () is a weak L2 -accumulation point of the sequence {uk} for k with state y := S(u). Then, Proposition 5.2iv) yields that we can pass over to subsequences of {uk} and {k} (still denoted by the same symbols) such that uk * u holds in L2 () and such that there exists a L () with = 0 a.e. in {y < 0}, = 1 a.e. in {y > 0}, [0, 1] a.e. in {y = 0}, and S0 k (uk)v * v in H1 0 () H2 () for all v L2 (), where v is again defined by -v + v = v. Since the left-hand side of (5.3) is linear in v, we may further invoke Lemma 4.4 to deduce that, for all k N, we can find a k RN with k n 0 n = 1, . . . , N, N X n=1 k n = 1, N X n=1 k n j0 n(Sk (uk)), S0 k (uk)v H1 0 H2 + n (uk, v)L2 = 0 v L2 (). (5.4) Note that the first line of (5.4) particularly implies that {k } is bounded, and that our assumptions on the functions jn and (5.2) yield that j0 n(Sk (uk)) converges strongly to j0 n(y) in (H1 0 () H2 ()) for k . This allows us to find a further subsequence (again not relabeled) such that k holds for some RN , and to pass to the limit k in (5.4) to obtain that there exists a RN with n 0 n = 1, . . . , N, N X n=1 n = 1, N X n=1 n hj0 n(y), viH1 0 H2 + n (u, v)L2 = 0 v L2 (). (5.5) Due to the bijectivity of the map L2 () 3 v 7 v H1 0 ()H2 () and the identity u = (-y +max(0, y)) = 1{y>0}(-y + max(0, y)) = 1{y>0}u, see Lemma 4.1, the last equation in (5.5) can be recast as N X n=1 n hj0 n(y), ziH1 0 H2 + n (u, -z + z)L2 = N X n=1 n j0 n(y) + n -u + 1{y>0}u , z H1 0 H2 = 0 z H1 0 () H2 (). If we now define p := - PN n=1 nnu, then it follows that u satisfies the strong stationarity system (4.6) and, as a consequence, the weak Pareto stationarity condition in Definition 3.2i). This completes the proof. We would like to point out that, in the single-objective case N = 1, we can use exactly the same arguments as in the proof of Theorem 5.3 to establish that the notion of C-stationarity that is obtained by passing to 20 C. CHRISTOF AND G. MULLER the limit with the regularization parameter in the first-order necessary optimality conditions of a regularized version of (P) is equivalent to the concept of strong stationarity. Indeed, we have: Corollary 5.4 (Equivalence of C- and strong stationarity in the case N = 1). Consider the single-objective case N = 1, i.e., the case where (P) has the form Minimize J(y, u) := j(y) + 2 kuk2 L2 w.r.t. u L2 (), y H1 0 () H2 (), s.t. - y + max(0, y) = u a.e. in (5.6) with a function j : H1 0 () H2 () R as in Assumption 2.1 and > 0. Then, the notions of C-stationarity in the sense of [14], Theorem 4.4, strong stationarity in the sense of [14], Theorem 4.12, and purely primal stationarity in the sense of [14], Proposition 4.10 are the same. Proof. The equivalence between strong and primal stationarity follows from [14], Proposition 4.13 and the remaining equivalences are straightforward consequences of the systems (26) and (32) in [14], the identity p = -u in the stationarity conditions, and the fact that the controls u = -y + max(0, y) L2 () in (5.6) always vanish a.e. in the set {y = 0} by Lemma 4.1. We remark that an effect similar to that in Corollary 5.4 has already been observed for optimal control problems governed by a class of quasilinear PDEs with Gateaux differentiable solution maps in [18], Section 5, cf. Remark 4.3. What is noteworthy about Corollary 5.4 is that, in the case of the problem (P), the notions of strong and C-stationarity are identical in spite of the fact that the solution operator S : u 7 y associated with (2.1) typically does not possess a Gateaux derivative, cf. Proposition 2.2iv). As we will see in the next section, this behavior can be explained with "hidden" smoothness properties of the problem (P) that only reveal themselves when (P) is considered as an optimization problem in the variable y. 6. Hidden smoothness properties and an alternative view on strong stationarity conditions One might ask at this point how the effects observed in the last two sections - the self-adjointness of the non-differentiable and non-linear operator S0 (u; ·)-1 in (1.2), the similarity of the strong stationarity conditions (4.6) and (4.7) in Theorem 4.5 to classical optimality conditions for smooth multiobjective optimal control problems, and the equivalence of the various notions of stationarity, e.g., in Corollary 5.4 ­ are possible. The answer to this question is that the problem (P) is actually not as non-smooth as it appears at first glance. To see this, let us consider an arbitrary but fixed index n {1, . . . , N} and the associated objective Jn(y, u) := jn(y) + n 2 kuk2 L2 (6.1) in (P). Using standard arguments, it is easy to check that the function Jn in (6.1) is typically non-differentiable when we reduce it to the variable u by expressing y in terms of the solution map S of the partial differential equation -y + max(0, y) = u in (2.1). Indeed, in the simple example d := 1, := (-1, 1), jn(y) := (y(0) + 1)2 , n := 1, (6.2) a short calculation shows S(1)(x) = 1 - 2e 1 + e2 cosh(x) if 0 1 2 (1 - x2 ) if < 0 , R, MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 21 where 1 denotes the element of L2 () that is identical one a.e. in , and we obtain Jn(S(1), 1) = (1 - e)2 1 + e2 + 1 2 + 2 if 0 1 2 + 1 2 + 2 if < 0 , R. The reduced objective function u 7 Jn(S(u), u) thus has a proper kink at the origin in the situation of (6.2) and is non-differentiable as claimed. Note that this example also demonstrates that it makes sense to refer to the problem (P) as a non-smooth multiobjective optimal control problem. In what follows, we will see that the latter term is, in fact, not entirely appropriate anymore when we reduce Jn not to the control u but to the state y. (A reduction that is rarely used in the context of optimal control since y is, per definition, not the quantity that one has under control in the context of the problem (P).) To begin our analysis, we observe that, by eliminating the variable u from Jn, we arrive at a reduced objective function of the form Jn(y, S-1 (y)) = jn(y) + n 2 k - y + max(0, y)k2 L2 = jn(y) + n 2 kyk2 L2 + n 2 k max(0, y)k2 L2 - n Z max(0, y)y dx. (6.3) Note that the first three terms on the right-hand side of (6.3) are trivially Gateaux differentiable as functions from H1 0 () H2 () to R, so that the only remaining potentially non-smooth term in (6.3) is the one involving the expression max(0, y)y. For this type of product, however, we have the following key result: Theorem 6.1 (Compensation of non-smoothness by weak derivatives). Let k N and r, q [1, ) be given such that r [1, q) and Wk,q () , Lqr/(q-r) () holds, and let g : R R be a globally Lipschitz continuous function. Suppose further that there exists a countable set N R such that g possesses a classical derivative at all points x R \ N. Then, the map G : Wk,q () Lr (), w 7 g(w) w, is well-defined and Gateaux differentiable for all multi-indices Nd 0 with || = k (in the sense that directional derivatives exist and depend linearly and continuously on the direction) and the Gateaux derivative of G in a point w in a direction z is given by G0 (w)z = 1{w / N}g0 (w) w z + g(w) z. (6.4) Proof. Suppose that k, q, r, and g satisfy the assumptions of the theorem, and assume that an arbitrary but fixed Nd 0 with || = k is given. Then, Holder's inequality, the triangle inequality, and the global Lipschitz continuity of g imply that there exists an absolute constant C > 0 with kg(w) wkLr k(g(w) - g(0)) wkLr + kg(0) wkLr C kw wkLr + C k wkLq CkwkLqr/(q-r) k wkLq + C k wkLq w Wk,q (). (6.5) This proves that G is well-defined as a function from Wk,q () to Lr (). It remains to show that G is Gateaux differentiable. To this end, we note that the same arguments as in (6.5), the dominated convergence theorem, 22 C. CHRISTOF AND G. MULLER and Lemma 4.1 yield that, for all w, z Wk,q (), we have 0 g(w + tz) (w + tz) - g(w) w t - 1{w / N}g0 (w) w z - g(w) z Lr g(w + tz) - g(w) t - 1{w / N}g0 (w)z w Lr + g(w + tz) - g(w) z Lr = g(w + tz) - g(w) t - g0 (w)z 1{w / N} w Lr + g(w + tz) - g(w) z Lr g(w + tz) - g(w) t - g0 (w)z 1{w / N} Lqr/(q-r) k wkLq + Ct kzkLqr/(q-r) k zkLq 0 for t 0+ , where C again denotes a generic constant. The function G is thus indeed directionally differentiable with the derivative in (6.4). Since the linearity and the continuity of the map Wk,q () 3 z 7 G0 (w; z) Lr () are trivial, the assertion of the theorem now follows immediately. If we consider the special case k = 2, q = 2, and r = 1, then Theorem 6.1 implies in particular that the function F : H1 0 () H2 () L1 (), y 7 max(0, y)y, is Gateaux differentiable with derivative F0 (y)z = 1{y>0}y z + max(0, y)z for all y, z H1 0 () H2 (). The map Jn(·, S-1 (·)) in (6.3) thus possesses a Gateaux derivative, and we arrive at the ­ quite counterintuitive ­ conclusion that, although typically non-smooth as a function of the control u, the objective Jn in (6.1) is always Gateaux differentiable when reduced to the state y. Note that the rather peculiar behavior that we observe here is the main reason why the analysis of Section 4 works as well as it does. Indeed, by invoking (6.4), we obtain that the Gateaux derivative of the map Jn(·, S-1 (·)) in (6.3) at a point y H1 0 () H2 () is precisely the functional j0 n(y) + n y + max(0, y) - 1{y>0}y - max(0, y) (H1 0 () H2 ()) , where the Laplacian is again understood in the very weak sense. If we now assume that we are given a state y which is Pareto stationary for the reduced multiobjective optimization problem Minimize J1(y, S-1 (y)) = j1(y) + 1 2 kS-1 (y)k2 L2 . . . JN (y, S-1 (y)) = jN (y) + N 2 kS-1 (y)k2 L2 w.r.t. y H1 0 () H2 (), (6.6) i.e., a y H1 0 () H2 () with the property that there is no z H1 0 () H2 () with j0 n(y) + n y + max(0, y) - 1{y>0}y - max(0, y) , z H1 0 H2 0 n = 1, . . . , N, j0 n(y) + n y + max(0, y) - 1{y>0}y - max(0, y) , z H1 0 H2 < 0 for at least one n {1, . . . , N}, (6.7) MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 23 then Lemma 4.4 yields that there exists a vector RN satisfying n > 0 n = 1, . . . , N, N X n=1 n = 1, N X n=1 n j0 n(y) + n y + max(0, y) - 1{y>0}y - max(0, y) = 0. (6.8) With the definitions u := -y + max(0, y) and p := - PN n=1 nnu, (6.8) takes precisely the form of the strong stationarity system (4.7). We may thus conclude: Theorem 6.2 (Alternative interpretation of strong stationarity conditions). The strong stationarity conditions (4.6) and (4.7) are precisely the conditions of weak and ordinary Pareto stationarity of the state y in the reduced multiobjective optimal control problem (6.6). In particular, a control u is a weakly/ordinarily Pareto stationary point (in the sense of Def. 3.2) for the problem (P) interpreted as a problem in u if and only if the state y := S(u) is a weakly/ordinarily Pareto stationary point (in the sense of (6.7)) for the problem (P) interpreted as a problem in y. Proof. The equivalence between the Pareto stationarity conditions of (6.6) and the strong stationarity conditions (4.6) and (4.7) follows straightforwardly from the arguments outlined after the proof of Theorem 6.1. (Note that we indeed obtain an "if and only if" here due to (4.4) and (4.5).) This proves the first part of the theorem. The second one is an immediate consequence of Theorem 4.5. Note that, in view of the above observations, it indeed makes sense that the strong stationarity conditions (4.6) and (4.7) have a structure similar to that of classical necessary optimality conditions for smooth multiobjective optimal control problems, cf., e.g., the results in [29], Theorem 2.6 and [22, 39]. We would like to point out that the identification results in Theorem 6.2 also suggest an alternative approach to the strong stationarity conditions in Theorem 4.5: Instead of the strategy pursued in Section 4, which consists of starting with the purely primal concepts in Definition 3.2 and subsequently applying the adjoint calculus of Lemma 4.2 for the derivation of (4.6) and (4.7), one can also start with the problem (6.6) formulated in the variable y, use the arguments after the proof of Theorem 6.1 to show that the necessary conditions for weak and ordinary Pareto stationarity in (6.6) are equivalent to (4.6) and (4.7), and subsequently prove that a state y is a weak/ordinary Pareto stationary point of (6.6) if and only if the associated control u is a weak/ordinary Pareto stationary point for the problem (P) when it is reduced to the control u. We remark that the last step in this alternative argumentation requires roughly the same effort as the analysis of Section 4. The amount of work needed for the derivation of Theorems 6.2 and 4.5 is thus essentially independent of the approach that one chooses to take here. We conclude this section with some further comments on how the results of the last three sections are related to the known literature: Remark 6.3. i) The differentiability properties in Theorem 6.1 are also the reason for the Gateaux differentiability of the solution map of the partial differential equation studied in [18], Section 5. Note that, in contrast to the problem in [18], where the weak formulation of the governing PDE already contains a term of the form g(y)y with a piecewise smooth g : R R, in the case of the problem (P), the smoothness effects that we have explored in this section only occur since the objective functions Jn in (P) have a composite structure, i.e., can be split into an observation part and an L2 -Tikhonov term. Compare, for instance, with the calculation in (6.3) in this context. Without such a structure, the Gateaux differentiability of the functions Jn(·, S-1 (·)) cannot be guaranteed and the situation is much less clear. (When considering general objective functions, one can, of course, still employ the subdifferential calculus of [14].) 24 C. CHRISTOF AND G. MULLER ii) The analysis of this section and, in particular, the Gateaux differentiability of the reduced objective function in (6.3) are also new and relevant in the single-objective setting. Using Theorem 6.1 and a calculation analogous to that in (6.3), we obtain, e.g., that the problems considered in [14], Section 5 and [21], Section 5, Case 1 are all Gateaux differentiable when reduced to the state y. To the best of the authors' knowledge, this observation has not been made so far in the literature. 7. Numerical experiments The results of the last sections suggest (at least) two different approaches for the numerical approximation of the set of weak/ordinary Pareto stationary points of a problem of the type (P): First, we can try to tackle the strong stationarity conditions (4.6) and (4.7) in Theorem 4.5 directly with a generalized Newton method or a comparable algorithm. Recall that, by Lemma 4.8, we know that the systems (4.6) and (4.7) have at least one solution for all with N > 0, and that Theorem 4.5 guarantees that all controls u that we determine by solving a system of the form (4.6) or (4.7), are Pareto stationary in the weak or ordinary sense, depending on the choice of the vector . It is thus indeed a reasonable strategy to try to solve the problems (4.6) or (4.7), respectively, for an a priori fixed selection of multipliers to get an approximation of the set of Pareto stationary points of (P) and the associated pseudo-Pareto front (Jn(y, u))n=1,...,N u L2 () (weakly) Pareto stationary for (P) and y = S(u) . (7.1) Note that this approach has the particular advantage that we can control precisely whether we calculate a weakly Pareto stationary or an ordinarily Pareto stationary point by choosing a with non-negative or positive components. This feature comes at the cost of having to deal with the indicator function 1{y>0} in the strong stationarity conditions, which is typically not easy to handle. Second, we can also simply mollify the problem (P) as described in Section 5 and then use the set of weakly Pareto stationary points of the regularized multiobjective optimal control problem (P) as an approximation of that of (P). Note that the convergence result in Theorem 5.3 suggests that the set that is obtained in this way can indeed be expected to approximate the set of weakly Pareto stationary points of (P) for small regularization parameters (provided the conditions in Corollary 4.7i) are satisfied). However, in contrast to our first approach, this strategy does not allow to guarantee that a calculated point is Pareto stationary in the ordinary sense, as Theorem 5.3 only ensures weak stationarity in the limit 0+ . Further, one has to deal with an additional regularization error. On the other hand, computing a weakly Pareto stationary point of (P) by solving the system (5.4) is, of course, much easier than the solution of one of the strong stationarity conditions in Theorem 4.5. In what follows, we will explore and compare both of the above approaches in numerical experiments. Before we begin with our investigation, we would like to emphasize that the subsequent analysis should be understood as a feasibility study. In particular, we postpone a detailed discussion of the solvability and the approximation properties of the discrete counterparts of the systems (4.6), (4.7), and (5.4) as well as the convergence behavior and reliability of the used numerical solution method to future research. For related results in the single-objective setting, see [14, 18]. As a model problem for our numerical tests, we consider a simple tri-criterial, tracking-type optimization problem of the form Minimize 1 2 ky - yD,1k2 L2 , 1 2 ky - yD,2k2 L2 , 2 kuk2 L2 w.r.t. u L2 (), y H1 0 () H2 (), s.t. - y + max(0, y) = u a.e. in (M) on the unit square := (0, 1)2 with two given desired states yD,1, yD,2 C(cl()) and a Tikhonov parameter > 0 (to be specified below). Note that this problem trivially satisfies all of the conditions in Assumption 2.1 MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 25 and Corollary 4.7i). Theorems 4.5 and 5.3 are thus applicable here. For the discretization of (M) (or its necessary optimality conditions, to be more precise), we use Friedrichs­Keller triangulations Th of with widths h = 1/n, n N, and the finite element spaces Vh := v C(cl()) v| is affine for all cells Th and v| = 0 H1 0 (). Recall that the stiffness and the mass matrix associated with Vh are given by A := Z i · kdx i,k=1,...,M and B := Z i kdx i,k=1,...,M , where {i}M i=1 denotes the nodal basis of Vh and M := dim(Vh). (Here and in what follows, we often suppress the dependency on the mesh width h in the notation for the sake of readability.) For later reference, we further define the lumped mass matrix L := diag 1 3 |supp(i)| , i = 1, . . . , M . Note that this matrix arises from an approximation of the L2 -scalar product on by means of a three point subdivided quadrature rule subordinate to Th. Indeed, for all v1, v2 C(cl()), we have Z v1v2dx = X Th Z v1v2dx X Th 1 3 || X x node of v1(x)v2(x) = vT 1 Lv2, (7.2) where v1, v2 RM are the coordinate vectors w.r.t. the basis {i}M i=1 of the Lagrange interpolants Ih(v1), Ih(v2) Vh of the functions v1, v2, respectively, i.e., the vectors containing the function values of v1 and v2 at the interior nodes of the mesh Th. Let us now first consider the strong stationarity conditions (4.6) and (4.7) of (M), i.e., the system u, p L2 (), y H1 0 () H2 (), -y + max(0, y) = u, -p + 1{y>0}p = 2 X n=1 n(y - yD,n), p + 3u = 0 (7.3) with a multiplier R3 that satisfies P3 n=1 n = 1 and either n 0 for n = 1, 2, 3 in the case of (4.6) or n > 0 for n = 1, 2, 3 in the case of (4.7). Then, by formulating the PDEs in (7.3) weakly, by passing over to the finite element space Vh, by applying the nodal quadrature rule in (7.2) to the L2 -scalar products involving the terms max(0, ·) and 1{y>0} in the resulting variational identities (thus discretizing them by means of the lumped mass matrix L), and by expressing the remaining L2 - and H1 -scalar products with the mass matrix B and the stiffness matrix A, respectively, we arrive at the discrete system of equations Ay + L max(0, y) = Bu, Ap + L diag(H(y))p = 2 X n=1 nB(y - yD,n), p + 3u = 0. (7.4) Here, y, p, u RM and yD,1, yD,2 RM are the coordinate vectors w.r.t. {i}M i=1 of the Vh-counterparts of the quantities y, p, u in (7.3) and the Lagrange interpolants of yD,1 and yD,2, respectively, H denotes the Heaviside function (with the convention H(0) = 0), and the notation max(0, y) and H(y) is understood componentwise. 26 C. CHRISTOF AND G. MULLER Note that, by eliminating u and by splitting y into a positive and a negative part, for all multipliers with 3 > 0, (7.4) can also be recast as Ay+ - Ay- + Ly+ + (3)-1 Bp = 0, Ap + L diag(H(y+ - y-))p - 2 X n=1 nB(y+ - y- - yD,n) = 0, min(y+, y-) = 0. (7.5) This reformulation has the advantage that the non-smoothness is completely removed from the state equation and transferred into a separate, standard complementarity constraint. Since the system (7.5) can ­ at least heuristically ­ be tackled with a standard Newton-type method with the pseudo-Jacobian A + L -A (3)-1 B -(1 + 2)B (1 + 2)B A + L diag(H(y+ - y-)) I - diag(H(y+ - y-)) diag(H(y+ - y-)) 0 R3M×3M , (7.6) we may now follow the first approach outlined at the beginning of this section and try to solve (7.5) for various choices of the multiplier R3 to compute an approximation of the Pareto front of (P). Before we demonstrate how this solution method performs in practice, we would like to remark the following: Remark 7.1. i) A strategy similar to the one above has also been used in [18], Section 6 for the calculation of first-order stationary points of single-objective optimal control problems governed by quasilinear elliptic PDEs. The finite element discretization and the mass-lumping scheme that we have employed for the derivation of (7.4) are further the same as those in [14], Section 5. ii) It is important to realize that the transition from (7.3) to (7.4) follows a pure "first optimize then discretize"-philosophy. Since the effects studied in Sections 4 and 6 rely on Stampacchia's lemma, which is inherently infinite-dimensional, it is a priori completely unclear if (7.4) can be identified with the necessary optimality condition of an appropriately defined discrete problem or if this system has a solution at all. As (7.4) arises from the continuous optimality condition (7.3) (which is known to possess a solution) by passing over to the finite element spaces Vh and by applying quadrature rules, one can only guarantee that there exist vectors y, p, u RM which satisfy (7.4) up to a certain error level r(h) that vanishes in the limit h 0+ . As we will see below, in practical applications, this deficit of the discretization (7.4) is rather unproblematic as r(h) turns out to be typically much smaller than the tolerances that are normally used in numerical solution algorithms (in our experiments this was 10-8 ). In fact, it can be observed that a Newton method based on the pseudo-Jacobian (7.6) only fails in exceptionally rare cases, and that the percentage of these cases decreases when the tolerance is fixed and h is driven to zero, see Table 1. However, even in view of the fact that calculating approximate solutions of the discrete system (7.4) works very well in practice, one should keep in mind that this approach (and, in extension, also the numerical procedure in [18], Sect. 6) is a heuristic. (Working with the matrix (7.6) is, of course, heuristically motivated as well as the system of equations in (7.5) is non-smooth and thus not covered by classical convergence results for Newton-type methods.) iii) As an alternative to the approach in (7.3), one could also first reduce the problem (M) to the state y and subsequently discretize the associated first-order necessary optimality conditions in (6.7). Note that, in this case, the obtained system of equations involves the bi-Laplacian so that some care has to be taken regarding the choice of the used finite element spaces. Compare, e.g., with [12] in this context where a similar technique is applied to state-constrained problems. We do not discuss this alternative discretization here to avoid overloading this paper. MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 27 Table 1. Performance of our Newton-type algorithm in the situation of (7.7). The first line shows the width h of the considered triangulation, the second one the average number of Newton iterations needed for a successful step over the course of the calculation of the front (7.1) for the given h, the third one the number/percentage of steps in which the Newton method failed to converge for the given h in less than ten iterations, and the fourth one the average number of seconds needed for one successful solution of the stationarity system (7.5). The width of the discretization of in (7.8) was chosen as 1/K = 1/100 so that, for each h, (7.5) was solved 5050 times. The linear systems of equations arising in our Newton-type method have been solved by means of the "backslash" operator implemented in Matlab, version R2020a, on a computer with eight IntelCore i7-7700 CPUs with 3.60GHz and 32GB of RAM. h 1/16 1/32 1/64 1/128 1/256 1/512 Average number of Newton iterations 2.0684 2.2321 2.5135 2.8294 2.9899 3.1407 Number/percentage of 19 9 10 10 5 5 unsuccessful steps 0.376% 0.178% 0.198% 0.198% 0.099% 0.099% Average number of seconds needed for a successful step 0.0050 s 0.0257 s 0.1737 s 0.8559 s 4.5399 s 27.767 s Table 1 and Figures 1 and 2 show the results that we have obtained by solving the system (7.5) in the case yD,1(x1, x2) := 1(1 4 ,1) 2 (x1, x2) sin 4 3 x1 - 1 4 sin 4 3 x2 - 1 4 , yD,2(x1, x2) := -1(0, 3 4 ) 2 (x1, x2) sin 4 3 x1 sin 4 3 x2 , := 10-4 , (7.7) with a Newton-type method based on the pseudo-Jacobian (7.6) up to the tolerance tol = 10-8 for various multipliers . Here, the parameter space for , i.e., the simplex := R3 | P3 n=1 n = 1, n 0 was discretized with an equidistant mesh of the form K := 1 K (m, n, K - m - n) R3 m, n = 0, . . . , K, m + n < K , K N. (7.8) Note that the vectors with 3 = 0 are excluded in (7.8) as for these multipliers the solvability of (7.3) cannot be guaranteed, cf. Lemma 4.8. We would like to emphasize that the choice of the equidistant mesh K in (7.8) does not imply that the computed points in the front (7.1) are distributed equidistantly as well since it is a priori unclear how the set (7.1) is parametrized by (see Fig. 1). To achieve a more homogeneous resolution of (7.1), one can employ adaptive discretization techniques or reference point methods analogous to those used for classical scalarization approaches, cf. [5, 34, 54] and also Remark 4.6iii). We omit a detailed discussion of this topic to avoid overloading the paper. As the results in Table 1 show, our Newton-type algorithm requires on average between two and three iterations to reduce the residue of (7.5) below the given threshold tol = 10-8 . (The starting point was always chosen as zero here.) We further see that the average number of iterations remains nearly constant as h is decreased. This mesh-independence of the solution procedure makes sense as our algorithm is based on the necessary optimality condition of the continuous problem (M) in (7.3). In the third line of Table 1, we can further see that our method only fails to solve the system (7.5) in exceptionally rare situations. Moreover, the number of unsuccessful solution steps (here defined as steps in which the Newton-type algorithm failed to converge for a given in less than ten iterations) decreases as the mesh width h tends to zero. This confirms the theoretical considerations made in Remark 7.1. 28 C. CHRISTOF AND G. MULLER Figure 1. Approximation of the set (7.1) for (M) in the situation (7.7) obtained by solving the system (7.5) for 5050 values of the multiplier . The spatial mesh width h was chosen as 1/512 and the width of the equidistant discretization of the simplex in (7.8) as 1/K = 1/100. The points in the scatter plot associated with multipliers that have a vanishing component and thus correspond to weakly but not ordinarily Pareto stationary points are depicted in red. Figure 2. State (left) and control (right) associated with the solution of (7.5) in the situation of (7.7) for the particular multiplier = (0.49, 0.5, 0.01). The width h was chosen as 1/64 here. For comparison, let us now consider the regularization approach that we have outlined at the beginning of this section: By starting from the necessary optimality condition (5.4) for the regularized multiobjective optimal control problem (P) and by proceeding along exactly the same lines as for (7.4), we arrive at the equations Ay + L max(y) = Bu, Ap + L diag(max0 (y))p = 2 X n=1 nB(y - yD,n), p + 3u = 0, (7.9) where is again an arbitrary but fixed element of , and where we again suppress the dependence on h and in the notation for y, A, B, etc. for the sake of readability. Note that, in contrast to (7.4), the system (7.9) is smooth so that a standard Newton algorithm can be used for its solution. Further, it is straightforward to check that (7.9) corresponds to the necessary optimality condition for weak Pareto optimality of a discrete version of (P), and that (7.9) admits at least one solution for all arbitrary but fixed with 3 > 0 (cf. the proof of Lem. 4.8 and also the results in [22], Sect. 3.3). The processes of discretization and optimization (or calculating the first-order necessary optimality conditions, to be more precise) thus commute in the case of (P) and the solvability of (7.9) is not an issue at all. This is an advantage over approaches based on the system (7.4) that, of course, again comes at the price of the additional regularization error. MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH PDE 29 Table 2. Average number of Newton iterations needed for the approximation of the set (7.1) by successive solution of the system (7.9) for various h and and K as in (7.8). The width of the mesh K was again chosen as 1/K := 1/100 here so that (7.9) was solved 5050 times in each configuration. The number of unsuccessfully solved systems and the average number of seconds needed for a successful step are denoted in parentheses, separated by a comma. The computations have been carried out on the same system as in Table 1. h 1/16 1/32 1/64 1/128 1/256 10-3 2.2771 (0, 0.0047 s) 2.2283 (0, 0.0193 s) 2.1671 (0, 0.0771 s) 2.0736 (0, 0.3927 s) 2.0392 (0, 1.9095 s) 10-6 2.0849 (9, 0.0044 s) 2.2063 (7, 0.0194 s) 2.3895 (3, 0.0848 s) 2.5165 (0, 0.4731 s) 2.4755 (0, 2.3120 s) 10-9 2.0588 (19, 0.0043 s) 2.1789 (9, 0.0194 s) 2.3645 (10, 0.0839 s) 2.4907 (8, 0.4689 s) 2.4085 (0, 2.2505 s) Figure 3. Approximations of the pseudo-Pareto front (7.1) obtained from the unregularized system (7.5) (blue) and the regularized system (7.9) with = 10 (red) and = 20 (green) for h = 1/256 and K = 100. The regularization parameter is chosen very large here because for smaller the sets are visually indistinguishable. Points on the front that correspond to weakly but not ordinarily Pareto stationary points are colored solidly. It can be seen that the sets calculated by solving (7.9) approximate that obtained from (7.5) as tends to zero. Table 2 and Figure 3 depict the results that are obtained when the function x 7 1 2 (x + x2 + 2) is chosen as max(·) in (P), the system (7.9) is solved with a classical Newton method, and the set is discretized as in (7.8). As Table 2 shows, the solution of (7.9) again requires between two and three Newton iterations on average over the course of a single approximation of the front (7.1), and, similarly to the behavior in Table 1, this number is largely independent of the mesh width h and the regularization parameter . Iterations, in which the Newton algorithm fails to converge, are encountered here as well (though less frequently than in the unregularized case if is moderate), and it can be seen that the behavior of our solution method for the regularized system (7.9) emulates that observed for the unregularized problem in Table 1 when tends to zero (with a notable exception for h = 1/256 most likely due to a favorable relationship between h and ). Figure 3 further shows that the front calculated by solving (7.9) approximates that obtained from the system (7.5) as decreases. This underlines that the solutions obtained from (7.5) are sensible, that the system (7.5) allows to calculate an approximation of the front (7.1) that does not suffer from a regularization error, and that the strong stationarity conditions derived in Section 4 are indeed not only interesting for their own sake and theoretical purposes but also for numerical solution algorithms. 30 C. CHRISTOF AND G. MULLER References [1] R.A. 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Rappaz, Approximation of a nondifferentiable nonlinear problem related to MHD equilibria. Numer. Math. 45 (1984) 117­133. [48] A.-T. Rauls and S. Ulbrich, Computation of a Bouligand generalized derivative for the solution operator of the obstacle problem. SIAM J. Optim. 57 (2019) 3223­3248. [49] A.-T. Rauls and G. Wachsmuth, Generalized derivatives for the solution operator of the obstacle problem. Set-Valued Var. Anal. (2019) 1­27. [50] R.T. Rockafellar and R.J.-B. Wets, Variational analysis. Vol. 317 of Grundlehren der Mathematischen Wissenschaften. Springer (1998). [51] M. Ruzicka, Nichtlineare Funktionalanalysis. Springer, Berlin/Heidelberg (2004). [52] W. Schirotzek, Nonsmooth Analysis. Springer, Berlin/Heidelberg (2007). [53] R. Temam, A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma. Arch. Rational Mech. Anal. 60 (1976) 51­73. [54] A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in Multiple Criteria Decision Making Theory and Application, edited by G. Fandel and T. Gal. Springer (1980) 468­486. [55] J. Xin, An Introduction to Fronts in Random Media. Vol. 5 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer (2009). [56] J.J. Ye, Necessary optimality conditions for multiobjective bilevel programs. Math. Oper. Res. 36 (2011) 165­184. [1] R. A. Adams, Sobolev Spaces. Academic Press, New York (1975). [2] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. SIAM, Philadelphia (2006). [3] H. Attouch, G. Garrigos and X. Goudou, A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions. J. Math. Anal. Appl. 422 (2015) 741–771. [4] S. Banholzer, S. Beermann and S. Volkwein, POD-based bicriterial optimal control by the reference point method. 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2016. IFAC-PapersOnLine 49 (2016) 210–215. [5] S. Banholzer and S. Volkwein, Hierarchical convex multiobjective optimization by the Euclidean reference point method. Preprint SPP1962-117 (2019). [6] V. Barbu, Optimal Control of Variational Inequalities. Research Notes in Mathematics. Pitman (1984). [7] D. Beermann, M. Dellnitz, S. Peitz and S. Volkwein, POD-based multiobjective optimal control of PDEs with non-smooth objectives. PAMM 17 (2017) 51–54. [8] D. Beermann, M. Dellnitz, S. Peitz and S. Volkwein, Set-oriented multiobjective optimal control of PDEs using proper orthogonal decomposition, in Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing, edited by W. Keiper, A. Milde and S. Volkwein. Springer (2018) 47–72. [9] S. Bellaassali and A. Jourani, Necessary optimality conditions in multiobjective dynamic optimization. SIAM J. Control Optim. 42 (2004) 2043–2061. [10] J. Bienvenido and N. Vicente, Optimality conditions in directionally differentiable Pareto problems with a set constraint via tangent cones. Numer. Funct. Anal. Optim. 24 (2003) 557–574. [11] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000). [12] S. C. Brenner, Finite element methods for elliptic distributed optimal control problems with pointwise state constraints (survey), in Advances in Mathematical Sciences: AWM Research Symposium, Houston, TX, April 2019, edited by B. Acu, D. Danielli, M. Lewicka, A. Pati, R. V. Saraswathy and M. Teboh-Ewungkem. Springer (2020) 3–16. [13] C. Christof, Sensitivity Analysis of Elliptic Variational Inequalities of the First and the Second Kind. Ph.D. thesis, Technische Universität Dortmund (2018). [14] C. Christof, C. Meyer, S. Walther and C. Clason, Optimal control of a non-smooth semilinear elliptic equation. Math. Control Relat. Fields 8 (2018) 247–276. [15] C. Christof and B. Vexler, New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints. To appear in: ESAIM: COCV (2020). Available from: . [16] C. Christof and G. Wachsmuth, On second-order optimality conditions for optimal control problems governed by the obstacle problem. To appear in: Optimization (2020). Available from: . [17] F. H. Clarke, Optimization and Nonsmooth Analysis. SIAM’s Classics in Applied Mathematics. SIAM, Philadelphia, PA (1990). [18] C. Clason, V. H. Nhu and A. Rösch, Optimal control of a non-smooth quasilinear elliptic equation. Preprint (2018). [19] J. C. De Los Reyes and C. Meyer, Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the second kind. J. Optim. Theory Appl. 168 (2016) 375–409. [20] S. Dempe. Foundations of Bilevel Programming. Vol. 61 of Nonconvex Optimization and Its Applications. Kluwer (2002). [21] O. Ebel, S. Schmidt and A. Walther, Solving non-smooth semi-linear optimal control problems with abs-linearization. Preprint SPP1962-093 (2018). [22] M. Ehrgott, Multicriteria Optimization, 2nd edn. Springer (2005). [23] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland (1976). [24] L. C. Evans, Partial Differential Equations, 2nd edn. AMS, Providence, RI (2010). [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edn. Springer, Berlin/Heidelberg/New York (2001). [26] G. Giorgi, B. Jiménez and V. Novo, On constraint qualifications in directionally differentiable multiobjective optimization problems. RAIRO: OR 38 (2004) 255–274. [27] G. Giorgi, B. Jiménez and V. Novo, Strong Kuhn-Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems. TOP 17 (2008) 288–304. [28] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). [29] L. Iapichino, S. Ulbrich and S. Volkwein, Multiobjective PDE-constrained optimization using the reduced-basis method. Adv. Comput. Math. 43 (2017) 945–972. [30] Y. Ishizuka, Optimality conditions for directionally differentiable multi-objective programming problems. J. Optim. Theory Appl. 72 (1992) 91–111. [31] Y. Ishizuka and K. Shimizu, Necessary and sufficient conditions for the efficient solutions of nondifferentiable multiobjective problems. IEEE Trans. Syst. Man Cybernet. SMC-14 (1984) 625–629. [32] B. N. Khoromskij and G. Wittum, Numerical Solution of Elliptic Differential Equations by Reduction to the Interface. Lecture Notes in Computational Science and Engineering. Springer, Berlin/Heidelberg (2012). [33] F. Kikuchi, K. Nakazato and T. Ushijima, Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria. Japan J. Appl. Math. 1 (1984) 369–403. [34] I. Y. Kim and O. L. De Weck, Adaptive weighted-sum method for bi-objective optimization: Pareto front generation. Struct. Multidiscip. Optim. 29 (2005) 149–158. [35] Y.-B. Lü and W. Zhong-Ping, A smoothing method for solving bilevel multiobjective programming problems. J. Oper. Res. Soc. China 2 (2014) 511–525. [36] M. M. Mäkelä, V. P. Eronen and N. Karmitsa, On nonsmooth multiobjective optimality conditions with generalized Convexities, in Optimization in Science and Engineering: In Honor of the 60th Birthday of Panos M. Pardalos, edited by T. M. Rassias, C. A. Floudas, and S. Butenko. Springer, New York, NY (2014) 333–357. [37] O. L. Mangasarian, Nonlinear Programming. Vol 10 of SIAM’s Classics in Applied Mathematics, 2nd edn. SIAM (1994). [38] C. Meyer and L. Susu, Optimal control of nonsmooth, semilinear parabolic equations. SIAM J. Control Optim. 55 (2017) 2206–2234. [39] K. M. Miettinen, Nonlinear Multiobjective Optimization. Kluwer, Boston/London/Dordrecht (1999). [40] F. Mignot, Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130–185. [41] F. Mignot and J.-P. Puel, Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466–476. [42] B. S. Mordukhovich, Multiobjective optimization problems with equilibrium constraints. Math. Program. 117 (2009) 331–354. [43] B. S. Mordukhovich, Variational Analysis and Applications. Springer Monographs in Mathematics. Springer (2018). [44] S. Ober-Blöbaum and K. Padberg-Gehle, Multiobjective optimal control of fluid mixing. PAMM 15 (2015) 639–640. [45] S. Peitz and M. Dellnitz, Gradient-based multiobjective optimization with uncertainties, in NEO 2016: Results of the Numerical and Evolutionary Optimization Workshop NEO 2016 and the NEO Cities 2016 Workshop held on September 20-24, 2016 in Tlalnepantla, Mexico, edited by Y. Maldonado, L. Trujillo, O. Schütze, A. Riccardi, and M. Vasile. Springer (2018) 159–182. [46] S. Peitz and M. Dellnitz, A survey of recent trends in multiobjective optimal control - surrogate models, feedback control and objective reduction. Math. Comput. Appl. 23 (2018) 1–33. [47] J. Rappaz, Approximation of a nondifferentiable nonlinear problem related to MHD equilibria. Numer. Math. 45 (1984) 117–133. [48] A.-T. Rauls and S. Ulbrich, Computation of a Bouligand generalized derivative for the solution operator of the obstacle problem. SIAM J. Optim. 57 (2019) 3223–3248. [49] A.-T. Rauls and G. Wachsmuth, Generalized derivatives for the solution operator of the obstacle problem. Set-Valued Var. Anal. (2019) 1–27. [50] R. T. Rockafellar and R. J.-B. Wets, Variational analysis. Vol. 317 of Grundlehren der Mathematischen Wissenschaften. Springer (1998). [51] M. Ruzicka, Nichtlineare Funktionalanalysis. Springer, Berlin/Heidelberg (2004). [52] W. Schirotzek, Nonsmooth Analysis. Springer, Berlin/Heidelberg (2007). [53] R. Temam, A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma. Arch. Rational Mech. Anal. 60 (1976) 51–73. [54] A. P. Wierzbicki, The use of reference objectives in multiobjective optimization, in Multiple Criteria Decision Making Theory and Application, edited by G. Fandel and T. Gal. Springer (1980) 468–486. [55] J. Xin, An Introduction to Fronts in Random Media. Vol. 5 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer (2009). [56] J. J. Ye, Necessary optimality conditions for multiobjective bilevel programs. Math. Oper. Res. 36 (2011) 165–184. COCV_2021__27_S1_A15_0c76b71e3-1b4b-4e16-bfba-8fb158685089 cocv190132 10.1051/cocv/202006310.1051/cocv/2020063 Shape derivatives for an augmented Lagrangian formulation of elastic contact problems* Chaudet-Dumas Bastien ** Deteix Jean Groupe Interdisciplinaire de Recherche en Éléments Finis de l’Université Laval, Départment de Mathématiques et Statistiques, Université Laval, Québec, Canada. **Corresponding author: [e-mail] 01 03 2021 01 03 2021 2021 cocv/2021/01 Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science OuverteSupplementS14 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF) Full (DJVU)

This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Because of the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem.

Shape and topology optimization unilateral contact elliptic variational inequalities conical differentiability augmented Lagrangian method 35J86 49K40 49Q10 74M15 74P15 idline ESAIM: COCV 27 (2021) S14 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S14 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020063 www.esaim-cocv.org SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS Bastien Chaudet-Dumas and Jean Deteix Abstract. This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formula- tion, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Because of the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem. Mathematics Subject Classification. 35J86, 49K40, 49Q10, 74M15, 74P15. Received August 7, 2019. Accepted September 28, 2020. 1. Introduction Structural optimization has become an integral part of industrial conception, with applications in more and more challenging mechanical contexts. Those contexts often lead to complex mathematical formulations involv- ing non-linearities and/or non-differentiabilities, which causes many difficulties when considering the associated shape optimization or optimal control problem. In this article, we study a shape optimization problem in the context of contact mechanics. Especially, the physical system models an elastic body (without restriction on its dimension) coming in sliding contact with a rigid foundation, which takes the mathematical form of an elliptic variational inequality (VI) of the first kind. As variational inequalities involve projection operators, differentiation with respect to the control parameter (in order to derive optimality conditions or use a gradient descent optimization method) is not an easy task. Using the terminology from [15], let us gather the approaches to treat optimal control or shape optimiza- tion problems for variational inequalities in two families: the ones following the first optimize then discretize paradigm, and the others following the first discretize then optimize paradigm. In the first one, the idea is to work with a weaker notion of differentiability, namely conical differentiability (see Def. 2.3), in order to get This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Keywords and phrases: Shape and topology optimization, unilateral contact, elliptic variational inequalities, conical differentia- bility, augmented Lagrangian method. Groupe Interdisciplinaire de Recherche en Éléments Finis de l'Université Laval, Départment de Mathématiques et Statistiques, Université Laval, Québec, Canada. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 B. CHAUDET-DUMAS AND J. DETEIX optimality conditions. Let us mention the pioneer work [24], as well as other works in the same direction [25], the series of works [28­35] and also [15, 19]. In those works, the conical derivative of the solution with respect to the control parameter is expressed by means of a variational inequality. Thus the optimality conditions obtained might be difficult to use in practice. However, in [27], for the two-dimensional Signorini problem, the authors give conditions for some specific functional to be shape differentiable. This allows them to get an explicit expression for the shape derivative of the functional using an adjoint state. In the second family of approaches, first the variational inequality is discretized, then tools from subdifferential calculus are used in the finite dimensional setting. The interested reader is referred to [11, 12, 21, 22] for shape optimization for the two-dimensional Sig- norini problem discretized using finite elements. The way the authors define the set of admissible shapes enable them to prove existence of an optimal design for the non-disctretized problem. Moreover, in [13], convergence analysis with respect to the discretization parameter is performed. In the same spirit, we finally mention the series of papers [1, 2, 14] dedicated to shape optimization for the contact problem with Coulomb friction, which is much more cumbersome. There, the authors manage to characterize an outer approximation of the shape subdifferential of the functional to minimize, then use a bundle algorithm. The approach proposed here follows the first optimize then discretize paradigm. The goal is to express first order optimality conditions for the shape optimization problem associated to some generic cost functional. It can somehow be understood as a continuation of [27]. In order to facilitate shape sensitivity of the contact problem, we write the sliding contact conditions using the normal to the rigid foundation instead of the normal to the body, as in [3]. Furthermore, the variational inequality arising from the basic formulation of sliding contact problems is transformed using an Augmented Lagrangian formulation (ALF) which we plan on solving using a basic iterative approach, the Augmented Lagrangian method (ALM). Aiming at practical use of the ALM in a numerical shape optimization process, we study consistency of this iterative process with respect to shape differentiation. In other words, we aim at deriving conditions for the shape derivatives of the iterates generated by this method to converge to the shape derivative of the solution to the original problem. This work is structured as follows. Section 2 presents the original problem, its different formulations as well as some notations and notions related to convex analysis and conical differentiability. We also introduce the augmented Lagrangian method (ALM) applied to this problem. Section 3 is dedicated to shape optimization and is divided in three parts. In the first one, we give a proof (adapted from the classical one) that the solution of this formulation is conically differentiable. In the second one, we prove the same property for each of the iterates generated by the iterative algorithm. Convergence analysis of those conical shape derivatives to the one obtained for the original formulation is studied in the third part. Finally, we give the expression of the derivative of a general cost functional J for a mechanical system in sliding contact with a rigid foundation. 2. Problem formulation 2.1. Geometrical setting Here, the body Rd , d {2, 3}, is assumed to have C1 boundary, and to be in contact with a rigid foundation rig, which has a C3 compact boundary rig, see Figure 1. Let D be the non-empty part of the boundary where a homogeneous Dirichlet condition applies (blue part), N the part where a Neumann condition applies (orange part), C the potential contact zone (green part), and the rest of the boundary, which is free of any constraint (i.e. homogeneous Neumann boundary condition). Those four parts are open, mutually disjoint and moreover: D N C = . In order to avoid technical difficulties, it is assumed that C D = . The outward normal to is denoted no. Similarly, the inward normal vector to rig is denoted n. 2.2. Notation, function spaces and preliminaries Throughout this article, for any O Rd , Lp (O) represents the usual set of pth power measurable functions on O, and (Lp (O)) d = Lp (O). The scalar product defined on L2 (O) or L2 (O) is denoted (without distinction) by (·, ·)O and its norm k · k0,O. SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 3 The Sobolev spaces, denoted Wm,p (O) with p [1, +], p integer are defined as Wm,p (O) = {u Lp (O) | D u Lp (O) || m} , where is a multi-index in Nd and Wm,p (O) = (Wm,p (O)) d . The spaces Ws,2 (O) and Ws,2 (O), s R, are denoted Hs (O) and Hs (O) respectively. Their norm are denoted k · ks,O. Without distinction for the dimension, we denote the duality pairing between H 1 2 (O) and its dual H- 1 2 (O) (or between H 1 2 (O) and H- 1 2 (O)) by h·, ·iO. More generally, for a space V and V its dual, we denote the duality pairing by h·, ·iV ,V . The subspaces of functions in Hs (O) and Hs (O) that vanish on a part of the boundary O are denoted Hs (O) and Hs (O). In particular, we denote the vector space of admissible displacements X := H1 D (). The sets T2 and T4 are the sets of real valued tensors of order 2 and 4 respectively. For any v vector in Rd or second order tensor in T2 , the product with the normal v · no (respectively with the normal to the rigid foundation v · n) is denoted vno (respectively vn). Similarly, the tangential part of v is denoted vto = v - vno no (respectively vt = v - vn n). Finally, we introduce some notations from convex analysis. The indicator function and the characteristic function of an arbitrary set S are denoted IS and S, respectively, that is IS(x) := 0 if x S , + if x / S , S(x) := 1 if x S , 0 if x / S . Let H be a Hilbert space, and C a non-empty closed convex subset of H, then for any x H, the unique projection of x onto C is denoted ProjC(x). Moreover, if b is a bilinear form inducing an inner product on H, the projection of x onto C with respect to this inner product is denoted Projb C(x). Let us now recall some notions related to conical differentiability, see [24]. If H denotes a Hilbert space and b a coercive symmetric bilinear form on H × H, then for any K H and y K, we define: · the polar cone of K with respect to b as [K] 0 b := {x H : z K, b(x, z) 0}; · the radial cone of K at y as Cy(K) := {w H : t > 0, y + tw K}; · the tangent cone of K at y as Sy(K) := Cy(K); · the cone Sy (K) := Sy(K) [R(v - y)] 0 b, where v y + [Sy(K)] 0 b and R(v - y) := {(v - y) | R}. Definition 2.1. Let K H be a closed convex set. K is said to be polyhedric at v H if, denoting y = Projb K(v), one has: Sy (K) = Cy(K) [R(v - y)] 0 b . Definition 2.2. Let K H be a closed convex set. K is said to be polyhedric in H if it is polyhedric at each v H. Definition 2.3. Let V1, V2 be two Banach spaces. A continuous function f : V1 V2 admits a conical derivative at x if there exists an operator Q : V1 V2 positively homogeneous such that: h V1, t > 0, f(x + th) = f(x) + tQ(h) + o(t) . Using these notations, we may recall one of the main results on the differentiability of projection operators, namely [24], Theorem 2.1. 4 B. CHAUDET-DUMAS AND J. DETEIX Figure 1. Elastic body in contact with a rigid foundation. Theorem 2.4. Let v H and y = Projb K(v). If K is polyhedric at v, then the projection Projb K is conically differentiable at v, with conical derivative Projb Sy(K). In other words, for all w H and t > 0: Projb K(v + tw) = y + t Projb Sy(K)(w) + o(t) . 2.3. Mechanical model In this work the material is assumed to verify the linear elasticity hypothesis (small deformations and Hooke's law, see for example [4]), associated with the small displacements assumption (see [20]). The phys- ical displacement is denoted u, and belongs to X. The stress tensor is defined by (u) = C : (u), where (u) = 1 2 (u + uT ) denotes the linearized strain tensor, and C is the elasticity tensor. This elasticity tensor is a fourth order tensor belonging to L (, T4 ), and it is assumed to be elliptic (with constant 0). Regarding external forces, the body force f L2 (), and traction (or surface load) H 1 2 (N ). 2.4. Non-penetration condition At each point x of C, let us define the gap gn(x), as the oriented distance function to the rigid foundation at x, see Figure 1. Due to the regularity of the rigid foundation, there exists h0 sufficiently small such that, for all h < h0 h rig := {x Rd | | gn(x)| < h} , is a neighborhood of rig where gn is of class C3 , see [5]. In particular, this ensures that n is well defined on h rig, and that n C2 (h rig, Rd ). Moreover, in the context of small displacements, it can be assumed that the potential contact zone C is such that C h rig. Hence there exists a neighborhood of C such that gn and n are of class C3 and C2 , respectively. Especially, this implies that the function gn n C2 (h rig, Rd ). It can thus be extended to a function g C2 (Rd ) such that g = 0 on Rd \ h0 rig for some h0 > h. Since C D = , one has that D (Rd \ h0 rig) for h, h0 small enough. Consequently, g X. The non-penetration condition can be stated as follows: un gn a.e. on C. Thus, we introduce the closed convex set of admissible deformations that realize this condition, see [8]: K := {v X | vn gn a.e. on C} = g + K0 , where K0 is a closed convex cone defined as K0 := {v X | vn 0 a.e. on C}. Remark 2.5. Our definition of K differs from the usual one since we compute the gap in the direction of the normal n to the rigid foundation instead of the normal no to . Actually, under the small displacements hypothesis, the normal vector n and the gap gn to the rigid foundation can be replaced by no and gno (we refer SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 5 to [20], Chap. 2 for the details). This ensures that in our context, using no or n to write the formulation has no impact on the solution. However, it will be seen in the next sections that using n will be really convenient when dealing with shape optimization. 2.5. Mathematical formulation of the problem Let us introduce the bilinear and linear forms a : X × X R and L : X R, such that: a(u, v) := Z C : (u) : (v) , L(v) := Z f v + Z N v . According to the assumptions of the previous sections, one is able to show (see [4]) that a is X-elliptic with constant 0 (ellipticity of C and Korn's inequality), symmetric, continuous, and that L is continuous (regularity of f and ). The unknown displacement u of the frictionless contact problem is the minimizer of the total mechanical energy of the elastic body, which reads, in the case of pure sliding (unilateral) contact problems: inf vK (v) := inf vK 1 2 a(v, v) - L(v) . (2.1) It is clear that the space X, equipped with the usual H1 norm, is a Hilbert space. Moreover, under the conditions of the previous section, since K is obviously non-empty and the energy functional is strictly convex, continuous and coercive, we are able to conclude that u solution of (2.1) exists and is unique, see for example [9]. It is well known that (2.1) may be rewritten as a variational inequality (of the first kind): a(u, v - u) L(v - u) , v K . (2.2) Even if this variational inequality is very well known from the theoretical point of view such formulation is not well suited for computational purposes. One approach to get around this difficulty consists in rewriting the formulation so that the constraint is implicitly verified. One of the most frequently used formulation (in practical applications) is the penalty formulation. In [3], we suggested a method based on directional derivatives in order to derive optimality conditions for the penalty formulation. The penalty formulation of (2.2) is numerically robust and relatively simple to implement, but its solutions depend on the penalty parameter. From a numerical point of view the simplest way to avoid this inconsistency while retaining the simplicity of the penalty approach is the augmented Lagrangian formulation. The rest of this work is related to the augmented Lagrangian formulation and the basic iterative process to solve this formulation, namely the augmented Lagrangian method. It should be noticed that both formulations write as non-linear non-differentiable variational equations (pos- sibly mixed), and thus lead to the same kind of technical difficulties (related to regularity) when studying the associated shape optimization problem. Here, an approach similar to [3] will be followed. 2.6. Lagrangian formulation The content of this paragraph is based on [36]. The reader is referred to the chapter 4 of this monograph for all proofs and technical details. A mixed formulation of problem (2.1) can be recovered in the framework of Fenchel duality theory by deriving the dual problem associated to this minimization problem. This result is formally stated in the following theorem. Theorem 2.6. If u K is the solution of (2.1), then there exists a unique dual variable H- 1 2 (C) such that: a(u, v) - L(v) + h, vniC = 0 , v X , (2.3a) 6 B. CHAUDET-DUMAS AND J. DETEIX h, iC 0 , H 1 2 (C) , 0 , (2.3b) h, un - gniC = 0 . (2.3c) Remark 2.7. It is important to note that, in general, the regularity of is only H- 1 2 (C). 2.7. Generalized Moreau­Yosida approximation The goal of this paragraph is to transform conditions (2.3b), (2.3c), into pointwise conditions using Moreau- Yosida regularizations. Since (2.1) is a non-smooth convex optimization problem, one may follow the approach from [18], Chapter 4 and introduce a consistent Moreau­Yosida regularization of (2.3), provided that the Lagrange multiplier belongs to a Hilbert space. From Theorem 2.6 this is not the case since H- 1 2 (C). However, it is known (see, e.g. [36], Chap. 4) that the additional regularity L2 (C) can be obtained if the set {un - gn = 0} is strictly contained in C, which can be formulated as follows: Assumption 2.8. {un - gn = 0} C . Remark 2.9. From the mechanical point of view, this assumption means that the points at the boundary of the potential contact zone C do not come in contact with the rigid foundation. Intuitively, in the context of small displacements, it should be the case when C is chosen large enough. Moreover, in practice, this assumption can be checked easily a posteriori. From now on, it is assumed that Assumption 2.8 holds, thus, one may follow the steps of [18], Chapter 4, which directly leads to the desired consistency result for the regularization. Before stating this result, we introduce a last notation, the projection onto R+ in R (also called the positive part function) will be denoted p+ (p+(y) := max{0, y}, for all y R). We are now ready to introduce an Augmented Lagrangian Formulation (or characterization of the solution) of the sliding contact problem. Theorem 2.10. Suppose Assumption 2.8 holds. If (u, ) X ×L2 (C) denotes the solution of (2.3), it verifies, for any > 0, a(u, v) - L(v) + (, vn)C = 0 , v X , (2.4a) - p+( + (un - gn)) = 0 a.e. on C . (2.4b) Conversely, if a pair (u, ) X ×L2 (C) satisfies (2.4) for some > 0, then u is the solution of (2.1). Proof. Let us start by rewriting (2.1) in a more suitable form. For this purpose, let L (X, L2 (C)) denote the normal trace operator, such that for all v X v = vn . We also introduce the convex closed set K := { L2 (C), gn a.e. on C}. Using these notations, problem (2.1) rewrites as an unconstrained non-smooth convex optimization problem: inf vX (v) + IK(v) . (2.5) IK being non-smooth, we introduce IK, its generalized Moreau-Yosida regularization, with regularization parameter > 0. Given , L2 (C), one has: IK,(, ) = inf L2(C ) n IK() + (, - )C + 2 k - k 2 0,C o . SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 7 Now, due to the indicator function in the inf, one gets: IK,(, ) = inf K ( 2 + - 2 0,C ) - 1 2 kk 2 0,C = 2 (Id - ProjK) + 2 0,C - 1 2 kk 2 0,C . Then, [18], Theorem 4.45 ensures that the complementarity conditions (2.3b), (2.3c) can equivalently be expressed as = IK,(, ) (u,) , (2.6) for any > 0, where the pair (u, ) is the solution of (2.3). Moreover, due to definition of K, one has for any L2 (C), ProjK() = min{, gn} . Therefore, condition (2.6) directly leads to the desired result. 2.8. Augmented Lagrangian method The augmented Lagrangian method consists in an iterative algorithm to solve formulation (2.4) using an update strategy for the multiplier. The reader is referred to [10] for further details about the application of such methods to the numerical resolution of problems in mechanics. We briefly recall the steps of the algorithm. Algorithm: ALM 1. Choose 0 L2 (C) and set k = 1. 2. Choose k > 0, then find uk X the solution of, a(uk , v) + p+ k-1 + k (uk n - gn) , vn C = L(v) v X , (2.7) 3. Update the multiplier following the rule: k = p+ k-1 + k (uk n - gn) a.e. on C . (2.8) 4. While a chosen convergence criterion is not satisfied, set k = k + 1 and go back to step 2. It has been shown (see, e.g. in [36] for a detailed proof) that one gets the following convergence result for this algorithm. Theorem 2.11. Suppose Assumption 2.8 holds, and (u, ) denotes the solution of (2.4). Then for any choice of parameters 0 < 1 2 · · · , the iterates uk converge to u strongly in X. Moreover, the iterates of the multiplier k converge to weakly in L2 (C). Remark 2.12. When 0 H 1 2 (C), the iterates generated by the ALM satisfy uk , k X ×H 1 2 (C) for all k 1. 8 B. CHAUDET-DUMAS AND J. DETEIX 3. Shape optimization Given a cost functional J() = J (, u()) depending explicitly on the domain , and also implicitly, through u() the solution of (2.2) on , the optimization of J with respect to or shape optimization problem associated to the original contact formulation reads: minimize J() over Uad , subject to u() solves (2.2). (3.1) where Uad stands for the set of admissible domains. Thanks to Theorem 2.11, from an analytical point of view, determining u and using the ALM or any other method has limited impact on the computation of the shape sensitivity of (2.4) (or even (2.3)). The shape sensitivities would simply be defined relatively to the converged results of the ALM process. However, in practice, given a convergence criterion, the ALM will stop at some iteration k, meaning that even though uk is assumed to be a good enough approximation of u, it is not the solution of (2.4). Therefore, the sensitivities of (2.4) do not correspond to the sensitivities of the equations defining uk . As mentioned in the introduction, the idea in this work is to perform shape optimization on the approximate formulation (2.7) instead of the original formulation (2.1). Then, we look for a domain in Uad that minimizes Jk , a cost functional defined by Jk () := J (, uk ()) where uk () is the solution of (2.7) defined on . The new shape optimization is thus given by: minimize Jk () over Uad , subject to uk () solves (2.7). (3.2) Obviously, since we are interested in the optimality conditions associated to those problems, we also want to study the consistency of the sensitivities of the iterates uk obtained from the ALM with respect to the sensitivity of u. Let D Rd be a fixed bounded smooth domain, and let D D be a part of its boundary which will be the "potential" Dirichlet boundary. This means that for any domain D, the Dirichlet boundary associated to will be defined on D := D. With these notations, we introduce the set Uad of all admissible domains, which consists of all smooth open domains such that the Dirichlet boundary D D is of strictly positive measure, that is: Uad := { D | is of class C1 and | D| > 0}. 3.1. Derivatives The shape optimization method followed in this work is the so-called perturbation of the identity, as presented in [26] and [16]. Let us introduce C C C1 b(Rd ) := (C1 (Rd ) W1, (Rd )) d , equipped with the d-dimensional W1, norm, denoted k·k1,. In order to move the domain , let C C C1 b(Rd ) be a geometric deformation vector field. The associated perturbed or transported domain in the direction will be defined as: (t) := (Id +t )() for any t > 0. It is known that for t sufficiently small, more specifically for t such that t kk1, < 1, Id +t is a diffeomorphism, see for example [16]. This enables to rely on the classical notion of differentiability in Banach spaces to define shape differentiability. To make things clear some basic notions of shape sensitivity analysis from [34] are briefly recalled. Let y() be the solution, in some Sobolev space denoted W(), of a variational formulation posed on . For any fixed , for any small t > 0, let y((t)) be the solution of the same variational formulation posed on (t). If the variational formulation is regular enough, which will be assumed to be true, it can be proved (see [34]) that y((t)) (Id +t ) also belongs to W(). SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 9 · The Lagrangian derivative or material derivative of y() in the direction is the element y()[] W() defined by: y()[] := lim t&0 1 t (y((t)) (Id +t ) - y()) . If the limit is computed weakly in W() (respectively strongly), we talk about weak material derivative (respectively strong material derivative). Moreover, when the map 7 y()[] is positively homogeneous, we will use the term conical material derivative. · If the additional condition y() W() holds for all C C C1 b(Rd ), then one may define a directional derivative called the Eulerian derivative or shape derivative of y() in the direction as the element dy()[] of W() such that: dy()[] := y()[] - y() . · The solution y() is said to be directionally shape differentiable if it admits a directional derivative for any admissible direction . If, in addition, the map 7 dy()[] is positively homogeneous from C C C1 b(Rd ) to W(), y() is said to be conically shape differentiable. Furthermore, if this map is linear continuous from C C C1 b(Rd ) to W(), y() is said to be shape differentiable. Remark 3.1. Linearity and continuity of 7 y()[] is actually equivalent to Gâteaux differentiability of the map 7 y(()) (Id + ). The reader is referred to [6], Chapter 8 for a complete review on the different notions of derivatives. Notation 3.2. Following the notations in [34], when there is no ambiguity, y() will be replaced simply by y, and for some fixed direction , we define yt := y((t)), yt := yt (Id +t ). In the same way, the material and shape derivatives of y at in the direction will be denoted y and dy, respectively. 3.2. Shape sensitivity analysis of the original formulation In this section, we prove that the solution u to problem (2.1) admits conical material/shape derivatives in some specific directions . It is well known that u is not classically shape differentiable because the projection operator onto the closed convex K is not Fréchet-differentiable. However, it is known from [24] that projection operators are conically differentiable. Using this result, it has been proved in [33] that the Signorini problem admits conical material/shape derivatives in 2d and 3d. Here, since formulation (2.1) is slightly different from the classical one due to our choice of n instead of no for the contact, the proof in [33] needs to be adapted. Therefore, we will redo it, for some specific directions . Then, we derive sufficient conditions for the conical material/shape derivative of (2.1) to be solution of a more regular optimization problem. Restriction on the directions . In view of the Zolésio­Hadamard structure theorem, we choose to limit ourselves to geometric deformation fields C C C1 b(Rd ) along the direction of the normal no, i.e. vector fields of the form: = no , where C1 b (Rd ) , (3.3) where no has been extended to C1 (Rd ), which is possible (see [16]) since is assumed to have C1 regularity. Moreover, this choice is well suited to our numerical algorithm, as it will be seen in the last section. The set of all these is denoted . Obviously, is a closed subspace of C C C1 b(Rd ), thus it is a Banach space. Let be a fixed direction. In order to perform sensitivity analysis with respect to the shape, let us first characterize ut = ut (Id +t ) for t > 0. This will be done by writing the problem solved by ut on the transported domain (t), then by bringing it back to by a change of variables. Before that, some additional 10 B. CHAUDET-DUMAS AND J. DETEIX assumptions on the data are required. Indeed, since the domain is transported, the functions C, f and have to be defined everywhere in Rd . They also need to enjoy more regularity for usual differentiability results to hold. In particular, we make the following regularity assumptions : Assumption 3.3. C C1 b (Rd , T4 ), f H1 (Rd ) and H2 (Rd ). Notation 3.4. For the solution ut to the transported problem on (t), we introduce Xt := H1 D(t)((t)) and the convex subset of admissible displacements: Kt := {v Xt | vn gn a.e. on C(t)} . Composition with the operator (Id +t ) will be denoted by (t), for instance, C(t) := C (Id +t ). Besides, the normal component associated to n(t) of a vector v is denoted vn(t). For integral expressions, the Jacobian and tangential Jacobian of the transformation give J(t) := Jac(Id +t ) and J(t) := Jac(t)(Id +t ). Finally, we introduce the transported strain tensor t , the bilinear form at on X × X and the linear form Lt on X, which are the versions of , a and L corresponding to the problem solved by ut , and are defined as follows: at (z, v) := Z C(t) : t (z) : t (v) J(t) z, v X , t (v) := 1 2 v (I +t ) -1 + (I +t T ) -1 vT v X , Lt (v) := Z f(t) v J(t) + Z N (t) v J(t) v X . Lemma 3.5. Let v Xt and , then v Kt if and only if v(t) g(t) + K0. Proof. The result is a direct consequence of our choice, (3.3), of direction . Indeed, n = no on C gives n(t) = n on C. The rest follows from the definitions and the fact that Id +t is an isomorphism: v Kt (v(y) - g(y)) · n(y) 0 for a.e. y C(t) , (v(x + t (x)) - g(x + t (x))) · n(x + t (x)) 0 for a.e. x C , v(t) g(t) + K0 . Remark 3.6. For t sufficiently small, D(t) (Rd \ h0 rig) and thus g Xt, which yields g(t) X. From Lemma 3.5, it follows that ut solves: find ut g(t) + K0 such that, at (ut , v - ut ) Lt (v - ut ) , v g(t) + K0 . Let us introduce the auxiliary unknowns w := u - g and wt := ut - g(t) which both belong to K0. Those functions satisfy the following variational inequalities: a(w, v - w) L(v - w) - a(g, v - w) , v K0 , (3.4a) at (wt , v - wt ) Lt (v - wt ) - at (g(t), v - wt ) , v K0 . (3.4b) Before stating the conical differentiability result for w and u, some additional notations are required. SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 11 Notation 3.7. If (u, ) denotes the solution of problem (2.4), let A, I and B be the subsets of C associated to the constraint u K in problem (2.1): A := {x C : (x) > 0, un(x) - gn(x) = 0} , I := {x C : (x) = 0, un(x) - gn(x) < 0} , B := {x C : (x) = 0, un(x) - gn(x) = 0} . The subsets A, I and B are usually referred to as the active, inactive and biactive sets. Note that they are at least measurable due to the regularities of , u and gn. Moreover, we only consider cases where contact occurs, which means that A is non-empty. The bilinear form a0 and the linear forms 0 , L0 , which will be naturally appear when differentiating (3.4b) with respect to the shape, are introduced: for any u, v X, a0 (u, v) := Z C : 0 (u) : (v) + C : (u) : 0 (v) + (div C + C ) : (u) : (v) , 0 (v) := - 1 2 v +T vT , L0 (v) = Z (div f + f ) v + Z N (div + ) v . Moreover, for any smooth function f defined on Rd , and that does not depend on , we denote f0 [] or simply f0 the following directional derivative: f0 [] := lim t&0 1 t (f(t) - f) = (f) . (3.5) Using this notation, n0 := (n) and for any v X one has vn0 := v ·n0 . For the gap (gn)0 := (gn) and since gn is the oriented distance function to the smooth boundary rig, gn = - n, which implies that (gn)0 = - · n. However, we will still use the notation (gn)0 to emphasize that this term comes from differentiation of the gap. Remark 3.8. From the definition of , one gets that any satisfies = n on C. This implies that: n0 := (n) = (n) n = 0 on C since n is a unit vector. Therefore, one automatically gets for the gap: (gn)0 = (g · n)0 = g0 · n + g · n0 = g0 · n on C . Thus, using the notations introduced earlier, it is possible to replace (gn)0 by g0 n = g0 · n on C. We can now use some of the results on the conical differentiability of projections on closed convex sets. In the next two results, material derivatives will be characterized using some concepts of capacity theory. More precisely the cones containing those derivatives will be defined up to a set of zero capacity (denoted q.e. for quasi-everywhere). The reader is referred to [24] and [7] for further details about capacity theory. Theorem 3.9. Under Assumption 3.3, the solution w of (3.4a) is conically shape differentiable on and its conical material derivative in the direction is given by the solution of the problem: find w Sw (K0) such that, a(w, -w) L0 ( -w) - a0 (u, -w) - a(g0 , -w) , Sw (K0) , (3.6) 12 B. CHAUDET-DUMAS AND J. DETEIX Moreover, one has the characterization Sw (K0) = { X | n 0 q.e. on A B and a(u, ) = L()}. Proof. We follow the same steps as in [23], Section 5.2, and adapt each of them to our specific formulation. The idea is to write wt as the projection onto K0 of some element in X. Then, using the conical differentiability of the projection (see [24]), one is able to perform a first order expansion of wt around t = 0+ . First, let us prove strong continuity of the map t 7 wt at t = 0+ in X. Taking respectively v = w and v = wt as test-functions in (3.4b) and (3.4a), then adding the two formulations, one obtains a(w - wt , w - wt ) (Lt - L)(w - wt ) - at (g(t), w - wt ) + a(g, w - wt ) . Using ellipticity of a, one deduces the following estimation: 0 w - wt X Lt - L X + C at - a + kg(t) - gkX , which yields the result since the right hand side goes to 0, due to the properties of at , Lt , and the regularity of g. Next, using this result and again the properties of at , Lt , we proceed to the expansion of each term in (3.4b), which leads to: at (wt , v - wt ) = a(wt , v - wt ) + ta0 (w, v - wt ) + o(t) , Lt (v - wt ) = L(v - wt ) + tL0 (v - wt ) + o(t) , at (g(t), v - wt ) = a(g, v - wt ) + ta0 (g, v - wt ) + ta(g0 , v - wt ) + o(t) . Plugging these expansions in (3.4b) yields: for all v K0, a(wt , v - wt ) L(v - wt ) - a(g, v - wt ) + t L0 (v - wt ) - a0 (u, v - wt ) - a(g0 , v - wt ) + o(t) . (3.7) Let l, l0 , a0 u X such that for all z X: a(l, z) = L(z) , a(l0 , z) = L0 (z) , a(a0 u, z) = a0 (u, z) . Then relation (3.7) can be equivalently rewritten as: wt = Proja K0 (l - g +t(l0 - a0 u - g0 ) + o(t)) = Proja K0 (l - g +t(l0 - a0 u - g0 )) + o(t) . As seen in Remark 2.5, one has the approximation n = no on C under the small displacements hypothesis. Thus we directly get from [23], Lemma 5.2.9 that K0 is polyhedric. Therefore, from Theorem 2.4, it follows that Proja K0 is conically differentiable and that: wt = w +t Proja Sw(K0) (l0 - a0 u - g0 ) + o(t) , which yields (3.6). Finally, the characterization of Sw (K0) is also given by [23], Lemma 5.2.9. Corollary 3.10. Under Assumption 3.3, the solution u of (2.1) is conically shape differentiable on and its conical material derivative in any direction is given by the solution of: find u S such that, a(u, -u) L0 ( -u) - a0 (u, -u) , S , (3.8) where S := g0 +Sw (K0) = { g0 + X | n g0 n q.e. on A B and a(u, - g0 ) = L( - g0 )}. Proof. This is a direct consequence of Theorem 3.9 since ut = wt + g(t) and therefore u = w + g0 . SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 13 As formulations (3.6) and (3.8) relies on zero capacity sets they are not easy to handle. However, under some mild additional regularity assumptions, it is possible to rewrite those formulations as standard optimization problems. In the same way as in [17], Section 4.1 for the obstacle problem, we introduce the following regularity assumption: Assumption 3.11. A B = int(A B). Remark 3.12. This assumption implies that not only A B is closed, but also it has a non-empty interior. Moreover, for every X it is obvious, that n 0 q.e. on int(A B) = n 0 a.e. on int(A B) . Regarding the converse implication, we invoke the result stated in [34], Lemma 4.31, from which we get that the subspace H := n n H 1 2 (C) | X o associated with the appropriate bilinear form (expressed in [34], Eq. (4.192)) is a Dirichlet space in the sense of [24], Definition 3.1. Therefore, n admits a unique quasi-continuous representative in the equivalence class related to the "q.e. on C equality". Considering this specific representative, one gets from [7], Theorem 5 that n 0 a.e. on int(A B) = n 0 q.e. on int(A B) , since int(A B) is an open subset of C. Quasi-continuity of n on C also implies that n 0 q.e. on int(A B) = A B. Before stating the optimization problem that w solves when Assumption 3.11 is fulfilled, let us define the subspace XA X, and the cone KA such that: XA := { X | n = 0 a.e. on A} , KA := { XA | n 0 a.e. on B} . Clearly, XA is a closed subspace of X, therefore it is a Hilbert space, and KA is a non-empty closed convex cone. Theorem 3.13. If Assumptions 3.3 and 3.11 hold, then w is the solution of (3.6) if and only if it solves: inf KA 1 2 a(, ) - L0 () + a0 (u, ) + a(g0 , ) . (3.9) Proof. It suffices to prove that Sw (K0) = KA. First of all, let us point out that in the characterization of Sw (K0) given in Theorem 3.9, since X and u solves (2.4a), condition a(u, ) = L() is equivalent to (, n)C = 0. As L2 (C), > 0 on A and = 0 on C \ A it follows that: X s.t. n 0 a.e. on A B and (, n)C = 0 KA . And Remark 3.12 gives us the equality of both sets. Finally, some obvious results are obtained, introducing the following assumption. Assumption 3.14. The biactive set B is of measure zero with respect to C (a (d-1)-dimensional submanifold of Rd ). 14 B. CHAUDET-DUMAS AND J. DETEIX Remark 3.15. Assumptions on the biactive set are often made when considering optimal control problems for variational inequalities (see, e.g. [17]) because this is the set of points where non-differentiabilities occur. From the mechanical point of view, a point x B is a point such that un(x) = gn(x) and (x) = 0, that is x is in contact but there is no contact pressure. The set B is often referred to as the set of weak contact points. For example, Assumption 3.14 is verified when all weak contact points represent a countable number of point in 2D or a countable number of curves in 3D. Corollary 3.16. If, in addition to the assumptions of Theorem 3.13, Assumption 3.14 holds, then w is the solution of (3.6) if and only if it solves: find w XA such that a(w, ) = L0 () - a0 (u, ) - a(g0 , ) , XA . (3.10) Proof. When B is of measure zero, KA = XA and the result follows as problem (3.9) can be equivalently rewritten as a linear variational formulation. Remark 3.17. Under the assumption of Corollary 3.16, the material derivative becomes linear with respect to , which implies that u is strongly shape differentiable in L2 (), and its shape derivative in any direction is given by: du = u - u = w + g0 - u , where w XA is the unique solution of (3.10). Note that this conclusion is similar to the one in [27], Remark 4.1. 3.3. Shape sensitivity analysis of the augmented Lagrangian formulation The goal of this section is to prove, at every iteration of the ALM algorithm, the differentiability of uk with respect to the shape. As p+ fails to be Frêchet differentiable it is not possible to rely on the implicit function theorem as in [16]. However, as it is a projection operator, it is conically differentiable, which enables us to show existence of conical material/shape derivatives for uk following the approach in [34]. Then, under assumptions on some specific subsets of C (this will be presented and referred to as Assumption 3.23), classical shape differentiability of uk is proved. First of all, let us briefly recall some properties of function p+. Lemma 3.18. The function p+ : R R is Lipschitz continuous and conically differentiable, with conical derivative at u in the direction v R: dp+(u; v) = 0 if u < 0, p+(v) if u = 0, v if u > 0. Lemma 3.19. The Nemytskij operator p+ : L2 (C) L2 (C) is Lipschitz continuous and conically differen- tiable. Remark 3.20. These results are well known if "conically" is replaced by "directionally", a proof can be found in [37], for example. Conical differentiability then follows directly because for any u R, dp+(u; ·) : R R is obviously positively homogeneous, and the same property holds for dp+(u; ·) : L2 (C) L2 (C) for any u L2 (C). Theorem 3.21. If Assumption 3.3 holds and 0 H 3 2 (C), then for any k 0, the pair uk , k defined by (2.7)­(2.8) is conically shape differentiable on , strongly in X ×L2 (C). SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 15 Proof. As the solution of (2.7) at each step k + 1 depends on the previous iterate, the result will be proved by induction. Base case For k = 0, given 1 > 0, the first iteration of the ALM gives u1 X the solution of a(u1 , v) + R1 n(u1 ), vn C = L(v) , where we have introduced the non-linear map R1 n : X H 1 2 (C) defined as, R1 n(v) := p+ 0 + 1 (vn - gn) v X . Obviously, R1 n(u1 ) = 1 , but we prefer here to denote R1 n(u1 ) to emphasize that 1 is defined explicitly from u1 and the data. Using the same arguments as in [3], one gets that, due to the regularity of the data, especially 0 , u1 admits a strong material derivative u1 X in each direction , which is the unique solution of a(u1 , v) + R1 n 0 (u1 ), vn C = L1 [](v) , where, for any v X, with material derivative v X, we have R1 n 0 (v) := dp+ 0 + 1 (vn - gn); (0 )0 + 1 (vn - g0 n) , L1 [](v) := L0 (v) - a0 (u1 , v) - Z C 1 vn div , where the notation introduced in (3.5) has been used for (0 )0 . Moreover, as 1 only depends on 0 and u1 , it is clear that it also admits a strong material derivative in L2 (C) in the direction , which is simply obtained by differentiating (2.8): 1 = R1 n 0 (u1 ) , (3.11) which finishes to prove that u1 , 1 is directionally shape differentiable. Moreover, one has that a is bilinear, the maps 7 L1 [], (0 )0 , g0 n are linear and dp+ 0 + 1 (u1 n - gn); · is positively homogeneous. Thus one deduces that the map 7 u1 is positively homogeneous. The same property holds for 7 1 due to (3.11). Inductive step Let k 1 and assume uk , k is conically shape differentiable, strongly in X ×L2 (C). This implies that k admits a strong material derivative k L2 (C). Hence, using the arguments of the case k = 0, one may differentiate (2.7), which leads to existence of a unique material derivative uk+1 X for uk+1 that solves: a(uk+1 , v) + Rk+1 n 0 (uk+1 ), vn C = Lk+1 [](v) , (3.12) where the same notation as in the case k = 0 has been used for Lk+1 [], while the notation for Rk+1 n 0 has been slightly adapted: Rk+1 n 0 (v) := dp+ k + k+1 (vn - gn); k + k+1 (vn - g0 n) . 16 B. CHAUDET-DUMAS AND J. DETEIX Especially, this also proves that k+1 admits a material derivative in L2 (C) defined as k+1 = Rk+1 n 0 (uk+1 ) . (3.13) Again, since a is bilinear, 7 Lk+1 [], g0 n are linear, and dp+ k + k+1 (uk+1 n - gn); · , 7 k are positively homogeneous, one deduces that 7 uk+1 , k+1 is positively homogeneous as well. Remark 3.22. Existence of directional (even conical) material/shape derivatives in all directions does not guarantee that the solution uk is shape differentiable. Indeed, as these derivative depends on p+, even when the solutions of the previous iteration uk-1 and k-1 are linear with respect to , the derivative uk may fail to have that property if the set k-1 + k (uk n - gn) = 0 is not of null measure. In light of the previous remark, we get interested in sufficient conditions for shape differentiability of uk . We introduce the corresponding subsets for problem (2.7), for each k 1, Ak := x C : k-1 + k uk n - gn (x) > 0 , Ik := x C : k-1 + k uk n - gn (x) < 0 , Bk := x C : k-1 + k uk n - gn (x) = 0 . Using these notations, for any k 1, we are able to state conditions that guarantee shape differentiability of uk , and will be referred to as Assumption 3.23 at rank k: Assumption 3.23. For each j {1, . . . , k}, the set Bj is of measure zero. Remark 3.24. This assumption is similar to Assumption 3.14, but in the case of the regularized formulation obtained at each iteration. Corollary 3.25. Let k 1, (uk , k ) still denotes the solution of (2.7)­(2.8) and (uk , k ) its material derivative, defined by (3.12)­(3.13). Under the assumptions of Theorem 3.21, if Assumption 3.23 holds at rank k, then the map 7 (uk , k ) is linear continuous from to X ×L2 (C). Therefore, uk is (strongly) shape differentiable in L2 (), and its shape derivative in any given direction writes: duk = uk - uk . Proof. Here again, let us proceed by induction. Base case Let k = 1. From Assumption 3.23 at rank 1, one gets that |B1 | = 0, which implies that 1 = R1 n 0 (u1 ) = A1 (0 )0 + 1 u1 n - g0 n a.e. on C . Therefore, u1 is the solution of a linear variational formulation. Moreover, due to the regularities of 0 , gn, n, and the expression of L1 [], the maps 3 7- (0 )0 L2 (C) , 3 7- g0 n L (C) , 3 7- L1 [] X , are all linear continuous. Therefore, 7 (u1 , 1 ) is also linear continuous, from to X ×L2 (C). SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 17 Inductive step Let k 1 and assume the result of the corollary is true at iteration k. Now, suppose Assump- tion 3.23 holds at rank k + 1. This exactly means that Assumption 3.23 holds at rank k and |Bk+1 | = 0. Thus, one has k+1 = Rk+1 n 0 (uk+1 ) = Ak+1 k + k+1 uk+1 n - g0 n a.e. on C . Hence, the variational formulation (3.12) is linear. Using the result at iteration k, one gets that 7 k is linear continuous from to L2 (C). This, combined with the properties of gn, n, and linear continuity of Lk+1 [·] from to X , enables us to conclude. 3.4. Convergence of the directional shape derivatives In this section, we aim at establishing the consistency of the augmented Lagrangian method with respect to shape differentiability. In other words, for any fixed, we get interested in convergence properties of the sequence n uk , k o k as k . Especially, we would like to derive sufficient conditions for n uk o k to converge to u the solution of (3.8). Remark 3.26. As we will manipulate in this section the material derivatives of uk and k , it will be required that Assumption 3.3 holds (see Theorem 3.9). As for the original problem, let us define wk := uk - g X. Since the existence of strong material derivatives in all directions has been established for uk (Thm. 3.21), we get that wk admits a strong material derivative in the direction given by wk = uk - g0 X. It follows from (3.12) and (3.13) that wk solves: a(wk , v) - Lk [](v) + a(g0 , v) + k , vn C = 0 , v X , (3.14a) k - dp+ k-1 + k wk n; k-1 + k wk n = 0 , a.e. on C . (3.14b) Note that (3.14b) can be equivalently rewritten as: k = k-1 + k wk n on Ak , 0 on Ik , p+ k-1 + k wk n on Bk . (3.15) In order to have a valid expression for all k 1, the notation 0 := 00 will be used. Finally, we give some straightforward but very useful properties of dp+: · for all u, v R, one has dp+(u; v)v = |dp+(u; v)|2 , · dp+ is continuous on R2 \ ({0} × R). Theorem 3.27. For any increasing sequence of strictly positive parameters k k , there exists a subsequence of n wk , k o k that is bounded in X ×H- 1 2 (C). Proof. For now, let k k R + be any increasing sequence, from which the ALM algorithm generates iterates (uk , k ). From Theorem 2.11, one gets that (uk , k ) k is bounded in X ×L2 (C), from which one deduces that Lk [] k is bounded in X . Thus, taking v = wk as test-function in (3.14a) yields 0 wk 2 X + k , wk n C C wk X . (3.16) 18 B. CHAUDET-DUMAS AND J. DETEIX Now, we rewrite the second term, then use the properties of dp+ and Young's inequality: k , wk n C = 1 k k , k-1 + k wk n C - 1 k k , k-1 C = 1 k k 2 0,C - 1 k k , k-1 C 1 2k k 2 0,C - k-1 2 0,C . (3.17) Plugging this into inequality (3.16) leads to 0 wk 2 X - C wk X + 1 2k k 2 0,C 1 2k k-1 2 0,C 1 2k-1 k-1 2 0,C , (3.18) since k k-1 . From here, let us distinguish the two possible cases. (i) Case 1 k k 2 0,C k bounded. Boundedness of the whole sequence n wk o k in X follows directly from (3.18). (ii) Case 1 k k 2 0,C k unbounded. In that case, let us sum (3.18) from 1 to k, with k > 1: k X l=1 0 wl 2 X - C wl X + 1 2k k 2 0,C 1 21 0 2 0,C . If we denote Wk the first term of the left hand side, it follows that the sequence Wk k is unbounded below. Thus there exists a subsequence (denoted (k)) such that W(k) k is strictly monotone and tends to -. Especially, strict monotonicity of this subsequence implies that, for all k 1, W(k+1) - W(k) = (k+1) X l=(k) 0 wl 2 X - C wl X < 0 . Hence, for all k 1, there exists (k) [[(k), (k + 1)]] such that 0 w(k) 2 X - C w(k) X < 0, which proves that n w(k) o k is bounded in X. Then, boundedness of n k o k or of a subsequence of n k o k in H- 1 2 (C) immediately follows from (3.14a) and the subjectivity of the trace operator from X to H 1 2 (C). Corollary 3.28. There exists an increasing sequence of strictly positive parameters k k such that the whole sequence n wk , k o k defined by (3.14) is bounded in X ×H- 1 2 (C). SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 19 Proof. From estimation (3.18), it is clear that if the sequence of parameters k k is built from some 1 > 0 following the rule, for k > 1, k = max k-1 , k-1 2 0,C , (3.19) then one automatically gets boundedness of the whole sequence n wk o k in X. In other words, such a choice of parameters ensures that we are in case (i) of the previous proof. Again, boundedness of n k o k in H- 1 2 (C) follows. Of course, the previous result implies that for any sequence of parameters, the associated iterates wk , k converge weakly in X ×H- 1 2 (C) to some limit (w, ), up to a subsequence. In order to specify this limit, we need to make the following additional assumption. Assumption 3.29. The iterates k generated by the ALM converge to a.e. on C. Remark 3.30. Because of the non-continuity of dp+ on {0} × R getting a convergence result without Assumption 3.14 seems rather difficult (maybe even impossible). Lemma 3.31. Let k k be any increasing sequence in R +, (uk , k ) still denotes the solution of (2.7)­(2.8), and (u, ) the solution of (2.4). When Assumptions 3.14 and 3.29 hold, the characteristic functions Ak (respectively Bk ) converge to A (respectively 0) strongly in Lp (C), for each 1 < p < +, up to a subsequence. Proof. First of all, note that since Ak and A are measurable, so are Ak and A, for any k 0. Since those functions are also bounded on C which is of finite measure, they both belong to L1 (C), and thus to any Lp (C) with 1 < p < +. Now, let us begin with proving pointwise convergence, from which we will deduce weak then strong convergence. Since uk u strongly in L2 (C), there exists a subsequence, still denoted uk , that converges a.e. on C. From now, we consider this subsequence. Then, from Assumption 3.14, for a.e. x C, x is either in A, or in I. · If x A, then there is some > 0 such that (x) > . From Assumption 3.29, one gets that k0 > 0, k k0, |k (x) - (x)| < /2. Especially, for all k k0, k+1 (x) = p+ k (x) + k+1 uk+1 n (x) - gn(x) > /2 . Thus, for all k k0 + 1, Ak (x) = 1 = A(x), and Bk (x) = 0. · If x I, then (x) = 0 and there is some 0 > 0 such that un(x) - gn(x) < -0 /1 . Moreover, k0 0 > 0 such that k k0 0, | uk n(x) - un(x)| < 0 /21 and k (x) < 0 /4. Consequently, for such values of k, one has k (x) + k+1 uk+1 n (x) - gn(x) < 0 /4 - k+1 0 /21 -0 /4 . Hence, for any k k0 0 + 1, one gets Ak (x) = 0 = A(x), and Bk (x) = 0. This proves that Ak A and Bk 0 a.e. on C. Let 1 < p < +. For {Bk }k, strong convergence follows directly from Lebesgue's dominated convergence theorem and the fact that |Bk |p 1 a.e. on C. For {Ak }k, since the sequence is bounded in Lp (C), weak convergence in Lp (C) follows. Taking for example p = 2, one gets that Z C Ak = Z C Ak · 1 - Z C A . 20 B. CHAUDET-DUMAS AND J. DETEIX Obviously, this also proves that kAk kLp(C ) kAkLp(C ). For such values of p, Lp (C) is uniformly convex, therefore weak convergence and convergence of the norms imply strong convergence. Theorem 3.32. Suppose Assumptions 3.14 and 3.29 hold. Then, choosing the parameters k as in Corol- lary 3.28 leads to one of the two following cases: (i) k k is bounded, then the whole sequence n wk o k converges to w, the solution of (3.10), strongly in X, (ii) k k is unbounded, then n wk o k converges weakly to some limit w XA, up to a subsequence. Proof. As mentioned above, a direct consequence of Theorem 3.27 is that, up to a subsequence, the iter- ates wk , k converge weakly to some limit (w, ) X ×H- 1 2 (C). In the following, let us consider that subsequence, which we still denote using the superscript k . Due to strong convergence uk , k (u, ) in X ×H- 1 2 (C), it is clear that Lk [] L[] strongly in X , where L[] is defined as: for all v X, L[](v) := L0 (v) - a0 (u, v) - Z C vn div . Consequently, the weak limit satisfies: a(w, v) - L[](v) + a(g0 , v) + h, vniC = 0 , v X . (3.20) From now, we distinguish two possible cases. (i) Case k k bounded. From the definition of k k (see (3.19)), it follows that n k o k is bounded in L2 (C). Therefore, there exists a subsequence that converges weakly in L2 (C) to (due to uniqueness of the weak limit), which also proves that L2 (C). As k k is an increasing bounded sequence, it converges, say to > 0. Now, let L4 (C), one has: k , C = k-1 + k wk n, Ak + k , Bk = k-1 , Ak C + k wk n, Ak C + k , Bk C . We know that k , wk n converge weakly in L2 (C), and using the results of Lemma 3.31 with p = 4, it follows that Ak , Bk converge strongly in L2 (C). Thus, passing to the limit on both sides of the equality yields: = A + wn in L 4 3 (C) . From this, one deduces that w XA and a(w, v) - L0 (v) + a0 (u, v) + a(g0 , v) = 0 , v XA , which means that w = w, since the solution of this problem is unique. Uniqueness also proves that the whole sequence converges weakly to w. SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 21 Finally, to get strong convergence, let us take v = wk as test-function in (3.14a). Since the embedding H 1 2 (C) , L2 (C) is compact, one obtains for the second term of (3.14a): k , wk n C - , wn C = 0 , from which we get that a(wk , wk ) a(w, w). The ellipticity of a finishes the proof. (ii) Case k k unbounded. Again, let us take v = wk as test-function in (3.14a), and rewrite the second term of the formulation. k , wk n C = k-1 + k wk n, wk n Ak + k , wk n Bk k wk n 2 0,Ak - C k-1 -1/2,C + k -1/2,C wk X . Using this estimation in (3.14a), one obtains that k wk n 2 0,Ak k is bounded. Using Lemma 3.31 with p = 2 and the continuous embedding H 1 2 (C) , L4 (C), one obtains, up to a subsequence, wk n 2 0,Ak = Z C Ak · |wk n|2 - kwnk 2 0,A . On the other hand, since k k is unbounded, there exists a subsequence (of the previous subsequence) that diverges to +. Hence, boundedness of k wk n 2 0,Ak k implies that kwnk 2 0,A = 0, that is w XA. Remark 3.33. The previous result relies on a specific choice of parameters k k , defined by (3.19), that may not be easy to build in practice. Moreover, Assumption 3.29 is essential in the proof of Lemma 3.31, and thus of Theorem 3.32. However it seems difficult to give an interpretation of Assumption 3.29 and, although Theorem 2.11 ensures weak convergence of {k }k in L2 (C), convergence almost everywhere seems to be a strong assumption. For those reasons, this (partial) convergence result should be understood as follows: there exist cases where we are able to prove consistency of the ALM with respect to shape sensitivity analysis. 3.5. Shape derivative of a generic criterion As we aim at expressing first order optimality conditions related to problem (3.2), we need to find an expression for the shape derivative of the functional Jk . From now, let us consider functionals which take the generic form: Jk () = J (, uk ()) := Z l(uk ()) + Z m(uk ()) . (3.21) Let us make the usual regularity assumptions: the functions l, m are C1 (Rd , R), and their derivatives, denoted l0 , m0 , are Lipschitz. It is also assumed that those functions and their derivatives satisfy, for all u, v Rd , |l(u)| C 1 + |u|2 , |m(u)| C 1 + |u|2 , (3.22a) |l0 (u) · v| C|u · v| , |m0 (u) · v| C|u · v| , (3.22b) 22 B. CHAUDET-DUMAS AND J. DETEIX for some constants C > 0. Let us state the well known shape differentiability result for such functionals Jk (see, e.g. [16]). Theorem 3.34. Let k 1. Under the assumptions of Theorem 3.21, Jk is conically shape differentiable at , and its derivative in the direction writes: dJk ()[] = Z l0 (uk ) · uk + l(uk ) div + Z m0 (uk ) · uk + m(uk ) div . (3.23) Corollary 3.35. Let k 1. In addition to the previous result, when the assumptions of Corollary 3.25 are verified, the map 7 dJk ()[] is linear from to R. Thus, Jk is classically shape differentiable at . Remark 3.36. It follows from Theorem 3.27 that dJk is bounded for suitable choices of parameters k . There- fore, formula (3.23) produces usable shape derivatives, regardless how many iterations of the ALM algorithm are performed. Moreover, a sufficient condition to get convergence of dJk k is that n uk o k converges weakly in X. 4. Conclusion In this work, we have expressed sufficient conditions for shape differentiability of u, the solution to the original Signorini problem. We also got interested in shape differentiability properties for the sequence of the iterates uk (approaching u) obtained when applying the augmented Lagrangian method to this problem. After proving that these iterates were always conically shape differentiable, we have given sufficient conditions for the associated shape derivatives to converge to the shape derivative of u. On the other hand, we also found conditions that guarantee shape differentiability of the iterates. A natural extension of the present work would be to consider the contact problem with Tresca friction. For this model, proving conical shape differentiabilty of uk and expressing sufficient conditions for uk to be shape differentiable is rather straightforward since the arguments from [3] apply. However, proving conical shape differentiability and deriving sufficient conditions for classical shape differentiability of the solution to the associated variational inequality seems more difficult. Indeed, for this problem, conical shape differentiability has been proved only in the two-dimensional case and for specific directions , see [33]. References [1] P. Beremlijski, J. Haslinger, M. Kocvara and J. Outrata, Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13 (2002) 561­587. [2] P. Beremlijski, J. Haslinger, J. Outrata and R. Pathó, Shape optimization in contact problems with Coulomb friction and a solution-dependent friction coefficient. SIAM J. Control Optim. 52 (2014) 3371­3400. [3] B. Chaudet-Dumas and J. 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Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Studies in Mathematics and Its Applications. Elsevier Science (2000). [11] J. Haslinger and A. Klarbring, Shape optimization in unilateral contact problems using generalized reciprocal energy as objective functional. Nonlinear Anal.: Theory Methods Appl. 21 (1993) 815­834. [12] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design: Theory and Applications. John Wiley & Sons (1988). SHAPE DERIVATIVES FOR AN AUGMENTED LAGRANGIAN FORMULATION OF ELASTIC CONTACT PROBLEMS 23 [13] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material, and Topology Design. John Wiley & Sons (1996). [14] J. Haslinger, J. Outrata and R. Pathó, Shape optimization in 2D contact problems with given friction and a solution-dependent coefficient of friction. Set-Valued Var. Anal. 20 (2012) 31­59. [15] C. Heinemann and K. Sturm, Shape optimization for a class of semilinear variational inequalities with applications to damage models. SIAM J. Math. Anal. 48 (2016) 3579­3617. [16] A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis. European Mathematical Society Publishing House (2018). [17] M. Hintermüller and A. Laurain, Optimal shape design subject to elliptic variational inequalities. SIAM J. Control Optim. 49 (2011) 1015­1047. [18] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM (2008). [19] J. Jarusek, M. Krbec, M. Rao and J. Sokolowski, Conical differentiability for evolution variational inequalities. J. Differ. Eq. 193 (2003) 131­146. [20] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Vol. 8. SIAM (1988). [21] A. Klarbring and J. Haslinger, On almost constant contact stress distributions by shape optimization. Struct. Optim. 5 (1993) 213­216. [22] M. Kocvara and J. Outrata, On optimization of systems governed by implicit complementarity problems. Numer. Funct. Anal. Optim. 15 (1994) 869­887. [23] A. Maury, Shape Optimization for Contact and Plasticity Problems Thanks to the Level Set Method. Ph.D. thesis, Université Pierre et Marie Curie-Paris VI (2016). [24] F. Mignot, Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130­185. [25] F. Mignot and J.-P. Puel, Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466­476. [26] F. Murat and J. Simon, Étude de problèmes d'optimal design, in IFIP Technical Conference on Optimization Techniques. Springer (1975), 54­62. [27] P. Neittaanmäki, J. Sokolowski and J.-P. Zolésio. Optimization of the domain in elliptic variational inequalities. Appl. Math. Optim. 18 (1988) 85­98. [28] J. Sokolowski and J.-P. Zolésio, Shape sensitivity analysis for variational inequalities, in System Modeling and Optimization. Springer (1982), 401­407. [29] J. Sokolowski and J.-P. Zolésio, Dérivée par rapport au domaine de la solution d'un problème unilatéral. C. R. Acad. Sc. Paris 301 (1985) 103­106. [30] J. Sokolowski and J.-P. Zolésio, Sensitivity analysis of elastic-plastic torsion problem, in System Modelling and Optimization. Springer (1986) 845­853. [31] J. Sokolowski and J.-P. Zolésio, Shape design sensitivity analysis of plates and plane elastic solids under unilateral constraints. J. Optim. Theory Appl. 54 (1987) 361­382. [32] J. Sokolowski and J.-P. Zolésio, Shape sensitivity analysis of unilateral problems. SIAM J. Math. Anal. 18 (1987) 1416­1437. [33] J. Sokolowski and J.-P. Zolésio, Shape sensitivity analysis of contact problem with prescribed friction. Nonlinear Anal.: Theory Methods Appl. 12 (1988) 1399­1411. [34] J. Sokolowski and J.-P. Zolésio, Introduction to shape optimization, in Introduction to Shape Optimization. Springer (1992). [35] J. Sokolowski and J.-P. Zolésio, Differential stability of solutions to unilateral problems. Free Bound. Probl.: Appl. Theory 4 (1993) 537­547. [36] G. Stadler, Infinite-dimensional semi-smooth Newton and augmented Lagrangian methods for friction and contact problems in elasticity. Selbstverl. (2004). [37] L.M. Susu. Optimal control of a viscous two-field gradient damage model. GAMM-Mitteilungen 40 (2018) 287­311. [1] P. Beremlijski, J. Haslinger, M. Kočvara and J. Outrata, Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13 (2002) 561–587. [2] P. Beremlijski, J. Haslinger, J. Outrata and R. Pathó, Shape optimization in contact problems with Coulomb friction and a solution-dependent friction coefficient. SIAM J. Control Optim. 52 (2014) 3371–3400. [3] B. Chaudet-Dumas and J. Deteix, Shape derivatives for the penalty formulation of elastic contact problems with Tresca friction. SIAM J. Control Optim. 58 (2020) 3237–3261. [4] P. G. Ciarlet (Ed.), Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. Studies in Mathematics and Its Applications. Academic Press, Elsevier (1988) 20. [5] M. C. Delfour and J.-P. Zolésio, A boundary differential equation for thin shells. J. Differ. Eq. 119 (1995) 426–449. [6] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimisation. Society for Industrial and Applied Mathematics, Philadelphia, PA (2001). [7] J. Deny, Théorie de la capacité dans les espaces fonctionnels. Sémin. Brelot-Choquet-Deny. Théor. potentiel 9 (1965) 1–13. [8] C. Eck, J. Jarusek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems. CRC Press (2005). [9] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Vol. 28. Siam (1999). [10] M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Studies in Mathematics and Its Applications. Elsevier Science (2000). [11] J. Haslinger and A. Klarbring, Shape optimization in unilateral contact problems using generalized reciprocal energy as objective functional. Nonlinear Anal.: Theory Methods Appl. 21 (1993) 815–834. [12] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design: Theory and Applications. John Wiley & Sons (1988). [13] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material, and Topology Design. John Wiley & Sons (1996). [14] J. Haslinger, J. Outrata and R. Pathó, Shape optimization in 2D contact problems with given friction and a solution-dependent coefficient of friction. Set-Valued Var. Anal. 20 (2012) 31–59. [15] C. Heinemann and K. Sturm, Shape optimization for a class of semilinear variational inequalities with applications to damage models. SIAM J. Math. Anal. 48 (2016) 3579–3617. [16] A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis. European Mathematical Society Publishing House (2018). [17] M. Hintermüller and A. Laurain, Optimal shape design subject to elliptic variational inequalities. SIAM J. Control Optim. 49 (2011) 1015–1047. [18] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM (2008). [19] J. Jarušek, M. Krbec, M. Rao and J. Sokołowski, Conical differentiability for evolution variational inequalities. J. Differ. Eq. 193 (2003) 131–146. [20] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Vol. 8. SIAM (1988). [21] A. Klarbring and J. Haslinger, On almost constant contact stress distributions by shape optimization. Struct. Optim. 5 (1993) 213–216. [22] M. Kočvara and J. Outrata, On optimization of systems governed by implicit complementarity problems. Numer. Funct. Anal. Optim. 15 (1994) 869–887. [23] A. Maury, Shape Optimization for Contact and Plasticity Problems Thanks to the Level Set Method. Ph.D. thesis, Université Pierre et Marie Curie-Paris VI (2016). [24] F. Mignot, Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130–185. [25] F. Mignot and J.-P. Puel, Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466–476. [26] F. Murat and J. Simon, Étude de problèmes d’optimal design, in IFIP Technical Conference on Optimization Techniques. Springer (1975), 54–62. [27] P. Neittaanmäki, J. Sokolowski and J.-P. Zolésio. Optimization of the domain in elliptic variational inequalities. Appl. Math. Optim. 18 (1988) 85–98. [28] J. Sokolowski and J.-P. Zolésio, Shape sensitivity analysis for variational inequalities, in System Modeling and Optimization. Springer (1982), 401–407. [29] J. Sokolowski and J.-P. Zolésio, Dérivée par rapport au domaine de la solution d’un problème unilatéral. C. R. Acad. Sc. Paris 301 (1985) 103–106. [30] J. Sokolowski and J.-P. Zolésio, Sensitivity analysis of elastic-plastic torsion problem, in System Modelling and Optimization. Springer (1986) 845–853. [31] J. Sokolowski and J.-P. Zolésio, Shape design sensitivity analysis of plates and plane elastic solids under unilateral constraints. J. Optim. Theory Appl. 54 (1987) 361–382. [32] J. Sokolowski and J.-P. Zolésio, Shape sensitivity analysis of unilateral problems. SIAM J. Math. Anal. 18 (1987) 1416–1437. [33] J. Sokołowski and J.-P. Zolésio, Shape sensitivity analysis of contact problem with prescribed friction. Nonlinear Anal.: Theory Methods Appl. 12 (1988) 1399–1411. [34] J. Sokolowski and J.-P. Zolésio, Introduction to shape optimization, in Introduction to Shape Optimization. Springer (1992). [35] J. Sokolowski and J.-P. Zolésio, Differential stability of solutions to unilateral problems. Free Bound. Probl.: Appl. Theory 4 (1993) 537–547. [36] G. Stadler, Infinite-dimensional semi-smooth Newton and augmented Lagrangian methods for friction and contact problems in elasticity. Selbstverl. (2004). [37] L. M. Susu. Optimal control of a viscous two-field gradient damage model. GAMM-Mitteilungen 40 (2018) 287–311. COCV_2021__27_S1_A16_0dab34875-5645-4a93-b9f5-99a11c5e0138cocv20007110.1051/cocv/202006410.1051/cocv/2020064 Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces Bahrouni Sabri 1 0000-0002-2106-7031 Salort Ariel M. 2* 1 Department of Mathematics, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia. 2 Departamento de Matemática, FCEyN - Universidad de Buenos Aires and IMAS - ^°ICET Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n. Buenos Aires, Argentina. *Corresponding author: [e-mail] SupplementS15 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF)Full (DJVU)

In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractional g-Laplacian (-Δ$$)$$ with different boundary conditions, namely, Dirichlet, Neumann and Robin.

Fractional Orlicz-Sobolev spaces Neumann and Robin problem three solutions eigenvalue problems 46E30 35R11 45G05 Conicet PIP 11220150100036CO idline ESAIM: COCV 27 (2021) S15 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S15 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020064 www.esaim-cocv.org NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS IN FRACTIONAL ORLICZ-SOBOLEV SPACES Sabri Bahrouni1 and Ariel M. Salort2,* Abstract. In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractional g-Laplacian (-g)s with different boundary conditions, namely, Dirichlet, Neumann and Robin. Mathematics Subject Classification. 46E30, 35R11, 45G05. Received March 30, 2020. Accepted September 29, 2020. 1. Introduction In the recent years has been an increasing interest in studying non-local problems with p-structure due to its accurate description of models involving anomalous diffusion. In several branches of science have been observed some phenomena having a non-local nature, which, nonetheless, do not obey a power-like growth law. See for instance [2, 3, 6, 20, 23] and references therein. The suitable operator to describe these kind of phenomena is the fractional g-Laplacian introduced in [20] and defined as (-g)s u := p.v. Z Rn g (|Dsu|) Dsu |Dsu| dy |x - y|n+s , (1.1) and defined in the principal value sense; here G is a Young function such that g = G0 and s (0, 1) is a fractional parameter. The quantity Dsu := u(x)-u(y) |x-y|s is the s-Holder quotient. Problems involving this operator have recently attracted some attention. We refer the readers to [3­5, 7­ 10, 16, 20, 21, 35]. Observe that when G(t) = tp /p, p > 1, (1.1) becomes the well-known fractional p-Laplacian operator. See also [13] for a non-singular version. Given an open bounded domain Rn with smooth boundary ( C0,1 is enough) the first aim of the present article is to study existence of nontrivial solutions of the following equation involving the nonlinearities Keywords and phrases: Fractional Orlicz-Sobolev spaces, Neumann and Robin problem, three solutions, eigenvalue problems. 1 Department of Mathematics, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia. 2 Departamento de Matematica, FCEyN - Universidad de Buenos Aires and IMAS - oICET Ciudad Universitaria, Pabellon I (1428) Av. Cantilo s/n. Buenos Aires, Argentina. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 S. BAHROUNI AND A.M. SALORT f and h with homogeneous Robin boundary condition on Rn \ ( (-g)s u + g(u) u |u| = f(x, u) + µh(x, u) in Ngu + (x)g(u) u |u| = 0 in Rn \ . (1.2) Here, we introduce a non-local normal derivative in this settings as Ngu(x) := Z g (|Dsu|) Dsu |Dsu| dy |x - y|n+s , x Rn \ , (1.3) which can be seen as the natural generalization of the non-local derivative introduced in [17]. Nonlocal equations for the fractional p-Laplacian with boundary conditions involving non-local normal derivatives have been recently developed in the literature; see for instance [1, 14, 15, 17, 18, 31, 37, 38]. Regarding existence of solutions to problem (1.2) in the particular case of the fractional p-Laplacian, there has been some recent develops. In [31], under suitable conditions on the nonlinearities, the authors obtain existence of at most one positive solution by following the celebrated paper of Brezis-Oswald. The authors in [30], for the same problem but with 0, and under suitable conditions on f, by using variational methods obtain existence of two positive solutions. It worths to be mention that the local counterpart of (1.2) for Orlicz functions in the Dirichlet case was studied in [12, 24, 32]. For some existence results in the nonlocal Orlicz case with Dirichlet boundary conditions see [5]. For problems with critical Trudinger-Moser nonlinearities see [28]. Finally, for an introduction to the theory of variational methods for nonlocal fractional problems we recommend [29]. Our first main scope is to provide conditions on the Young function G, on the nonlinearities f and h, and over , µ and to ensure existence of at least three nontrivial (weak) solutions of (1.2). Our arguments are based in the celebrated result [34] by B. Ricceri together with an integration by parts formula related to the operator (-g)s . The Young function G = R t 0 g(t) dt is assumed to satisfy the following growing condition 1 < p- tg(t) G(t) p+ < t > 0, (G1) for fixed constants p± . Moreover, the following structural condition is assumed t 7 G( t), t [0, [ is convex. (G2) To ensure compactness we restrict ourselves to the sub-critical case of the fractional Orlicz-Sobolev embeddings: Z 1 0 G-1 () n+s n d < and Z + 1 G-1 () n+s n d = . (G3) Here, and µ are two positive real parameters in a suitable range and L (Rn \) is strictly positive. The nonlinearities f, h: × R R will be suitable Caratheodory functions assumed to belong to the class A defined as follows: f A if it fulfills the growth condition |f(x, t)| w(x)(1 + m(|t|)) for a.e x and for all t R, (f1) where w is a positive function such that w L () and m = M0 , being M a Young function decreasing essentially more rapidly than the critical Sobolev function G, i.e., M G, being G the critical function in the fractional Orlicz-Sobolev embedding (see Sect. 2.2 for details). We remark that (f1) is fulfilled, for instance, if |f(x, t)| w(x)(1 + |u|)q-1 for some q (1, p- ), being p- := np- n-p- . NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 3 From now on, we denote F(x, t) = Z t 0 f(x, s) ds, H(x, t) = Z t 0 h(x, s) ds, F(u) = Z F(x, u) dx, and we anticipate that the natural space to look for (weak) solutions of (1.2) is given by (see Sect. 3 for details and motivations) X = ( u measurable: ZZ R2n\(c)2 G(|Dsu(x, y)|) dµ + Z G(|u|) dx + Z Rn\ G(|u|) dx < ) , where we have denoted dµ := dx dy |x-y|n . With these preliminaries, our first result reads as follows. Theorem 1.1. Let G be a Young function satisfying the structural hypotheses (G1), (G2) and (G3), let L (Rn \ ) and let f, h A be such that max ( lim sup |u|0 supx F(x, u) G(u) , lim sup |u|+ supx F(x, u) G(u) ) 0, (F1) sup uX Z F(x, u) dx > 0. (F2) Then, if we set = inf J (u) F(u) : u X, F(u) > 0 , where J (u) := ZZ R2n\(c)2 G(|Dsu|) dµ + Z G(|u|) dx + Z Rn\ G(|u|) dx, for each compact interval [a, b] (, ) there exists > 0 with the following property: for every [a, b] and h, there exists > 0 such that, for each µ [0, ], problem (1.2) has at least three weak solutions whose norms in X are less than . We also prove the following result characterizing the geometry involved in the class of admissible nonlinearities. Theorem 1.2. Let G be a Young function satisfying (G1),(G2) and (G3), let L (Rn \ ) and let f, h A such that (i) there exists a Young function B(t) = R t 0 b() d such that tb(t) B(t) b+ < p- and B G, and a constant c1 > 0 for which F(x, t) c1(1 + B(t)) for all (x, t) × R; 4 S. BAHROUNI AND A.M. SALORT (ii) there exist a constant c2 > 0, 1 > 0 and a Young function D(t) = R t 0 d() d such that p+ < d- td(t) D(t) and G D for which F(x, t) c2D(t) for all (x, t) × [-1, 1]; (iii) there exists 2 R \ {0} such that F(x, 2) > 0 and F(x, t) 0 for all (x, t) × [0, 2]. Then there exists > 0 such that for every compact interval [a, b] (, ) there exists a real number such that, for every [a, b] and every continuous function h there exists > 0 such that, for each µ [0, ], then problem (1.2) has at least three weak solutions whose norms in X are less than . We remark that the class of admissible nonlinearities in Theorem 1.1 includes perturbations of powers and concave-convex type combinations, among other. See Section 4 for further examples. Very close to (1.2), as a second aim, we will study eigenvalues and minimizers of several nonlocal problems with non-standard growth involving different boundary conditions. For the case of powers, that is, for fractional p-Laplacian type operators, the Dirichlet case was studied for instance in [27, 36], for the Neumann case see for instance [15, 30], the Robin case was dealt in [19]. For general Orlicz functions and Dirichlet boundary conditions we refer to [35]. To be more precise, we consider the following Dirichlet eigenvalue problem ( (-g)s u + g(|u|) u |u| = g(|u|) u |u| in u = 0 in Rn \ , (1.4) the following Neumann problem in terms of the nonlocal normal derivative Ng ( (-g)s u + g(|u|) u |u| = g(|u|) u |u| in Ngu = 0 in Rn \ , (1.5) the following problem, which, from a probabilistic point of view can be seen also as a Neumann eigenvalue problem (see [15]) ( (-g)s u + g(|u|) u |u| = g(|u|) u |u| in u Ws,G reg , (1.6) and finally, the following Robin eigenvalue problem ( (-g)s u + g(|u|) u |u| = g(|u|) u |u| in Ngu + g(|u|) u |u| = 0 in Rn \ . (1.7) Here, for 0 < s < 1 we have denoted the regional fractional g-Laplacian as (-g)s u := 2 p.v. Z × g(|Dsu|) Dsu |Dsu| dy |x - y|n+s , NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 5 which is naturally defined in the space Ws,G reg () := u: Z G(|u|) dx + ZZ × G (Dsu) dµ < . A substantial difference which contrasts with the case of powers is that, in general, eigenvalues of (1.4), (1.5), (1.6) and (1.7) are not variational, i.e., they cannot be obtained by minimizing some Rayleigh quotient on a suitable space. For this reason, it is very interesting to study also the natural variational minimization problem related to Dirichlet, Neumann, regional Neumann and Robin boundary conditions. In order to not extend considerably the length of this introduction, we anticipate that the corresponding minimizers exist, are well defined (see Prop. 5.4) and are denoted as D, N , N and R, respectively, but we will not define them here (see Eqs. (5.5), (5.6), (5.7) and (5.8) for the precise definition). In spite of the fact that eigenvalues and minimizers are different quantities in general, in light of Proposi- tion 5.8 they are comparable, with equality in the case of powers (i.e., when G(t) = tp /p, p > 1). Regarding the relation among the different minimizers, in Proposition 5.6 we prove that they are ordered as N N R D. In view of the aforementioned Proposition 5.8, eigenvalues are consequently ordered as N c2 N c4 R c6 D, where c = p+ /p- . In Theorem 5.5 we prove that a function reaching the minimization problem for {N , N , R, D} is an eigenfunction for {N , N , R, D}, respectively. A considerable difference with the case of powers is that, due to the non-homogeneous nature of the problems, both eigenvalues and minimizers strongly depend on the energy level: for each µ > 0, if the eigenfunction/minimizing function is normalized such that R G(|u|) = µ, then and depend on µ. Nevertheless, in Proposition 5.7 we prove that and are uniformly bounded by below independently of µ. Before concluding this introduction, we mention some interesting issues we not deal and let as open questions: to establish positivity of eigenfunctions, to obtain its boundedness, and to study its interior/up to the boundary regularity. This paper is organized as follows. In Section 2 we introduce some preliminary results and definitions, as well as a proof of an integration by parts formula related to the operator (-)s g. Section 3 deals with the proof of our existence results. Some examples of nonlinearities which illustrate Theorems 1.1 and 1.2 are given in Section 4. Finally, Section 5 is devoted to study the eigenvalue problems (1.4), (1.5), (1.6) and (1.7). 2. Preliminaries In this section we introduce the classes of Young function and fractional Orlicz-Sobolev functions, the suitable class where the fractional g-Laplacian is well defined. 2.1. Young functions An application G: R+ R+ is said to be a Young function if it admits the integral formulation G(t) = R t 0 g() d, where the right continuous function g defined on [0, ) has the following properties: g(0) = 0, g(t) > 0 for t > 0, g is nondecreasing on (0, ), lim t g(t) = . 6 S. BAHROUNI AND A.M. SALORT From these properties it is easy to see that a Young function G is continuous, nonnegative, strictly increasing and convex on [0, ). We will assume from now on that the Young functions satisfy the growth behavior given in (G1). Roughly speaking, this condition indicates that G remains between two power functions. The following properties on Young functions are well-known. See for instance [25] for a proof. Lemma 2.1. Let G be a Young function satisfying (G1) and a, b 0. Then min{ap- , ap+ }G(b) G(ab) max{ap- , ap+ }G(b), (L1) G(a + b) C(G(a) + G(b)) with C := 2p+ , (L2) G is Lipschitz continuous. Condition (L2) is known as the 2 condition or doubling condition and, as it is showed in ([25], Thm. 3.4.4), it is equivalent to the right hand side inequality in (G1). The complementary Young function G of a Young function G is defined as G(t) := sup{tw - G(w) : w > 0}. From this definition the following Young-type inequality holds ab G(a) + G(b) for all a, b 0, and the following Holder's type inequality Z |uv| dx kukGkvkG, for all u LG () and v LG (). Moreover, it is not hard to see that G can be written in terms of the inverse of g as G(t) = Z t 0 g-1 () d, (2.1) see ([33], Thm. 2.6.8). Since -1 is increasing, from (2.1) and (G1) it is immediate the following relation. Lemma 2.2. Let G be an Young function satisfying (G1) such that g = G0 and denote by G its complementary function. Then G(g(t)) (p+ + 1)G(t), holds for any t 0. The following convexity property will be useful. Lemma 2.3. ([26], Lem. 2.1) Let G be a Young function satisfying (G1) and (G2). Then for every a, b R, G(|a|) + G(|b|) 2 G a + b 2 + G a - b 2 . NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 7 2.2. Fractional Orlicz-Sobolev spaces Given a Young function G, a parameter s (0, 1) and an open and bounded set Rn we consider the spaces LG () := {u: R measurable : G,(u) < } , Ws,G () := u LG (): s,G,Rn (u) < , Ws,G reg () := {u LG (): s,G,(u) < } where the modulars G, and s,G are defined as G,(u) := Z G(|u(x)|) dx s,G,Rn (u) := ZZ Rn×Rn G(|Dsu(x, y)|) dµ, s,G,(u) := ZZ × G(|Dsu(x, y)|) dµ, and the s-Holder quotient is defined as Dsu(x, y) := u(x) - u(y) |x - y|s , with dµ(x, y) := dx dy |x-y|n . These spaces are endowed with the so-called Luxemburg norms kukLG() := inf n > 0: G, u 1 o , kukW s,G() := kukLG() + [u]W s,G(Rn), kukW s,G reg () := kukLG() + [u]W s,G reg (), where the (s, G)-Gagliardo semi-norms are defined as [u]W s,G(Rn) := inf n > 0: s,G,Rn u 1 o , [u]W s,G reg () := inf n > 0: s,G, u 1 o . The space Ws,G () is a reflexive Banach space. Moreover C c is dense in Ws,G (Rn ). See ([20], Prop. 2.11) and ([16], Prop. 2.9) for details. We also consider the following space Ws,G 0 () := u Ws,G (Rn ) : u = 0 a.e. in Rn \ . Observe that Ws,G 0 () Ws,G (Rn ) LG (Rn ). In order to state some embedding results for fractional Orlicz-Sobolev spaces we introduce the following notation. Given two Young functions A and B, we say that B is essentially stronger than A or equivalently that A decreases essentially more rapidly than B, and denoted by A B, if for each a > 0 there exists xa 0 such that A(x) B(ax) for x xa. 8 S. BAHROUNI AND A.M. SALORT When the Young function G fulfills condition (G3), the critical function for the fractional Orlicz-Sobolev embedding is given by G-1 (t) = Z t 0 G-1 () n+s n d. The following result can be found in [8]. See also [3] for further generalizations. Theorem 2.4. Let G be a Young function satisfying (G3) and s (0, 1). Let Rn be a C0,1 bounded open subset. Then (i) the embedding Ws,G reg () , LG () is continuous; (ii) for any Young function B such that B G, the embedding Ws,G reg () , LB () is compact. From (2.1) it follows the following relation between modulars and norms. See ([7], Lem. 3.1) or ([22], Lem. 2.1). Lemma 2.5. Let G be a Young function satisfying (G1) and let - (t) = min{tp- , tp+ }, + (t) = max{tp- , tp+ }, for all t 0. Then, given Rn , (i) - (kukG) G,(u) + (kukG) for u LG (), (ii) - ([u]s,G) s,G,(u) + ([u]s,G) for u Ws,G (). 2.3. The fractional g-Laplacian operator Let G be a Young function such that G0 = g and s (0, 1). As anticipated, the fractional g-Laplacian operator is defined as (-g)s u := 2 p.v. Z Rn g(|Dsu|) Dsu |Dsu| dy |x - y|n+s , where p.v. stands for in principal value. This operator is well defined between Ws,G (Rn ) and its dual space W-s,G (Rn ). In fact, in ([20], Thm. 6.12) the following representation formula is provided h(-g)s u, vi = ZZ Rn×Rn g(|Dsu|) Dsu |Dsu| Dsv dµ, for any v Ws,G (Rn ). On the other hand, the censored or regional fractional g-Laplacian is well defined between Ws,G reg () and its dual space and it is defined as (-g)s u := 2 p.v. Z g(|Dsu|) Dsu |Dsu| dy |x - y|n+s , which acts as h(-g)s u, vi = ZZ × g(|Dsu|) Dsu |Dsu| Dsv dµ, for any v Ws,G reg (). NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 9 2.4. Integration by parts formula Here we prove an integration by parts formula in our settings which exploits the divergence form of the operator. We introduce the following notation h(-g)s u, vi = 1 2 Z R2n\(c)2 g(|Dsu|) Dsu |Dsu| Dsv dµ, the modular s,G,(u) = ZZ R2n\(c)2 G(|Dsu(x, y)|) dµ, and the corresponding Luxemburg semi-norm [u]W s,G (Rn) = inf n > 0: s,G, u 1 o . Of course, it is naturally defined the space Ws,G () := {u LG (): s,G,(u) < }. Proposition 2.6. Given u X, the following holds. (i) The following version of the divergence theorem is true Z (-g)s u = - Z Rn\ Ngu. (ii) More generally, we have the following integration by parts formula h(-g)s u, vi = Z v(-g)s u dx + Z Rn\ vNgu dx v X. Proof. In light of ([16], Prop. 2.9), it suffices to prove the result for u C2 c (Rn ). Let us prove (i). Observe that, since the role of x and y are symmetric, we get Z Z g (|Dsu|) u(x) |Dsu| dxdy |x - y|n+s = Z Z g (|Dsu|) u(y) |Dsu| dxdy |x - y|n+s , from where it is immediate that Z Z g (|Dsu|) Dsu |Dsu| dxdy |x - y|n+s = 0. Hence, we have that Z (-g)s u(x) dx = Z Z Rn g (|Dsu|) Dsu |Dsu| dydx |x - y|n+s = Z Z Rn\ g (|Dsu|) Dsu |Dsu| dydx |x - y|n+s 10 S. BAHROUNI AND A.M. SALORT = Z Rn\ Z g (|Dsu|) Dsu |Dsu| dx |x - y|n+s dy = - Z Rn\ Ngu(y) dy, as desired. Now, let us prove (ii). Since R2n \ (c )2 = ( × Rn ) [(Rn \ ) × ], we get h(-g)s u, vi = Z v(x) Z Rn g(|Dsu|) Dsu |Dsu| dy |x - y|n+s dx + Z Rn\ v(x) Z g(|Dsu|) Dsu |Dsu| dy |x - y|n+s dx. In light of (1.1) and (1.3) we obtain the desired relation. Remark 2.7. If we consider the function ws,(x) = R R Rn\ g(|x - y|-s )|x - y|n+s dy and the normalization of Ng given by Ng(x) := Ng(x) ws,(x) , if Ng(x) = 1 for any x Rn \ , we can define a generalization of the fractional perimeter defined in [11] as follows Z Rn\ Ng dx = Z Rn\ ws, dx = Z Z Rn\ g 1 |x - y|s dxdy |x - y|n+s := Pers,g(). 3. Variational setting and proofs of Theorems 1.1 and 1.2 We start defining the notion of weak solution for problem (1.2). With that end it will be useful introducing the following functional settings. Let us denote X := {u: Rn R measurable s.t.: kukX < }, where kukX := [u]W s,G (Rn) + kukLG() + kukLG, (c), and kukLG, (c) = inf ( > 0: Z Rn\ G u dµ 1 ) . By following standard arguments it can be seen that X is a reflexive Banach space with respect to the norm k · kX . See for instance [16]. The integration by parts formula given in Proposition 2.6 leads to the following definition. Definition 3.1. We say that u X is a weak solution of (1.2) if h(-g)s u, vi + Z g(|u|) u |u| v dx = Z fv dx + µ Z hv dx - Z Rn\ g(|u|) u |u| v dx, for all v X. NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 11 As anticipated in the introduction, we will approach problem (1.2) through the machinery of variational methods, and in particular, it will be done by using the abstract multiplicity result given in Theorem A.2. With that aim, we consider the functional : X R defined as (u) := J (u) - F(u) - µH(u), for every u X, where J , F, H: X R are defined as J (u) := Z R2n\(c)2 G(|Dsu|) dµ + Z G(|u|) dx + Z Rn\ G(|u|) dx, F(u) = Z F(x, u) dx and H(u) = Z H(x, u) dx. The following compact embedding for the space X holds. Lemma 3.2. Given a Young function A such that A G, then the embedding X , LA () is compact. Proof. Let u X. Observe that [u]W s,G reg () [u]W s,G (Rn) as a consequence of the inequality Z × G |Dsu| kuk dµ Z R2n\(c)2 G |Dsu| kuk dµ 1, together with the definition of the Luxemburg norm. Then, from Theorem 2.4, there exists a constant c > 0 such that kukLA() c[ukW s,G reg () c([u]W s,G (Rn) + kukLG()) ckukX concluding the proof due to the compactness of Ws,G reg () into LA (). The next proposition proves the well-posedness of . Proposition 3.3. Let f, h A, then the functional is well defined on the space X. Proof. First, we notice that given u X, from Lemma 2.5 it follows that J (u) C+ (kukX ) for some constant C = C(p± ). Moreover, by (f1) and the fact that m is increasing we get Z F(x, u) dx Z w(x) Z u 0 m(|t|) dt dx kwk Z |u|m(|u|) dx. In light of Lemma 2.2, m(|u|) LM (), and then, by applying Holder's inequality for Young function we get that Z |u|m(|u|) dx kukLM ()km(u)kLM (). Observe that ([25], Thm. 3.17.1) and Lemma 2.2 give that km(u)kLM () ckukLM (). Moreover, from Lemma 3.2 it follows that kukLM () ckukX , and therefore F is well defined. The well-posedness of H follows analogously, concluding that is well defined on X. Next, we prove some useful properties of the functional J . 12 S. BAHROUNI AND A.M. SALORT Lemma 3.4. Assume that (G1), (G2) and (G3) hold. Then, (i) the functional J : X R is C1 with derivative given by hJ 0 (u), vi = h(-g)s u, vi + Z g(|u(x)|) u |u| v(x) dx + Z Rn\ g(|u(x)|)v(x) dx for all u, v X; (ii) J is coercive, sequentially weakly lower semicontinuous; (iii) J WX , where the class WX is given in Definition (A.1); (iv) J is bounded on each bounded subset of X and its derivative admits a continuous inverse on X . Proof. (i) From ([35], Prop. 4.1), it is easy to see that J is class C1 . (ii) Let u X with kukX > 1. In view of Lemma 2.5, J is coercive since J (u) - (kukLG()) + - ([u]W s,G (Rn)) + - (kukLG, (c)) c- (kukX ), where c > 0 depends only on p± . Moreover, the sequential weak lower semicontinuity of J follows by ([8], Lem. 19). (iii) Let {uk}kN be a sequence in X such that uk * u in X and lim infk J (uk) J (u). Then, by the sequential weak lower semicontinuity of J proven in (ii) we get that, up to a subsequence, J (uk) J (u) as k +. Since uk+u 2 converges weakly to u, and modulars are lower semicontinuous with respect to the weak convergence, we get J (u) lim inf k J uk + u 2 . (3.1) We assume by contradiction that uk does not converge to u in X. Hence, there exists > 0 such that uk+u 2 X > . Then, by Lemma 2.5 J uk + u 2 > - (). (3.2) On the other hand, by applying Lemma 2.3 it follows that 1 2 (G(u) + G(uk)) - G uk + u 2 G uk - u 2 , which together with (3.2) leads to 1 2 (J (u) + J (uk)) - J uk + u 2 J uk - u 2 > - (). Taking limsup in the above inequality we obtain that J (u) - - () lim sup k+ J uk + u 2 , which contradicts (3.1). Therefore uk u strongly in X, and then J WX . NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 13 (iv) When kukX , in light of Lemma 2.5 we have that J (u) + (), i.e., J is bounded on any bounded subset of X. We prove now that J admits a continuous inverse J -1 : X X by means of the monotone operator method introduced by Browder and Minty (see ([39], Thm. 26.A (d))). Therefore, it suffices to verify that J 0 is coercive, hemicontinuous and uniformly monotone. Observe that since G is convex, J also is convex. Thus J (u) hJ 0 (u), ui for all u X, and, by using Lemma 2.5, for any u X such that kukX > 1 we have hJ 0 (u), ui kukX J (u) kukX min{kukp- -1 X , kukp+ -1 X }, from where the coercivity of J 0 follows by taking kukX . Furthermore, since the real function t 7 hJ 0 (u + tv), wi is continuous in [0, 1] for any u, v, w X, we have that J 0 is hemicontinuous. Let us finally prove that J 0 is uniformly monotone. Since G is convex we have that for every u, v X it holds G(|u|) G u + v 2 + g(|u|) u |u| u - v 2 and G(|v|) G u + v 2 + g(|v|) v |v| v - u 2 . Adding the above two relations and integrating over we find that 1 2 Z g(|u|) u |u| - g(|v|) v |v| (u - v) dx Z G(|u|) dx + Z G(|v|) dx - 2 Z G u + v 2 dx u, v X. On the other hand, we deduce by Lemma 2.3 that Z (G(|u|) + G(|v|)) dx 2 Z G u + v 2 dx + 2 Z G u - v 2 dx u, v X. From the last two relations it follows that Z (g(|u|) u |u| - g(|v|) v |v| )(u - v) dx 4 Z G |u - v| 2 dx u, v X. Similarly, for any u, v X it holds that Z Rn\ (g(|u|) u |u| - g(|v|) v |v| )(u - v) dx 4 Z Rn\ G |u - v| 2 dx, and h(-g)s (u - v), u - vi 4 Z R2n\(c)2 G |Dsu - Dsv| 2 dµ. Gathering the last three inequalities one gets that hJ 0 (u) - J 0 (v), u - vi 4J u - v 2 u, v X. 14 S. BAHROUNI AND A.M. SALORT Define now the function : [0, +) [0, +) by (t) = 1 p+ - 2 ( tp+ -1 for t 1 tp- -1 for t 1. It is easy to check that is an increasing function with (0) = 0 and (t) as t . Taking into account the above information and Lemma 2.5, we deduce that hJ 0 (u) - J 0 (v), u - vi (ku - vkX ), that is, J 0 is uniformly monotone, which concludes our proof. Lemma 3.5. F : X R is C1 with derivative given by hF0 (u), vi = Z f(x, u)v dx, for all u, v X. Moreover, F : X X is compact. Proof. Usual arguments show that F C1 (X, R). In order to verify the compactness of F, let {uk}kN X be a bounded sequence. Then up to a subsequence uk weakly converges in X to u X. Moreover, in light of Lemma 3.2, uk u strongly in LM () and a.e. in . Fixed v X with kvkX 1, thanks to the Holder's inequality for Young functions and the embedding of Lemma 3.2 we have |hF0 (uk), vi - hF0 (u), vi| = Z (f(x, uk) - f(x, u))v dx kf(·, uk(·)) - f(·, u(·))kLM () kvkLM () c kf(·, uk(·)) - f(·, u(·))kLM () kvkX , for some c > 0. Thus, taking supremum for kvkX 1, we get kF0 (uk) - F0 (u)kX ckf(·, uk(·)) - f(·, u(·))kLM (). Being f A we deduce immediately that f(x, uk(x)) - f(x, u(x)) 0 as k , for almost all x and |f(x, uk(x)) - f(x, u(x))| |f(x, uk(x))| + |f(x, u(x))| kwk(m(|uk(x)|) + m(|u(x)|)). Note that the majorant function in the previous relation is uniformly bounded in LM (). Hence, by applying the dominate convergence theorem we get that Z M(|f(x, uk(x)) - f(x, u(x))|) dx 0 as k . NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 15 Since M satisfies (G1), M-mean convergence is equivalent to norm convergence (see [25], Lem. 3.10.4), that is, kf(·, uk(·)) - f(·, u(·))kLM () 0 as k . Therefore kF0 (uk) - F0 (u)kX 0 as k , giving that F0 is a compact operator. Remark 3.6. Combining Lemmas 3.4 and 3.5, we deduce that C1 (X, R) with the derivative given by h0 (u), vi = h(-g)s u, vi + Z Rn\ g(|u(x)|) u |u| v(x) dx + Z g(|u(x)|) u |u| v(x) dx - Z f(x, u(x))v(x) dx - µ Z g(x, u(x))v(x) dx, for every v X. Then, critical points of are weak solutions of problem (1.2). Having proved these preliminaries, we are in position to prove our first main theorem. Proof of Theorem 1.1. Fix , µ and f, h A, we check the conditions needed to apply Theorem A.2. Fixed > 0, in light of (F1) there exist intervals I1 = [-r2, -r1] and I2 = [r1, r2] such that F(x, t) G(|t|) (x, t) × R \ (I1 I2). (3.3) In I1 I2, F(x, ·) is bounded in , then there exist d > 0 and a Young function B such that b = B0 , G B G and p+ < b- (here b- denotes a constant such that b- < tb(t) B(t) ) for which F(x, t) dB(|t|) (x, t) × (I1 I2). Then, from the inequalities above we obtain that F(u) J (u) R F(x, u) dx R G(|u|) dx + d R B(|u|) dx R G(|u|) dx . Observe that, assuming that kukX 1, from Lemma 2.5 and ([25], Thm 3.17.1) it holds that lim u0 R B(|u|) dx R G(|u|) dx lim u0 kukb- LB() kukp+ LG() c lim u0 kukb- -p+ LG() = 0. From the previous computations it follows that J1 := lim sup u0 F(u) J (u) . Moreover, assuming that kukX 1, by using again (3.3) and Lemma 2.5 we get F(u) J (u) R {x: |u(x)|r2} F(x, u) dx J (u) + R {x: |u(x)|>r2} F(x, u) dx R G(|u|) dx || kukp+ X sup{F(x, u(x)): (x, u(x)) × [-r2, r2]} + , 16 S. BAHROUNI AND A.M. SALORT from where we obtain that J2 := lim sup kuk F(u) J (u) . Therefore, since is arbitrary we obtain that max{0, J1, J2} = 0. Finally, since we are assuming (F2) it follows that the quantity sup{F(u)/J (u): u J -1 ([0, ])} is strictly positive. Finally, gathering Lemmas 3.4, 3.5 and the last computations, we are in position of applying Theorem A.2 to obtain our conclusion. Finally, we prove our second existence result. Proof of Theorem 1.2. Since G G, hypothesis (i) implies (f1). Note that hypothesis (i) also implies that F(u) J (u) R F(x, u) dx R G(|u|) dx c1 R (1 + B(|u|) dx) R G(|u|) dx . Assuming that kukX 1, from Lemma 2.5 and ([25], Thm. 3.17.1) it holds that lim kukX R B(|u|) dx R G(|u|) dx lim u kukb+ LB() kukp- LG() c lim u kukb+ -p- LG() = 0. From where J1 := lim sup kukX F(u) J (u) = 0. Similarly, assuming that kukX 1, hypothesis (ii) implies that J2 := lim sup kukX 0 F(u) J (u) c2 lim kukX 0 R D(|u|) dx R G(|u|) dx c lim kukX 0 kukd- -p+ LG() = 0. From these relations it follows that max{0, J1, J2} = 0. Now, without loss of generality we assume that 2 > 0 and choose a function u X such that u(x) 0 in and such that there exists x0 with u(x0) > 2. It follows that U := {x : u(x) > 2} is a nonempty open subset of . Let k: R R defined by k(t) = min{t, 2}. Then k(0) = 0 and k is Lipschitz with Lipschitz constant 1. Therefore, the function u1 = k u X satisfies that u1(x) = t for every x U and 0 u1(x) 2 for every x . Then, by hypothesis (iii) we obtain that F(x, u1(x)) > 0 for any x U, F(x, u1(x)) 0 for every x . From this we conclude that F(u1) > 0 and thus -1 = sup F(u) J (u) : u J -1 ((0, )) > 0. NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 17 Therefore, from Lemma 3.4, Lemma 3.5 and the last computations, the result follows by applying Theorem A.2. 4. Some examples of nonlinearities Let G be a Young function satisfying (G1), (G2) and (G3). Let us prove that the following examples of nonlinearities belong to the class A and satisfy the hypothesis of Theorem 1.1. (i) Consider the function f(t) = p| sin t|p-2 sin t cos t with p+ < p < p+ and observe that |f(t)| p(1 + |t|p+ -1 ), and since F(t) = | sin t|p we obtain lim |t|0 supx F(t) G(t) lim |t|0 | sin t|p |t|p+ = 0, lim |t| supx F(t) G(t) lim |t| | sin t|p |t|p- = 0. Finally, given a compact set C of positive measure, we consider a function v X such that v(x) = 2 in C and 0 v(x) 2 in \ C. Then sup uX Z F(u) dx Z | sin v(x)|p dx = |C| + Z \C | sin v(x)|p dx > 0. (ii) More generally, let M be a Young function such that p- < p+ < m- < m+ , where 1 < m- < tm(t) M(t) < m+ < for all t 0. Consider the function f(t) = m(| sin t|) cos t for t 0, and observe that this function fulfills that |f(t)| max{m(1), 1} + m(|t|). Moreover, taking = sin r, we get Z t 0 m(sin r) cos r dr = Z t 0 m() d = M(| sin t|), from where lim |t|0 supx F(t) G(t) lim |t|0 M(| sin t|) G(|t|) = 0 lim |t|0 | sin t|m- |t|p+ = 0. and lim |t| supx F(t) G(t) lim |t| M(| sin t|) G(|t|) = 0 lim |t| | sin t|m+ |t|p- = 0. As before, given a compact set C of positive measure, we consider a function u X such that v(x) = 2 in C and 0 v(x) 2 in \ C. Then sup uX Z F(u) dx Z G(| sin v(x)|) dx = |C| + Z \C G(| sin v(x)|) dx > 0. 18 S. BAHROUNI AND A.M. SALORT (iii) We consider the following concave-convex combination f(t) = tp-1 - tq-1 with p+ < p < q < p+ := np+ n - sp+ . Note that for some positive constant c = c(p± ) it holds that |f(t)| c(1 + |t|p-1 ) c(1 + |t|p+ -1 ). Moreover, lim |t|0 supx F(t) G(t) lim |t|0 |t|p p - |t|q q |t|p+ = 0, lim |t| supx F(x, t) G(t) lim |t| |t|p p - |t|q q |t|p- = -. Finally, let a compact set C large enough and v X such that v(x) = in C and 0 v(x) in \ C, where is chosen such that q q - p p > 0. Then sup uX Z F(u) dx Z F(v) dx = 1 q Z vq dx - 1 p Z vp dx 1 q Z C vq dx - 1 p Z C vp dx - 1 p Z \C vp dx |C| q q - p p - p p | \ C| > 0. The following example satisfies the hypothesis of Theorem 1.2. (iv) Let 0 < < p- p+ < . Consider the function f1(t) = ( |t|-2 t if |t| 1 |t|-2 t if |t| > 1. Then, it easily follows that F1(t) = ( |t| if |t| 1 1 - 1 + 1 |t| if |t| > 1, and conditions (i)­(iii) from Theorem 1.2 are fulfilled. 5. Eigenvalues and minimizers We start this section by defining the notion of eigenvalues. Definition 5.1. We say that is an eigenvalue of (1.4) with eigenfunction u Ws,G 0 () if h(-g)s u, vi = ( - 1) Z g(|u|) u |u| v dx v Ws,G 0 (). (5.1) NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 19 We say that is an eigenvalue of (1.5) with eigenfunction u Ws,G () if h(-g)s u, vi = ( - 1) Z g(|u|) u |u| v dx v Ws,G (). (5.2) We say that is an eigenvalue of (1.6) with eigenfunction u Ws,G reg () if h(-g)s u, vi = ( - 1) Z g(|u|) u |u| v dx v Ws,G reg (). (5.3) We say that is an eigenvalue of (1.7) with eigenfunction u X if h(-g)s u, vi = ( - 1) Z g(|u|) u |u| v dx - Z Rn\ g(|u|) u |u| v dx v X. (5.4) In order to prove our eigenvalues and minimizers results we will consider the following functionals defined in Sections 2.2 and 2.4 s,G,Rn (u): Ws,G 0 () R, s,G,(u): Ws,G reg () R, s,G,(u): Ws,G () R. and G,(u): W() R. where its definition domain W() is either Ws,G 0 (), Ws,G reg () or Ws,G (). Following ([35], Prop. 4.1) it is straightforward to see that these functionals are well-defined and are Frechet derivable. Moreover, the following expressions can be deduced. Proposition 5.2. We have that (s,G,Rn )0 is defined from Ws,G 0 () onto its dual, (s,G,)0 from Ws,G reg () onto its dual, (s,G,)0 from Ws,G () onto its dual, and (G,)0 from W() onto its dual, are C1 and their Frechet derivatives are given by h(s,G,Rn )0 (u), vi = h(-g)s u, vi v Ws,G 0 (), h(s,G,)0 (u), vi = h(-g)s u, vi v Ws,G reg (), h(s,G,)0 (u), vi = h(-g)s u, vi v Ws,G (), h(G,)0 (u), vi = Z g(|u|) u |u| v dx v W(). Proof. See ([35], Prop. 4.1) with the pertinent changes. Given µ > 0, we consider the minimization problems D := inf uMD µ s,G,Rn (u) + G,(u) G,(u) with MD µ = {u Ws,G 0 : G,(u) = µ}, (5.5) N := inf uMN µ s,G,(u) + G,(u) G,(u) with MN µ = {u Ws,G (): G,(u) = µ}, (5.6) 20 S. BAHROUNI AND A.M. SALORT N := inf uMN µ s,G,(u) + G,(u) G,(u) with MN µ = {u Ws,G reg (): G,(u) = µ}, (5.7) and R := inf uMR µ s,G,(u) + G,(u) + G,,c (u) G,(u) , (5.8) with MR µ = {u X : G,(u) = µ} and G,,c (u) := Z Rn\ G(|u(x)|) dx. Note the subindex refers to Dirichlet, Neumann, regional Neumann and Robin, respectively. Moreover, due to the possible lack of homogeneity, in general the quantities defined above depend on the energy level µ. Remark 5.3. By using the Poincare's inequality ([35], Prop. 3.2) it follows that [·] is an equivalent norm in Ws,G 0 (). Proposition 5.4. For each µ > 0 there exist solutions of the minimization problems (5.5), (5.6), (5.7) and (5.8), respectively. Proof. It follows just by applying the direct method of the calculus of variations. (See [35], Prop. 5.1). Existence of minimizers allow us to prove existence of eigenvalues. Theorem 5.5. For every µ > 0 there exist positive numbers D, N , N and R which are eigenvalues of (1.4), (1.5), (1.6) and (1.7), respectively, with non-negative eigenfunctions uD Ws,G 0 (), uN Ws,G (), uN Ws,G reg () and uR X, respectively, normalized such that G,(uD) = G,(uN ) = G,(uN ) = G,(uR) = µ. Proof. Given a fixed µ > 0, in light of Proposition 5.4 there exist functions uD Ws,G 0 (), uN Ws,G (Rn ), uN Ws,G () and uR X, respectively, normalized such that their modular G, is equal to µ, which attain the minimization problems (5.5), (5.6), (5.7) and (5.8), respectively. Therefore, by the Lagrange multipliers rule, since the involved functionals are C1 , there exist numbers D, N , N and R for which the corresponding function uD, uN , uN and uR satisfy the weak formulations (5.1), (5.2), (5.3) and (5.4), respectively. Proposition 5.6. The following relations among the minimizers of (5.5), (5.6) and (5.7) holds N N R D. Proof. Observe that since s,G,(u) s,G,(u) it follows that Ws,G () Ws,G reg (). Then, given a minimizer u MN µ of N we get that min uW s,G () 1 µ (µ + s,G,(u)) min uWs,G,() 1 µ µ + s,G (u), = N , but since minimizing over a small set enlarges the minimum, we conclude that N = min uW s,G reg () 1 µ (µ + s,G,(u)) min uW s,G () 1 µ (µ + s,G,(u)) N . NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 21 Moreover, since X Ws,G () it follows that N R. Finally, note that u Ws,G 0 () if s,G,(u) < and u = 0 in Rn \ . Therefore, Ws,G 0 () X and s,G,Rn (u) = s,G,(u). Then, proceeding as before, R D. The following proposition claims that minimizers are uniformly bounded away from zero independently of the energy level. Proposition 5.7. Given µ > 0, the minimizers N , N , R and D are positive and bounded by below independently on µ. Proof. The Dirichlet case is treated in ([35], Thm. 4.2). We deal here with the general case. Given µ > 0, let u MN µ be a minimizer of N , that is, u Ws,G reg () is such that G,(u) = µ and N = s,G,(u) + G,(u) G,(u) . Denote by u = 1 || R u(x) dx the average of u on . By using the 2 condition we have that Z G(|u|) dx C Z G(|u - u|) dx + C Z G(|u|) dx. By using Jensen's inequality and (L1) we get Z G(|u - u|) dx = Z G 1 || Z (u(x) - u(y)) dy dx 1 || Z Z G(|u(x) - u(y)|) dy dx 1 || Z Z G |u(x) - u(y)| |x - y|s diam ()s dy dx c(||)s,G,(u). Finally, since again the Jensen's inequality gives Z G(|u|) Z G 1 || Z |u(y)| dy dx 1 || Z Z G(|u(y)|) dy dx = G,(u) we obtain that G,(u) c(C, ||)(s,G,(u) + G,(u)), which implies a lower bound for N : N = s,G,(u) + G,(u) G,(u) 1 c(||) . In view of Proposition 5.6, the same lower bound is admissible for N , R and D. The following proposition states that, although eigenvalues and minimizers differ in general, both quantities are indeed comparable. Proposition 5.8. It holds that p- p+ p+ p- , 22 S. BAHROUNI AND A.M. SALORT where {D, R, N , N } and {D, R, N , N }, respectively. As a direct consequence, denoting c = p+ /p- , we have N c2 N c4 R c6 D. Proof. These first chain of inequalities just follow by testing in the definition of eigenvalue with the eigenfunction itself and using the fact that condition (G1), for all t 0, relates tg(t) with G(t) up to the constants p± . The second chain of inequalities are obtained just gathering the first one together with Proposition 5.6. As a consequence of Proposition 5.7 and 5.8 we obtain a lower bound for eigenvalues. Theorem 5.9. D, R, N , N are bounded by below by a positive constant independent on µ. Appendix A. An abstract existence result Definition A.1. We introduce the following definitions. (i) If X is a real Banach space, we denote by WX the class of all functionals J : X R possessing the following property: if {uk}kN is a sequence in X converging weakly to u X and lim infk J (uk) J (u), then {uk}kN has a subsequence converging strongly to u. (ii) We say that the derivative of J admits a continuous inverse on X we mean that there exists a continuous operator T : X X such that T(J (x)) = x for all x X. The above property is somehow a compactness property, stating the existence of a convergent subsequence of a given sequence. Theorem A.2 ([34]). Let X be a separable and reflexive real Banach space; J : X R a coercive, sequentially weakly lower semicontinuous C1 functional, belonging to WX, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X , and F : X R a C1 functional with compact derivative. Assume that has a strict local minimum x0 with J (x0) = F(x0) = 0. Finally, setting = max ( 0, lim sup kxk+ F(x) J (x) , lim sup kxkx0 F(x) J (x) ) , = sup xJ -1(]0,+[) F(x) J (x) , and assume < . Then, for each compact interval [a, b] ( 1 , 1 ) (with the conventions 1 0 = +, 1 + = 0) there exists > 0 with the following property: for every [a, b] and every C1 functional J : X R with compact derivative, there exists > 0 such that, for each µ [0, ], the equation J 0 (x) = F0 (x) + µH0 (x) has at least three solutions whose norms are less than . References [1] N. Abatangelo, A remark on nonlocal Neumann conditions for the fractional Laplacian. Preprint arXiv:1712.00320 (2020). [2] K.B. Ali, M. Hsini, K. Kefi and N.T. Chung, On a nonlocal fractional p(., .)-Laplacian problem with competing nonlinearities. Complex Anal. Oper. Th. 13 (2019) 1377­1399. [3] A. Alberico, A. Cianchi, L. Pick and L. Slavikova, Fractional Orlicz-Sobolev embeddings. Preprint arXiv:2001.05565 (2020). NEUMANN AND ROBIN TYPE BOUNDARY CONDITIONS 23 [4] A. Alberico, A. Cianchi, L. Pick and L. Slavikova, On the limit as s 0+ of fractional Orlicz-Sobolev spaces. Preprint arXiv:2002.05449 (2020). [5] E. Azroul, A. 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Electron. J. Qual. Theory Differ. Equ. 76 (2017) 1­10. [24] A. Kristaly, M. Mihailescu and V. Radulescu, Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz­ Sobolev space setting. Proc. Roy. Soc. Edinb. 139 A (2009) 367­379. [25] A. Kufner, O. John and S. Fucik, Function Spaces, Vol. 3. Springer Science Business Media (1979). [26] J. Lamperti, On the isometries of certain function-spaces. Pacific J. Math. 8 (1958) 459­466. [27] E. Lindgren and P. Lindqvist, Fractional eigenvalues. Cal. Var. Partial Differ. Equ. 49 (2014) 795­826. [28] X. Mingqi, V.D. Radulescu, B. Zhang, Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58 (2019) 57. [29] G. Molica Bisci, V.D. Radulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems. Cambridge University Press, Cambridge (2016). [30] D. Mugnai and E.P. Lippi, Neumann fractional p-Laplacian: Eigenvalues and existence results. Nonlinear Anal. 188 (2019) 455­474. [31] D. Mugnai, A. Pinamonti and E. Vecchi, Towards a Brezis-Oswald-type result for fractional problems with Robin boundary conditions. Calc. Var. 59 (2020) 43. [32] T.C. Nguyen, Three solutions for a class of nonlocalproblems in Orlicz-Sobolev spaces. Appl. Math. Comput. 218 (2013) 1257­1269. [33] M. Rao and Z. Ren, Applications of Orlicz Spaces, Vol. 250. CRC Press (2002). [34] B. Ricceri, A further three critical points theorem. Nonlinear Anal. 71 (2009) 4151­4157. [35] A.M. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator. J. Differ. Equ. 268 (2020) 5413­5439. [36] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinb. Sect. A Math. 144 (2014) 831­855. [37] B. Volzone, Symmetrization for fractional Neumann problems. Nonlinear Anal. 147 (2016) 1­25. [38] M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional p-Laplacian. Nonlinear Differ. Equ. Appl. 23 (2016) 1. [39] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer (1985). [1] N. Abatangelo, A remark on nonlocal Neumann conditions for the fractional Laplacian. Preprint (2020). [2] K. B. Ali, M. Hsini, K. Kefi and N. T. Chung, On a nonlocal fractional $p(.,.)$-Laplacian problem with competing nonlinearities. Complex Anal. Oper. Th. 13 (2019) 1377–1399. [3] A. Alberico, A. Cianchi, L. Pick and L. Slavíková, Fractional Orlicz-Sobolev embeddings. Preprint (2020). [4] A. Alberico, A. Cianchi, L. Pick and L. Slavíková, On the limit as s → 0⁺ of fractional Orlicz-Sobolev spaces. Preprint (2020). [5] E. Azroul, A. Benkirane and M. Srati, Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces. Adv. Oper. Th. 5 (2020) 1350–1375. [6] A. Bahrouni and V. D. Radulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin. Dyn. Syst. Ser. S. 11 (2018) 379–389. [7] S. Bahrouni, H. Ounaies and L. S. Tavares, Basic results of fractional Orlicz-Sobolev space and applications to non-local problems. Topol. Methods Nonlinear Anal. 55 (2020) 681–695. [8] S. Bahrouni and H. Ounaies, Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discret Cont. Dyn. Syst. 40 (2020) 2917–2944. [9] S. Bahrouni, Infinitely many solutions for problems in fractional Orlicz-Sobolev spaces. Rochy Mt. J. Math. 50 (2020) 1151–1173. [10] A. Bahrouni, S. Bahrouni and M. Xiang, On a class of nonvariational problems in fractional Orlicz-Sobolev spaces. Nonlinear Anal. 190 (2020) 111595. [11] L. Caffarelli, J. M. Roquejoffre and O. Savin, Nonlocal minimal surfaces. Comm. Pure Appl. Math. 63 (2009) 1111–1144. [12] F. Cammaroto and L. Vilasi, Multiple solutions for a nonhomogeneous Dirichlet problem in Orlicz–Sobolev spaces. Appl. Math. Comput. 218 (2012) 11518–11527. [13] E. Correa and A. De Pablo, Remarks on a nonlinear nonlocal operator in Orlicz spaces. Adv. Nonlinear Anal. 9 (2020) 305–326. [14] L. M. Del Pezzo, J. D. Rossi and A. M. Salort, Fractional eigenvalue problems that approximate Steklov eigenvalues. Proc. R. Soc. Edinb. Sect. A 148 (2018) 499–516. [15] L. Del Pezzo and A. M. Salort, The first non-zero Neumann $p$-fractional eigenvalue. Nonlinear Anal. Theory Methods Appl. 118 (2015) 130–143. [16] P. De Nápoli, J. Fernández Bonder and A. Salort, A Pólya–Szegö principle for general fractional Orlicz–Sobolev spaces, To appear in: Complex Var. Elliptic Equations (2020). [17] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam 33 (2017) 377–416. [18] S. El Manouni, H. Hajaiej and P. Winkert, Bounded solutions to nonlinear problems in ℝ$$ involving the fractional Laplacian depending on parameters. Minimax Theory Appl. 2 (2017) 265–283. [19] A. El Khalil On the spectrum of Robin boundary $p$-Laplacian problem, Moroccan J. Pure Appl. Anal. 5 (2019) 279–293. [20] J. Fernández Bonder and A. M. Salort, Fractional order Orlicz-Sobolev space. J. Funct. Anal. 277 (2019) 333–367. [21] J. Fernández Bonder, M. Pérez-Llanos and A. M. Salort, A Hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians. Preprint (2018). [22] N. Fukagai, M. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on ℝ$$. Funkcialaj Ekvacioj 49 (2006) 235–267. [23] U. Kaufmann, J. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p\left(x\right)$-Laplacians. Electron. J. Qual. Theory Differ. Equ. 76 (2017) 1–10. [24] A. Kristály, M. Mihăilescu and V. Rădulescu, Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz–Sobolev space setting. Proc. Roy. Soc. Edinb. 139 A (2009) 367–379. [25] A. Kufner, O. John and S. Fucik, Function Spaces Vol. 3. Springer Science Business Media (1979). [26] J. Lamperti, On the isometries of certain function-spaces. Pacific J. Math. 8 (1958) 459–466. [27] E. Lindgren and P. Lindqvist, Fractional eigenvalues. Cal. Var. Partial Differ. Equ. 49 (2014) 795–826. [28] X. Mingqi, V. D. Radulescu, B. Zhang, Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58 (2019) 57. [29] G. Molica Bisci, V. D. Radulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems. Cambridge University Press, Cambridge (2016). [30] D. Mugnai and E. P. Lippi, Neumann fractional p-Laplacian: Eigenvalues and existence results. Nonlinear Anal. 188 (2019) 455–474. [31] D. Mugnai, A. Pinamonti and E. Vecchi, Towards a Brezis-Oswald-type result for fractional problems with Robin boundary conditions. Calc. Var. 59 (2020) 43. [32] T. C. Nguyen, Three solutions for a class of nonlocalproblems in Orlicz-Sobolev spaces. Appl. Math. Comput. 218 (2013) 1257–1269. [33] M. Rao and Z. Ren, Applications of Orlicz Spaces, Vol. 250. CRC Press (2002). [34] B. Ricceri, A further three critical points theorem. Nonlinear Anal. 71 (2009) 4151–4157. [35] A. M. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator. J. Differ. Equ. 268 (2020) 5413–5439. [36] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinb. Sect. A Math. 144 (2014) 831–855. [37] B. Volzone, Symmetrization for fractional Neumann problems. Nonlinear Anal. 147 (2016) 1–25. [38] M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional $p$-Laplacian. Nonlinear Differ. Equ. Appl. 23 (2016) 1. [39] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer (1985). COCV_2021__27_S1_A17_0839309f4-4ee7-47a3-91ae-8f8f157c2e25cocv19009510.1051/cocv/202006710.1051/cocv/2020067 A hierarchy of multilayered plate models de Benito Delgado Miguel 0000-0003-4805-9055 Schmidt Bernd * Universität Augsburg, Augsburg, Germany. *Corresponding author: [e-mail] SupplementS16 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF)Full (DJVU)

We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of Γ-convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the Γ-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von Kármán, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.

Multilayers dimension reduction effective plate theories Gamma-convergence 74K20 49J45 74G65 Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659 285722765 idline ESAIM: COCV 27 (2021) S16 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S16 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020067 www.esaim-cocv.org A HIERARCHY OF MULTILAYERED PLATE MODELS Miguel de Benito Delgado and Bernd Schmidt* Abstract. We derive a hierarchy of plate theories for heterogeneous multilayers from three dimen- sional nonlinear elasticity by means of -convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the -limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von Karman, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials. Mathematics Subject Classification. 74K20, 49J45, 74G65. Received June 4, 2019. Accepted October 6, 2020. 1. Introduction The derivation of effective theories for thin structures such as beams, rods, plates and shells is a classical problem in continuum mechanics. Fundamental results in formulating adequate dimensionally reduced theories for three-dimensional elastic objects have already been obtained by Euler [18], Kirchhoff [30] and von Karman [55], cf. also [9, 10, 37]. A physical plate, given by a domain h = × (-h/2, h/2) R3 , is identified with a hyperelastic body of height h "much smaller" than the lengths of the sides of . The plane domain R2 constitutes the mid-layer of the plate. We assume that the body has a (possibly non-homogeneous) stored energy density W (precise conditions on W will be specified later) and, after deformation by y : h R3 , the total elastic energy Eh(y) = Z h W(z, y(z)) dz. The problem amounts to identifying effective functionals in the limit h 0 operating on dimensionally reduced deformations of the mid-plane. In spite of its long history, rigorous results in this direction relating classical models for plates to the parent three-dimensional elasticity theory have only been obtained comparatively recently. In order to avoid working on a changing domain, a rescaling x3 = z3/h is performed to obtain a fixed 1. We set zh(x1, x2, x3) = (x1, x2, hx3) and we consider instead of a deformation y : h R3 , the rescaled one Keywords and phrases: Multilayers, dimension reduction, effective plate theories, Gamma-convergence. Universitat Augsburg, Augsburg, Germany. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 M. DE BENITO DELGADO AND B. SCHMIDT yh : 1 R3 , yh(x) = y(zh(x)). We define the energy per unit volume as Jh = 1 h Eh, which after a change of variables becomes Jh(y) = Z 1 W(x, hy(x)) dx, where h = (1, 2, 3/h). After first results in linear elasticity had been established, see [2, 4], perhaps the first work to derive a non- linearly elastic, lower dimensional theory with a rigorous analysis using variational convergence was [3] for the case of strings. In the context of nonlinear plates we consider the rescaled functionals J h (y) = 1 h Z 1 W(x, hy(x)) dx. Inspired by the work in [3], a non-linear membrane theory is derived in [33] for = 0. The range (0, 2) is the so-called constrained membrane regime, which still is not fully explored, except under certain kinds of boundary conditions or assumed admissible deformations, see, e.g., [5] and the work in [13]. For minimizers under body forces, convergence results have been obtained in [11]. Most significant in view of our setup are the contributions to the cases 2. In [22] Friesecke, James and Muller prove the fundamental geometric rigidity estimate which carries Korn's inequality to the nonlinear setting and utilise it to obtain the non-linear Kirchhoff theory of pure bending under an isometry constraint in the case = 2. This estimate is at the core of most of the later developments in this area. In their seminal paper [23], the same authors exploit the quantitative geometric rigidity estimate of [22] in a systematic investigation of limits for the whole range of scalings [2, ), deriving the first hierarchy of limit models. They also provide a thorough (albeit succinct) overview of the state of the art around 2006. The lecture ([41], Chap. 2) provides a nice walkthrough of this paper, as well as abundant references and open problems as of 2017. This variational approach has been extended and revisited in a variety of different contexts, among them more complex shell geometries [21, 36], more basic atomistic models [8, 50], or more complicated material properties as incompressibility [12], brittleness [53] or oscillatory dependence on the space variable [27, 28, 44]. Moreover, the convergence of equilibria [40, 43] and even dynamic solutions [1] have been established. The focus in this contribution is on materials whose reference configuration is subjected to stresses (one speaks of pre-strained or pre-stressed bodies) and whose energy density exhibits a dependence on the out-of-plane direction (modelling multilayered plates). Examples of these situations are heated materials, crystallisations on top of a substrate and multilayered plates. For = 2 the second author derived in [51, 52] an effective Kirchhoff theory for stored energy densities of the form W(x3, F) = W0(x3, F(I + hBh (x3))), depending explicitly on the out-of-plane coordinate x3 and a "mismatch tensor" Bh (x3) which measures the deviation of the energy well argmin W(x3, ·) from the rigid motions argmin W0(x3, ·) = SO(3). We remark that the regime = 2 is precisely adapted to capture the effects of a misfit hBh scaling linearly in h. In the simplest case with linearly changing Bh (x3) = ax3I one obtains a -limit IKi with IKi(y) = 1 24 Z Q(II -a1I) - a2 dx, if y A (and IKi(y) = + if not), where A is a suitable class of admissible deformations (isometric immersions). Q is a quadratic form acting on the shape tensor II (the second fundamental form of y). The coefficients of Q and the numbers a1, a2 can be explicitly computed. In [51, 52] also a thorough investigation of the shape of energy minimisers (for free boundary conditions) is provided which shows that the optimal configurations are rolled-up portions of cylinders whose winding directions and radii are determined by the material parameters and the misfit tensor. A HIERARCHY OF MULTILAYERED PLATE MODELS 3 The main goal of our work is to extend such an analysis to the energy regimes > 2. This will in particular allow for a more subtle investigation of the effect of general pre-strain scalings of the form h-1 Bh (x3), > 2. Indeed, as we will see, such pre-strains will result in contributions to the limiting functional which are both finite and non-trivial precisely if = 2 - 2. A main source of motivation are physical experiments which show that there are situations in which optimal configurations are spherical caps (paraboloids with positive Gauß curvature) rather than cylinders, [17, 19, 20, 29, 39, 49]. We will see that indeed this discrepancy can be explained in terms of different energy scaling regimes, where the von Karman scaling = 4 is critical. In the present paper we lay the foundation for this by deriving effective plate theories for pre-strained multilayers. We analyse the functionals obtained here in depth in our companion paper [15]. From a more general perspective, a thorough theoretical understanding of multilayers is not only interesting from a mathematical point of view. Such structures are of great interest in engineering applications, see [48]. In particular, the mechanism through which a pre-strain provokes mechanical displacement has been of great interest: It allows to access and manipulate objects even at the nanoscale in a convenient and feasible way, see, e.g., [24, 45, 47, 54], where a misfit of the equilibria of the constituents is used to achieve a self-organised fabrication of nano-scrolls. On a macroscopic scale one also observes rather complicated material behaviour that is caused by pre-strains which induce a non-Euclidean target metric, see e.g. [31]. In [17] recent experiments are reported on elastomer polydimethylsiloxane bilayers which fold into complex three dimensional configurations in response to programmed misfit distributions. In fact, the derivation of effective models for thin plates with a pre-strain has received a lot of attention in recent contributions. With no attempt to be exhaustive we mention [6, 16, 32, 34, 35, 38, 51, 52]. Yet, as we are primarily interested in modeling multilayers, our set-up is quite different from those contributions, where the pre-strain and elastic moduli depend only on the in-plane variables. In particular, in [35] the authors derive the von Karman functional with a spontaneous curvature term for pre-stressed plates in the case = 4. The same functional also arises in a special case of our setting, cf. (1.1) with = 1 below. However, their setup is not comparable to our situation. On the one hand, it is even more general as an explicit (x1, x2) dependence of the misfit is allowed. On the other hand, there is no explicit x3 dependence, neither on the pre-strain nor on the material properties, as would be necessary to model multilayers. As discussed above, in the bending dominated regime, models in which both pre-strain and material parameters were allowed to vary in the thin film direction were already discussed in [51, 52]. Generalizing [35, 52], the recent paper [34] considers a variety of scaling regimes also for thickness dependent pre-strains. However, our treatment of energy densities which may vary considerably in the thin film direction is, as we will see, quite subtle. A main source of technical difficulties is the fact that in our situation we can no longer expect the mid plane to follow the limiting plate deformation exactly. This phenomenon can be observed already in the simplest situation of a bilayer with one layer being much softer than the other. If rolled up, the unstretched plane will move into the stiffer layer, to an extent which depends on the local curvature. From a technical point of view, it turns out that in our setup the von Karman case = 4 is in fact a rather straightforward extension of [23, 52]. The regime > 4 is however a bit more involved. In contrast to the homogeneous case in [23], the dependence of the resulting limiting theory on the in-plane displacements may be non-trivial so it cannot be discarded by setting it to 0 without loss of generality. The scaling in the linearised Kirchhoff case (2, 4) turns out to be the most difficult. In order to construct recovery sequences we need to provide a representation result for symmetric tensor fields on in terms of symmetrised gradients and solutions to the non-elliptic Monge-Ampere equation det 2 v = 0, cf. Theorem 7.3. From a modelling point of view, a main novelty is in our introducing a new interpolating regime in between the linearised Kirchhoff case < 4 and the fully linear case > 4. This is motivated by our findings in [15] which show that minimisers (after rescaling) coincide for all (2, 4) (parts of a cylinder) and for all (4, ) (parts of a parabolic cap). In this sense, the exponent = 4 in the von Karman case is critical. We introduce an additional fine scale inducing a new scaling regime h4 with (0, ) and obtain von Karman functionals that upon varying continuously connect the extreme cases 0 and , which turn out to reduce to the functionals obtained for > 4 and < 4, respectively. In the simplest non-trivial example, the prototypical 4 M. DE BENITO DELGADO AND B. SCHMIDT limit functional is of von Karman type: I vK(u, v) = 2 Z Q2(su + 1 2 v v) dx + 1 24 Z Q2(2 v - I) dx. (1.1) In contrast to the cases 6= 4 minimisers of this functional are not explicit. We discuss their behaviour in detail in [15], in particular, how they interpolate in between < 4 and > 4. With a view to the applicability of our results in engineering problems, we emphasize that our effective models are completely explicit. The effective elastic moduli and spontaneous curvature terms in the limiting plate theory are explicitly computable from the elastic moduli and pre-strain of the constituents of the multilayer. Explicit formulae identifying the limiting homogenised material constants in terms of the moments in x3 of the corresponding parameters within the individual layers are provided in our companion paper [15]. There, we also discuss a BGaAs/InGaAs bilayer in order to give a concrete and explicit example which connects our theory to experimental results. Outline Having fixed the precise setup in Section 2, in Section 3 we present our main results: Theorem 3.1 on - convergence in a hierarchy of energy scalings and Theorem 3.3 on the asymptotic behaviour of the interpolating von Karman functional for 0 or . We then recall some basic results on compactness and explicit representations for the limit strains from [23] in Section 4. Proofs of lower and upper bounds in Theorem 3.1 are collected in Section 5, where we obtain (1.1) and more general functionals. In Section 6 we show how the von Karman functional interpolates between different theories. Finally, in Section 7 we prove some density and matrix representation theorems essential for the construction of recovery sequences and identification of minimisers in the linearised Kirchhoff regime. Notation We denote by e1, e2, e3 the standard basis vectors in R3 and write x = (x0 , x3) R3 , x0 R2 . The spaces of symmetric and antisymmetric n × n matrices are Rn×n sym and Rn×n ant , respectively. Asym = sym A = 1 2 (A + A> ) is the symmetric part and Aant = ant A = 1 2 (A - A> ) the antisymmetric part of a square matrix A. Attaching a row and a column of zeros to a matrix G R2×2 leads to G := P2 ,=1 Ge e R3×3 , conversely, B R2×2 is the matrix resulting from the deletion of the third row and column of any B R3×3 . If Q(·) is a quadratic form, we denote the associated bilinear form by Q[·, ·]. For a scalar function f : R3 R, f = (1f, 2f, 3f)> is a column vector, whereas for y : R3 R3 we have y R3×3 with rows > yi, i.e., (y)ij = yi,j = jyi, i, j {1, 2, 3}. Its left 3 × 2 submatrix is 0 y, its rescaled gradient hy = (1y, 2y, 1/h 3y). Moreover, su = 1 2 (u + > u), is the symmetrised gradient of u : R2 R2 , 2 v the Hessian matrix of v : Rn R. Let R2 . We set ^ v := (1v, 2v, 0)> R3 for v : R, ^ u := P2 ,=1(0 u)e e R3×3 for u : R2 and ^ b := P3 =1 P2 =1(0 b)e e R3×3 for b : R3 . The norm on Sobolev spaces is k · kk,p, = k · kW k,p() and, in particular, k · k0,p, is the Lp -norm. We will omit the domain when it is clear from the context. We abbreviate A := su + 1 2 v v, mostly in Section 6 and set (f) := 1 || R f(x0 ) dx0 is the average of f over . A HIERARCHY OF MULTILAYERED PLATE MODELS 5 2. The setting As described in Section 1, we consider a sequence of increasingly thin domains h := × (-h/2, h/2) R3 and rescale them to 1 := × (-1/2, 1/2) R3 , where R2 is a bounded domain with Lipschitz boundary. As a consequence of the rescaling, instead of maps y : h R3 , we consider the rescaled deformations y : 1 R3 , x 7 y(x) = y(x1, x2, hx3), belonging to the space Y := W1,2 (1; R3 ). For each scaling1 (2, ), and for all deformations y Y , define the scaled elastic energy per unit volume: Ih (y) = 1 h2-2 Z 1 Wh (x3, hy(x)) dx, (2.1) where h = (1, 2, 3/h)> is the gradient operator resulting after the change of coordinates described in Section 1. For the sake of conciseness, we will present most results below for all scalings simultaneously, adding the parameter to much of the notation. The energy density for 6= 3 is given by Wh (x3, F) = W0(x3, F(I + h-1 Bh (x3))), F R3×3 , where Bh : (-1/2, 1/2) R3×3 describes the internal misfit and W0 the stored energy density of the reference configuration. In the regime = 3 we include an additional parameter > 0 controlling further the amount of misfit in the model: Wh =3(x3, F) = W0 x3, F I + h2 Bh (x3) , F R3×3 , and we later write Bh = Bh . Note that given the choice h-1 for the scaling of the misfit, the fact that in the limit it will be again scaled quadratically entails our choice of rescaling the energy with h-2(-1) . Indeed on coarser energy scales h , < 2( - 1), such a misfit will not contribute to the limiting functional. (This case is indeed also covered in our analysis below by considering Ih 1+/2 with Bh 0.) On the other hand, on finer energy scales h , > 2( - 1), the pre-strain will typically lead to diverging energy terms in the limit (see (2.5), (2.6) and the computation of the lower bounds in Thm. 5.1). Our assumptions for Bh and W0 are those of ([52], Asm. 1.1): Assumption 2.1. (a) For a.e. t (-1/2, 1/2), W0(t, ·) is continuous on R3×3 and C2 in a neighbourhood of SO(3) which does not depend on t. (b) The map t 7 Q3(t, ·) = D2 W0(t, I)[·, ·] is in L (-1/2, 1/2) ; R9×9 . (c) The map (s) := ess sup -1/2<t<1/2 sup |F |6s |W0(t, I + F) - 1 2 Q3(t, F)| shall satisfy (s) = o(s2 ) as s 0. 1In the notation of Section 1 we have = 2 - 2. 6 M. DE BENITO DELGADO AND B. SCHMIDT (d) For all F R3×3 and all R SO(3) W0(t, F) = W0(t, RF). (e) For a.e. t (-1/2, 1/2) , W0(t, F) = 0 if F SO(3) and ess inf -1/2<t<1/2 W0(t, F) > c dist2 (F, SO(3)), for all F R3×3 and some c > 0. (f) Bh B in L (-1/2, 1/2) ; R3×3 . The Hessian Q3(t, F) := D2 W0(t, I)[F, F] = 2 W0(t, I) FijFij FijFij, for t (-1/2, 1/2) , F R3×3 is twice the quadratic form of linear elasticity theory, which results after a linearisation of W0 around the identity. By Assumption 2.1.e it is positive definite on symmetric matrices and vanishing on antisymmetric matrices. We note in passing two consequences of the above conditions. First, frame invariance (Assumption 2.1.d) extends to the second derivative where defined, i.e. D2 W0(t, R)[FR, FR] = D2 W0(t, I)[F, F] = Q3(t, F). Second, the energy W0 grows at most quadratically in a neighbourhood of SO(3), i.e., there exists a C > 0 such that for sufficiently small |F| it holds that for all t (-1/2, 1/2): W0(t, I + F) 6 C dist2 (I + F, SO(3)). Define Q2 to be the quadratic form on R2×2 obtained by relaxation of Q3 among stretches in the x3 direction: Q2(t, G) := min cR3 Q3(t, G + c e3), for t (-1/2, 1/2) , G R2×2 , (2.2) where e3 = (0, 0, 1) R3 . (See the last paragraph of Sect. 1 for the definition of G.) This process effectively minimises away the effect of transversal strain. Solving the minimisation problem yields a map L : (-1/2, 1/2) × R2×2 R3 , linear in its second argument, which attains the minimum: Q2(t, G) = Q3(t, G + L(t, G) e3). (2.3) In particular, also the Q2(t, ·) are positive definite on symmetric matrices and vanishing on antisymmetric matrices. In fact, by Assumptions 2.1.b and 2.1.e we have the bounds Q2(t, F) & |F|2 F R2×2 sym and |L(t, F)| . |F| F R2×2 , (2.4) uniformly in t (-1/2, 1/2). For the regimes > 3, we define the effective form Q2(E, F) := Z 1/2 -1/2 Q2(t, E + tF + B(t)) dt, (2.5) A HIERARCHY OF MULTILAYERED PLATE MODELS 7 with E, F R2×2 (see the last paragraph of Sect. 1 for the definition of B). For (2, 3) we consider its relaxation Q ? 2(F) := min ER2×2 Q2(E, F) = min ER2×2 sym Z 1/2 -1/2 Q2(t, E + tF + B(t)) dt. (2.6) For the case = 3, we include an additional parameter > 0 as discussed in page 5 and later write B = B. Both Q2 and Q ? 2 are non-negative quadratic polynomials. Solving the quadratic minimization problems in (2.2) and (2.6) and computing the t-integrations in (2.5) and (2.6) one obtains explicit formulae for their coefficients in terms the first moments Z 1/2 -1/2 tk Q3(t, ·) dt, Z 1/2 -1/2 tk B(t) dt, k = 0, 1, 2, in the small film direction of the elastic moduli and the misfit tensor, respectively. This is made explicit in Section 2.2 of our companion paper [15]. For fixed (2, ) we say that a sequence (yh )h>0 Y has finite scaled energy if there exists some constant C > 0 such that lsup h0 Ih (yh ) 6 C. After some corrections we will have precompactness of such sequences, thus essentially proving that the family Ih is equicoercive, the essential condition for the fundamental theorem of -convergence showing convergence of minimisers and energies. This compactness takes place in adequate target ambient spaces X = W1,2 (; R) if (2, 3), W1,2 (; R2 ) × W1,2 (; R) if > 3, (2.7) equipped with the weak topology. An essential ingredient in arguments with -convergence is the choice of sequential convergence to obtain (pre-)compactness. For the lower bounds we may suppose that a sequence (yh )h>0 has finite scaled energy, which enables Lemma 4.1 for the identification of the limits, cf. Section 4. This requires us to work with the corrected deformations (yh ) := (R h )> yh - ch , for some constants R h SO(3) and ch R3 depending on yh , see (4.1).2 We choose to encode this transformation into the definition of -convergence via maps Ph (Def. 2.3) for general transformations with arbitrary Rh SO(3) and ch R3 . Despite adding clutter to the notation, this helps to highlight and isolate the technical requirement of the sequences involved with special rigid transformations.3 Definition 2.2. Let Y := W1,2 (1; R3 ) and X as in (2.7). We say that a sequence (yh )h>0 Y Ph -converges to some w X if and only if there exist constants Rh SO(3), ch R3 which define maps : Y Y, yh 7 (yh ) := (Rh )> yh - ch , such that Ph (yh ) * w weakly in X, 2These maps "remove" rigid movements from the yh bringing them close to the identity. Note that the energy is not affected by this change because of frame invariance (Assumption 2.1.d). 3We only require that there be some constants Rh, ch for Ph-convergence. In order to obtain compactness and in the lower bounds we will take the specific ones given in Lemma 4.1 whereas for the recovery sequences we will use Rh = I, ch = 0. 8 M. DE BENITO DELGADO AND B. SCHMIDT where Ph : Y X, yh 7 vh , if (2, 3), (uh , vh ) if = 3, (uh , vh ), if > 3, and we defined: For 6= 3 and x0 , the scaled, averaged and corrected in-plane and out-of-plane displacements: ( uh (x0 ) := 1 h R 1/2 -1/2 ((yh )0 (x0 , x3) - x0 ) dx3, vh (x0 ) := 1 h-2 R 1/2 -1/2 (yh )3(x0 , x3) dx3, (2.8) where (yh )0 (x0 , x3) = ((yh )1, (yh )2)(x0 , x3) and = 2( - 2) if (2, 3), - 1 if > 3. For = 3 and x0 , introducing the additional parameter > 0: ( uh (x0 ) := 1 h2 R 1/2 -1/2 [(yh )0 (x0 , x3) - x0 ] dx3 vh (x0 ) := 1 h R 1/2 -1/2 (yh )3(x0 , x3) dx3. (2.9) For = 3, we overload the notation with the parameter writing (uh , vh ) and Ph instead of (uh , vh ) or Ph , letting the letter used in the subindex resolve ambiguity. With Definition 2.2 we can specify precisely what we mean by -convergence of the energies (2.1):4 Definition 2.3. Let > 2. We say that the family of scaled elastic energies {Ih : Y R}h>0, h > 0, - converges via maps Ph to I : X R iff: (a) Lower bound: For every w X and every sequence (yh )h>0 Y which Ph -converges to w as h 0 it holds that linf h0 Ih (yh ) > I(w). (b) Upper bound: For every w X there exists a recovery sequence (yh )h>0 Y which Ph -converges to w as h 0 and lsup h0 Ih (yh ) 6 I(w). Finally, we identify what the space of admissible displacements for the limit theories will be: X0 := W2,2 sh (; R) if (2, 3), W1,2 (; R2 ) × W2,2 (; R) if > 3, where the space of out-of-plane displacements with singular Hessian W2,2 sh () := v W2,2 (; R) : det 2 v = 0 a.e. , 4We refer to the notes [7] for a quick introduction to -convergence. A HIERARCHY OF MULTILAYERED PLATE MODELS 9 will be central in the linearised Kirchhoff theory. We will define the functionals to be + for inadmissible displacements in X\X0 . 3. Main results Our first goal is to prove that in the pre-strained setting described above one has a hierarchy of plate models a la [23]. The proof is split into several theorems in Section 5. For notation we refer to the end of Section 1, for details on our particular use of -convergence, see Definition 2.3. Theorem 3.1 (Hierarchy of effective theories). Let Ih (y) = 1 h2-2 Z 1 Wh (x3, hy(x)) dx. If (2, 3) and is convex,5 then the elastic energies Ih -converge to the linearised Kirchhoff energy IlKi(v) := 1 2 R Q ? 2(-2 v) if v W2,2 sh (), otherwise, (3.1) where Q ? 2 is defined in (2.6). See Theorems 5.1 and 5.2. If = 3 and > 0 then the energies Ih := 1 Ih =3 -converge to the von Karman type energy6 I vK(u, v) := 1 2 R Q2(1/2 (su + 1 2 v v), -2 v) if (u, v) W1,2 (; R2 ) × W2,2 (; R), , otherwise, (3.2) where Q2 is defined in (2.5). See Theorems 5.1 and 5.4. Finally, if > 3 then Ih -converges to the linearised von Karman energy IlvK(u, v) := 1 2 R Q2 su, -2 v , if (u, v) W1,2 (; R2 ) × W2,2 (; R) , otherwise. (3.3) See Theorems 5.1 and 5.5. Moreover, in all cases > 2 for every sequence (yh )h>0 of finite scaled energy there exists a subsequence (not relabelled) such hat (yh )h>0 Ph -converges to v X (if (2, 3)), respectively (u, v) X (if 3), see Lemma 4.1. Remark 3.2. 1. We will not be considering body forces for simplicity, but including them in the analysis as in [23] is straightforward. 2. A standard argument shows that almost minimisers of Ih Ph -converge (up to subsequences) to minimisers of the limiting functional IlKi, respectively IvK, respectively IlvK. 3. Explicit formulae for Q ? 2, Q2 can be found in ([15], Sect. 2.2). 4. Convexity of the domain is required for the representation theorems in Section 7 which are used in the construction of the recovery sequence for (2, 3). Thanks to the results in [25, 26] the theorem is also true for general simply connected domains whose boundary satisfies a (mild) regularity assumption, more 5Also certain non-convex domains can be treated, see Remark 3.2.4 below. 6Again, we slightly overload the notation in what would be a double definition of Ih 3 , trusting the letter used in the subindex to dispel the ambiguity. 10 M. DE BENITO DELGADO AND B. SCHMIDT precisely, if there exists = with H1 () = 0 such that on its complement \ the outer unit normal to exists and is continuous. This is verified, e.g., if is piecewise continuously differentiable. For the sake of clarity we explicitly address the case of convex , but we do include all the necessary arguments to cover the other case as well. The functional IlKi is said to model a linearised Kirchhoff regime because the isometry condition > yy = I of the Kirchhoff model is replaced by det 2 v = 0, a necessary and sufficient condition for the existence of an in-plane displacement u such that u + > u + v v = 0. This condition is to leading order equivalent to > yy = I for deformations y = (h2-4 u, h-2 v).7 The functional I vK is of von Karman type with in-plane and out-of-plane strains interacting in a membrane energy term, and a bending energy term. For simple choices of Q2 and Bh , one recovers the classical functional (1.1). Finally, we say that the third limit IlvK, models a linearised von Karman (or fully linear) regime by analogy with the classical equivalent, but it is of a different kind than the one expected from the hierarchy derived in [23], since it again features an interplay between in-plane and out-of-plane components.8 Our second goal is to show that the limit energy I vK interpolates between IlKi and IlvK as the parameter moves from to 0, so that one can say that the theory of von Karman type bridges the other two. More precisely, in Section 6 we prove: Theorem 3.3 (Interpolating regime). The following two -limits hold: I vK - IlKi, if is convex9 (Thms. 6.4 and 6.5) and: I vK - 0 IlvK (Thm. 6.6 and 6.7). Furthermore, sequences (u, v)>0 of bounded energy I vK are precompact in suitable spaces as or 0 (Thm. 6.3). Example. The easiest non-trivial situation is given by a linear internal misfit in a homogeneous material with B(t) := tI3 R3×3 and Q2(t, ·) = Q2(·). Then I vK(u, v) = 2 Z Q2(su + 1 2 v v) + 1 24 Z Q2(2 v - I). for (u, v) W1,2 (; R2 ) × W2,2 (; R). We refer to [14, 15] for more worked out examples. 4. Compactness and identification of limit strain We collect here some basic results proving compactness of sequences of finite scaled energy and providing explicit representations for the limit strains, as required for the proofs of -convergence in Section 3. These 7In the numerical analysis literature, the denomination linear Kirchhoff is sometimes used for a pure bending regime without constraints. 8This is in contrast to [23]. In our setting with the additional dependence on the x3 coordinate, it is not possible to simply drop terms while bounding below the energy in the proof of the lower bound as is done in ([23], p. 211). Indeed, our keeping track of both in-plane and out-of-plane displacements is essential to capture the effect of pre-stressing with the internal misfit Bh. 9Or simply connected with satisfying the condition in Remark 3.2.4 A HIERARCHY OF MULTILAYERED PLATE MODELS 11 results are direct consequences of the homogeneous case treated in ([23], Lem. 1). We recall the definition of the scaled elastic energies (2.1): Ih (y) = 1 h2-2 Z 1 W0(x3, hy(x)(I + h-1 Bh (x3))) dx. Lemma 4.1. Let (2, ) and let (yh )h>0 Y have finite scaled energy Ih . For every h > 0 there exist constants R h SO(3) and ch R3 such for the corrected deformations yh = (yh ) := (R h )> yh - ch , (4.1) there exist rotations Rh : SO(3) (extended constantly along x3 to all of 1 outside {0} × ) approximating hyh in L2 (1). Quantitatively: khyh - Rh k0,2,1 6 Ch-1 . Furthermore, kRh - Ik0,2,1 6 Ch-2 . Finally there exists a subsequence (not relabelled) such that for the scaled and averaged in-plane and out-of-plane displacements from (2.8) there exist (u, v) W1,2 (; R2 ) × W2,2 () such that, if 6= 3: uh * u in W1,2 (; R2 ) and vh v in W1,2 (), If = 3 an analogous result holds with uh and vh from (2.9). In particular, in the sense of Definition 2.2 we have that (yh )h>0 Ph -converges to v X (if (2, 3)), respectively (u, v) X (if 3). Proof. This is exactly a particular case of ([23], Lem. 1), estimates (84) and (85) and estimates (86) and (87), once we prove that if (yh )h>0 have finite scaled Ih energy, then they have finite scaled energy in the sense of [23]. Note first that among all choices we can make for the energy density W which fulfil the assumptions in [23], we can pick dist2 (·, SO(3)). Therefore we will bound this quantity. Write d(F) := dist(F, SO(3)). We begin by using Assumption 2.1.e: Ch2-2 > Z 1 W0(x3, hy(x)(I + h-1 Bh (x3))) & Z 1 d2 (hy(x)(I + h-1 Bh (x3))). Consider now the following: d2 (F(I + h-1 Bh )) > 1 2 d2 (F) - |Fh-1 Bh |2 > 1 2 d2 (F) - Ch2-2 |1 + d2 (F)| > 1 4 d2 (F) - Ch2-2 . 12 M. DE BENITO DELGADO AND B. SCHMIDT But then we are done since: h2-2 & Z 1 1 4 d2 (hy). Lemma 4.2. 10 Let (2, ) and let (yh )h>0 be a sequence in Y which Ph -converges to (u, v) X in the sense of Theorem 3.1 and Rh : SO(3) (extended constantly along x3 to all of 1 outside {0} × ) such that khyh - Rh k0,2,1 6 Ch-1 . Then: Ah := 1 h-2 (Rh - I) - A if = 3, A else, in L2 (; R3×3 ), where A = e3 ^ v - ^ v e3, and Gh := (Rh )> hyh - I h-1 * G in L2 (1; R3×3 ), where the submatrix G R2×2 is affine in x3: G(x0 , x3) = G0(x0 ) + x3G1(x0 ) and G1 = - 2 v if = 3, -2 v else, (4.2) sym G0 = su + 1 2 v v if = 3, su if > 3, (4.3) and su + 1 2 v v = 0, if (2, 3). Proof. See ([23], pp. 208­209). 10This is almost word for word ([23], Lem. 2) with the very minor addition of the factors , . For other scaling choices see ([23], p. 208). Note that this is inspired by ([9], Thm. 5.4.2) (itself based in ([9], Thm. 1.4.1.c)). A HIERARCHY OF MULTILAYERED PLATE MODELS 13 5. -convergence of the hierarchy This section proves the lower (Thm. 5.1) and upper bounds (Thm. 5.2, 5.4 and 5.5) required for deriving the hierarchy of models in Theorem 3.1. Recall that we are always using weak convergence in the spaces X. Theorem 5.1 (Lower bounds). Let (2, 3). If (yh )h>0 Y is a sequence Ph -converging to v X, then linf h0 Ih (yh ) > IlKi(v). Now let = 3. If (yh )h>0 Y is a sequence Ph -converging to (u, v) X, then for all > 0 linf h0 1 Ih (yh ) > I vK(u, v). Finally, let > 3. If (yh )h>0 Y is a sequence Ph -converging to (u, v) X, then linf h0 Ih (yh ) > IlvK(u, v). Proof. If = 3, we define Bh := Bh and B = B, otherwise B := B and Bh := Bh . Following closely the techniques in [22, 23, 51, 52] we use a Taylor expansion of the energy around the identity which allows us to cancel or identify its lower order terms. For this we must correct the deformations with an approximation by rotations and work in adequate sets where there is control over higher order terms. Upon passing to a subsequence (not relabelled) which realises linfh0Ih (yh ) as its limit, we may w.l.o.g. assume that (yh )h>0 has finite scaled Ih energy and pass to further subsequences in the following. Step 1: Approximation by rotations. We will be working with the corrected deformations (yh ) := (R h )> yh - ch , as given in Lemma 4.1. For simplicity we use the same notation yh for these functions. Also by Lemma 4.1 there exist rotations Rh : SO(3) (extended constantly along x3 to all of 1 outside × {0}) which approximate hyh in L2 (1) and are close to the identity, as required for the identification of the limit strain in Lemma 4.2. Step 2: Rewriting of the deformation gradient. The functions Gh := (Rh )> hyh - I h-1 , are uniformly bounded in L2 by invariance of the norm by rotations: kGh k0,2,1 = h1- khyh - Rh k0,2,1 6 C. (5.1) Now, by the frame invariance of Wh (x3, ·) Wh (x3, hyh ) = Wh (x3, (Rh )> hyh ) = W0(x3, (Rh )> hyh (I + h-1 Bh (x3))) = W0(x3, I + h-1 Ah ), where we have set Ah (x) := (Rh )> hyh (x) - I h-1 + (Rh )> hyh (x)Bh (x3) = Gh + (Rh )> hyh Bh . 14 M. DE BENITO DELGADO AND B. SCHMIDT Step 3: Cutoff function. Let h be the characteristic function of the "good set" {x 1 : |Gh | 6 h-1/2 }, where (Rh )> hyh is close to I. Here we have: h1/2 h-3/2 > h |h-1 Gh | = h |(Rh )> hyh - I| = h |hyh - Rh |, which, because |Rh | 3, implies that h |hyh | 6 C. Consequently, since the Bh are uniformly bounded as well: h |h-1 Ah | = h |h-1 Gh + h-1 (Rh )> hyh Bh | 6 h |h-1 Gh | + O(h-1 ) = o(h1/2 ), and then dist I + h-1 h Ah , SO(3) 6 |I + h-1 h Ah - I| = o(h1/2 ), so in the good sets we may indeed expand around I for small values of h. Now, the sequence (Gh )h>0 is bounded in L2 by (5.1) so we may extract a subsequence converging weakly in L2 to some G L2 (1), which we consider from now on without relabelling. Furthermore the sequence (h )h>0 is essentially bounded and h 1 in measure in 1. Indeed |{|h - 1| > }| = |{|Gh | > h-1/2 }| 0 as h 0 because kGh k0,2,1 6 C uniformly. Consequently we have h Gh * G in L2 (1). Analogously, the sequence (h Bh )h>0 is essentially bounded and converges in measure to B because |{|h Bh - B| > }| 6 |{|Bh - B| > }| + |{h = 0} {|B| > }| 0. Hence, using again the strong convergence (Rh )> hyh I in L2 (1) (Lem. 4.1): (Rh )> hyh h Bh * B in L2 (1). So we conclude h Ah * A := G + B in L2 (1). Step 4: Taylor expansion. Because W0(x3, ·)| SO(3) 0, for any fixed x3 the lower order terms of its Taylor expansion W0(x3, I + E) = W0(x3, I) + DW0(x3, I)[E] + 1 2 D2 W0(x3, I)[E, E] + o(|E|2 ) vanish and we have (for small enough h, as explained above) W0 x3, I + h-1 h Ah = 1 2 Q3 x3, h-1 h Ah + h x3, h-1 h Ah , where h (x3, h-1 h Ah ) = o(h2-2 |h Ah |2 ) represents the higher order terms. Defining the uniform bound (s) := ess sup -162r61 sup |M|6s |h (r, M)|, A HIERARCHY OF MULTILAYERED PLATE MODELS 15 we have (s) = o(s2 ) by Assumption 2.1.c, and integrating over the rescaled domain 1 we obtain the estimate: 1 h2-2 Z 1 Wh (x3, hyh ) dx > 1 h2-2 Z 1 Wh (x3, I + h h-1 Ah ) dx > 1 h2-2 Z 1 h2-2 2 Q3(x3, h Ah ) - (|h-1 h Ah |) dx = 1 2 Z 1 Q3(x3, h Ah ) - 1 h2-2 Z 1 (|h-1 h Ah |) dx. (5.2) Step 5: The limit inferior. In order to pass to the limit, for the first integral on the right hand side of (5.2) we use that Q3 is positive semidefinite, therefore convex and continuous, and the integral is sequentially weakly lower semicontinuous. For the second integral we use again Assumption 2.1.c and the fact that |h-1 h Ah | 6 h1/2 to obtain the bound (uniform over 1): (|h-1 h Ah |) |h-1hAh|2 6 sup |s|6h1/2 (s) s2 - 0 as h 0. But then, because h Ah converges weakly in L2 , we have kh Ah k2 0,2,1 6 C and 1 h2-2 Z 1 h-1 h Ah dx = Z 1 h-1 h Ah |h-1hAh| 2 h-1 h Ah 2 h2-2 dx 6 sup |s|6h1/2 (s) s2 Z 1 h Ah 2 dx | {z } uniformly bded. - 0 as h 0. Taking the lim inf at both sides of (5.2) we have: linf h0 1 h2-2 Z 1 Wh (x3, hyh ) dx > linf h0 1 2 Z 1 Q3 x3, h Ah dx -lim h0 1 h2-2 Z 1 (|h-1 Ah |) dx > 1 2 Z 1 Q3(x3, G + B) dx > 1 2 Z 1 Q2(x3, G + B) dx, where the last estimate follows trivially from the definition of Q2. By Lemma 4.2 the limit strain G has the representation G(x) = G0(x0 ) + x3G1(x0 ), 16 M. DE BENITO DELGADO AND B. SCHMIDT with G1 and sym G0 as in Lemma 4.2. We plug both into the last integral and use the fact that Q2(x3, ·) vanishes on antisymmetric matrices to obtain linf h0 1 h2-2 Z 1 Wh (x3, hyh ) dx > 1 2 Z 1 Q2(x3, G0(x0 ) + x3G1(x0 ) + B(x3)) dx = 1 2 Z Q2(sym G0, G1) dx0 with with G1 and sym G0 given respectively by (4.2) and (4.3). In particular, if = 3, we have again: linf h0 1 h4 Z 1 Wh (x3, hyh ) dx > 1 2 Z Q2(sym G0, G1) dx0 = 1 2 Z Q2(1/2 (su + 1 2 v v), -2 v). If (2, 3), then sym G0 is unknown, so we must further relax the integrand. With the definition of Q ? 2 we see that the final integral above is 1 2 Z 1 Q2(x3, G0 - x32 v + B) dx > 1 2 Z Q ? 2(-2 v) dx0 . We proceed now with the computation of the recovery sequences for each of the three regimes discussed. Theorem 5.2 (Upper bound, linearised Kirchhoff regime). Assume is convex11 , let (2, 3) and v X := W1,2 (). There exists a sequence (yh )h>0 Y which Ph -converges to v such that lsup h0 Ih (yh ) 6 IlKi(v), with IlKi defined as in (3.1) by IlKi(v) := 1 2 R Q ? 2(2 v(x0 )) dx0 if v W2,2 sh (), otherwise. Proof. We set = h-2 , so that h 1 and h2 h 1. Step 1: Setup and recovery sequence. The functional IlKi is strongly continuous on W2,2 sh () by the continuity and 2-growth of Q ? 2. By Theorem 7.1 we have a set V0 of smooth maps with singular Hessian which is W2,2 - dense in W2,2 sh , see (7.2). Therefore, by a standard argument (see, e.g., [7]) it is enough to construct here the recovery sequence. Take then a smooth function v V0. Because kvk < C, for small enough there exist by ([23], Thm. 7) in-plane displacements u W2,2 (; R2 ) W2, (; R2 ) with uniform bounds in such that the deformations y(x0 ) := x0 + 2 u(x0 ) v(x0 ) , 11Instead of convexity one may assume that be simply connected and satisfy the assumption detailed in Remark 3.2.4, because Theorem 7.1, Corollary 7.4 and ([23], Thm. 7) also apply in this situation. A HIERARCHY OF MULTILAYERED PLATE MODELS 17 are isometries.12 That is: > yy = I2, where y = I2 0 0 + 02 > v + 2 u 0 0 R3×2 . Additionally the following normal vectors are unitary in R3 : b(x0 ) := y,1(x0 ) y,2(x0 ) = - v 0 + 3 u2 · (v,2, -v,1) 3 u1 · (-v,2, v,1) 1 + 2 tr u + 4 det u = e3 - ^ v(x0 ) + r(x0 ), where the rest r satisfies krk1, = O(2 ), by virtue of kuk2, 6 C and kvk 6 C. Consequently the matrices R := (y, b) = I + 0 -v > v 0 + r e3 + 2 ^ u | {z } =:r are in SO(3) for every x0 , with the remaining matrix r satisfying krk1, = O(2 ), by the same arguments as before. Now, for some smooth functions , g1, g2 C (; R), g := (g1, g2) and d L (1; R3 ) with 0 d L (1; R3×2 ) and D C (1; R3 ) to be determined later, set yh (x0 , x3) := y(x0 ) + h(x3 - (x0 ))b(x0 ) + h(g(x0 ), 0) +h2 Z x3 0 d(x0 , ) d + h2 D(x0 , x3). (5.3) We will prove Ih (yh ) - h0 IlKi(v), as well as Ph (yh ) v in W1,2 for Rh I SO(3), ch 0 R3 . Step 2: Preliminary computations. In order to compute the limit of 1 h2-2 R 1 W0(x3, hyh (I + hBh )) we start with the gradient of the recovery sequence: hyh = (y, 0) + hh[(x3 - )b] +h[ ^ g + d e3] + hD,3 e3 + o(h). 12The uniform bounds for kuk2,2 follow from ([23], Thm. 7), equation (181), and those for kuk2, from the explicit construction done in the proof, in particular equations (183), (186) and (190). 18 M. DE BENITO DELGADO AND B. SCHMIDT For the first term in h we have h[(x3 - )b] = (x3 - ) ^ b - b ^ + 1 h b e3 = ( - x3)( ^ 2 v - ^ r) - b ^ + 1 h b e3. Substituting back into the gradient yields: hyh = R + h [( - x3) ^ 2 v + ^ g + d e3 + o(1)] | {z } =:Ah -hb ^ + hD,3 e3. Because we intend to use the frame invariance of the energy, we will need the product of hyh with R> = I + O(). First we have: hR> Ah = hAh + o(h) = hAh , where we have subsumed terms o(h) into the o(1) inside Ah . Therefore R> hyh = I3 + hAh -he3 ^ + R> hD,3 e3 | {z } =:hF h . (5.4) Step 3: Convergence of the energies. The next step is a Taylor expansion around the identity. Given that the energy is scaled by (h)-2 , only those terms scaling as h in (5.4) will remain: anything beyond that will not be seen and anything below will make the energy blow up. This means that we must choose D so that Fh = D,3 e3 + (v,1D3,3, v,2D3,3, -v,1D1,3 - v,2D2,3) e3 -e3 ^ + o() = o(). Although these equations have no solution the symmetrised version does,13 so that for every smooth choice of we can pick a bounded D such that Fh s = 0, and Fh = O(1), (5.5) a fact that we will exploit next. By frame invariance and (5.4), we can write W0(x3, hyh (I + hBh )) = W0(x3, R> hyh (I + hBh )) = W0(x3, (I + hAh + hFh )(I + hBh )) = W0(x3, I + h ((Ah + Bh ) + Fh + o()) | {z } =:Ch ). 13Dividing by h we arrive at: D1,3 + v,1D3,3 = ,1 + o(), D2,3 + v,2D3,3 = ,2 + o(), D3,3 - v,1D1,3 - v,2D2,3 = o(), with solution: D(x0 , x3) = x3 ^ + x3v · e3. A HIERARCHY OF MULTILAYERED PLATE MODELS 19 Because of (5.5) by our choice of D we need to subtract the antisymmetric part of Fh , which we do by means of another rotation and frame invariance: W0(x3, I + hCh ) = W0(x3, e-hF h a (I + hCh )) = W0(x3, I + hCh - hFh a + O(h2 )) = W0(x3, I + h(Ah + Bh ) + o(h)). Now whenever h is small enough that I + hCh belongs to the neighbourhood of SO(3) where W0 is twice differentiable, we can apply Taylor's theorem and the fact that Q3 vanishes on antisymmetric matrices to see that, as h 0: 1 2h2 W0(x3, hyh (I + hBh )) = 1 2 Q3(x3, (Ah + Bh )s) + o(1) 1 2 Q3(x3, As + Bs), where As = ( - x3) ^ 2 v + ^ sg + (d e3)s. We choose d(x0 , x3) = L(x3, ( - x3)2 v + sg + Bs) - B·3, with L the map from (2.3), which by (2.3) and (2.4) is linear in the second component and satisfies |L(t, A)| . |A| uniformly in t, and B·3 the third column of B. Because the matrix ( -x3)2 v +sg +Bs is bounded uniformly in x0 , by the bound (2.4) the map x 7 Z x3 0 L(, ( - ) ^ 2 v + ^ sg + Bs()) d, is in W1, (1; R3 ) and yh W1,2 as required (for the derivatives with respect to x0 note that v, g are smooth and B independent of x0 ). Now, all quantities being bounded, by dominated convergence: Ih (yh ) 1 2 Z 1 Q3(x3, ( - x3) ^ 2 v + ^ sg + (d e3)s + Bs) = 1 2 Z 1 Q2 x3, ( - x3)2 v + sg + Bs . Note that a final step is required to obtain convergence to IlKi(v). Step 4: Convergence of the deformations: Ph (yh ) v in W1,2 . Choose Rh I SO(3), ch 0 R3 in the definition of for (2.8). We have Ph (yh ) = 1 Z 1/2 -1/2 yh 3 (x0 , x3) dx3, 20 M. DE BENITO DELGADO AND B. SCHMIDT where in (5.3) we defined yh 3 (x0 , x3) = v(x0 ) + h(x3 - (x0 ))b3(x0 ) + O(h). Then: |Ph (yh ) - v|2 = 1 Z 1/2 -1/2 [v + h(x3 - )b3 + O(h)] dx3 - v 2 = O(-2 h2 ), and consequently kPh (yh ) - vk0,2 0. An analogous computation for the derivatives shows strong convergence in W1,2 . Step 5: Simultaneous convergence. Finally, as in ([52], Thm. 3.2), in order for the energy to converge to the true limit, we must pick and g in (5.3) so as to approximate the minimum Q2. This is done with Corollary 7.4, substituting sequences of smooth functions (k)kN, (gk)kN for the functions , g. Then, for each fixed k we have: Ih (yh k ) h0 1 2 Z 1 Q2 x3, (k - x3)2 v + sgk + Bs = 1 2 Z Q ? 2(-2 v) dx0 + o(1)k, and kPh (yh k ) - vk2 1,2 6 C(k)-2 h2 . And by a diagonal argument we can find (yh )h>0 whose energy converges to IlKi(v) while maintaining the convergence of the deformations. Remark 5.3. The recovery sequence defined in (5.3) is substantially different from the recovery sequence constructed in [23] for homogeneous layers. While the summand depending on d, which allows for relaxation of the strain in the thin film direction to account for the Poisson effect, is of a more complicated form, it still has a direct counterpart in the homogeneous case. The contributions involving , g and D, however, are particular to heterogeneous layers. The mapping is used to re-adjust the height x3 = 0 to x3 = (x0 ) at which the film follows the isometry y to leading order. Considering, e.g., a bilayer consisting of a soft material on top of a material with large moduli, one typically has < 0. This, however, leads to new terms in the strain depending on . The last term involving D, in combination with the microrotation e-hF h a introduced in the above proof, is used to compensate for these terms. Finally, the summand involving the in-plane term g, together with the strain induced by b, by Corollary 7.4 allows for a full relaxation of the x3-independent strain contributions and thus to pass to the relaxed quadratic form Q ? 2, which only depends of the linearized curvature term 2 v. Theorem 5.4 (Upper bound, von Karman regime). Let = 3 and consider displacements (u, v) X=3 := W1,2 (; R2 ) × W1,2 (; R). There exists a sequence (yh )h>0 Y which Ph -converges to (u, v) such that lim h0 1 Ih (yh ) = I vK(u, v), with I vK defined as in (3.2) by I vK(u, v) := 1 2 Z Q2(1/2 (su + 1 2 v v), -2 v) over X0 =3 = W1,2 (; R2 ) × W2,2 (; R) and as elsewhere. A HIERARCHY OF MULTILAYERED PLATE MODELS 21 Proof. In order to build the recovery sequence (yh )h>0 we will use the map L : (-1/2, 1/2) × R2×2 R3 given by (2.3), which for each t realises the minimum of Q3(t, A + c e3), A R2×2 , i.e. Q2(t, A) = Q3(t, A + L(t, A) e3) = Q3(t, A + (L(t, A) e3)s), where the last equality follows from the fact that Q3 vanishes on antisymmetric matrices. Recall from (2.4) that L(t, ·) is linear for every t and that |L(t, A)| . |A| uniformly in t. The functional I vK is clearly continuous in X0 = W1,2 (; R2 ) × W2,2 (; R) with the strong topologies, so a standard argument [7] shows that it is enough to consider (u, v) C (; R2 ) × C (; R), which is dense in X0 . We define: yh (x0 , x3) := x0 hx3 + h2 u(x0 ) hv(x0 ) - h2 x3 v(x0 ) 0 +h3 d(x0 , x3), where d W1, (1; R3 ) is a vector field to be determined along the proof. Step 1: Approximation of the energy. A direct computation yields hyh = I + h2 u -h v h > v 0 - h2 x3-1/2 2 v 0 0 0 +h2 3d e3 + O(h3 ) = I + h (e3 ^ v - ^ v e3) | {z } E +h2 ^ u - x3-1/2 ^ 2 v + 3d e3 | {z } F + O(h3 ). For later use we note here the product: > h yh hyh = I + h E> + h2 F> I + h E + h2 F + O(h3 ) = I + h 2Es | {z } =0 + h2 (2Fs + E> E) | {z } N + O(h3 ), where we used that E is antisymmetric. For any matrix M with positive determinant we have the polar decom- position M = U M>M = U I + P, with U SO(3) and P = M> M - I. By the frame invariance of the energy and a Taylor expansion around the identity of the square root W0(x3, M) = W0(x3, M>M) = W0 x3, I + 1 2 (M> M - I) + o(|M> M - I|) , and, assuming that a Taylor expansion of W0 around the identity can be carried, i.e. that M is close enough to SO(3), this is equal to: 1 2 Q3 x3, 1 2 (M> M - I) + o(|M> M - I|2 ). 22 M. DE BENITO DELGADO AND B. SCHMIDT In view of the definition of W0, we set Mh := hyh (I + h2 Bh ), where Bh = Bh B = B in L . Then (Mh )> Mh := [hyh (I + h2 Bh )]> [hyh (I + h2 Bh )] = (I + h2 (Bh )> )(I + h2 N)(I + h2 Bh ) + O(h3 ) = I + h2 N + h2 2Bs + o(h2 ). To compute the first term in h2 , N = 2Fs + E> E, we have 2Fs = 2 ^ su - x3-1/2 ^ 2 v + (3d e3)s , and: E> E = ( ^ v e3 - e3 ^ v)(e3 ^ v - ^ v e3) = ^ v ^ v + | ^ v|2 e3 e3. Since these quantities are independent of h, for sufficiently small h the product (Mh )> Mh does lie close enough to SO(3) and we can perform the desired Taylor expansion: Wh (x3, hyh ) = W0(x3, hyh (I + h2 Bh )) = W0 x3, ((Mh )> Mh )1/2 = 1 2 Q3 x3, 1 2 [(Mh )> Mh - I] + o(|(Mh )> Mh - I|2 ). Define now G0 := ^ su + 1 2 ^ v ^ v , G1 := -1/2 ^ 2 v as in Lemma 4.2. Bringing the previous computations together we obtain: 1 2 [(Mh )> Mh - I] = h2 [G0 - x3G1 + b Bs + (B(t)·3 e3)s + 2 | ^ v|2 e3 e3 + (3d e3)s | {z } H ] +o(h2 ), hence 1 h4 Q3 x3, 1 2 ((Mh )> Mh - I) + o(|(Mh )> Mh - I|2 ) = Q3 x3, G0 - x3G1 + b Bs + H + o(1). We now choose the vector field d to cancel one term and attain the minimum for the others by solving for 3d in: H ! = L x3, G0 - x3G1 + Bs(x3) e3 s , A HIERARCHY OF MULTILAYERED PLATE MODELS 23 that is: -1/2 B(t)·3 + 1 2 | ^ v|2 e3 + 3d(x0 , x3) = 1 L x3, G0 - x3G1 + Bs(x3) . Consequently, we set: d(x0 , x3) := - 1 2 | ^ v|2 x3e3 + 1 Z x3 0 L t, G0 - tG1 + Bs(t) - B(t)·3 dt, and we obtain Q3 x3, G0 - x3G1 + Bs(x3) + H = Q2 x3, G0 - x3G1 + Bs(x3) . As in the proof of Theorem 5.2, (2.3) and (2.4) imply that d W1, (1; R3 ). Step 2: Convergence. By the previous step we have 1 h4 W0(x3, hyh ) 1 2 Q2 x3, G0 + x3G1 + Bs a.e. as h 0, and the sequence is uniformly bounded so we can integrate over the domain and pass to the limit: 1 h4 Z 1 Wh (x3, hyh ) 1 2 Z 1 Q2 x3, G0 - x3G1 + Bs = 1 2 Z Q2(1/2 (su + 1 2 v v), -2 v). Step 3: Convergence of the recovery sequence. Note that Ph (yh ) (u, v) in X as h 0 with the choice Rh = I SO(3), ch = 0 R3 in Definition 2.2 since 1 h2 Z 1/2 -1/2 (yh (·, x3) - x0 ) dx3 - u in W1,2 (; R2 ), 1 h Z 1/2 -1/2 yh 3 (·, x3) dx3 - v in W1,2 (; R). In the next result, there is a departure from the analogous functional in [23] beyond the dependence on the out-of-plane component x3. In the preceding cases, if one sets Q2(t, A) Q2(A), and B 0 then the same functionals are obtained as in that work. However, in the regime > 3 their limit has no membrane term, but we have Q2 su, -2 v = 1 2 R Q2 (su) + 1 24 R Q2(2 v), with the membrane term. The reason is that [23] discard the in-plane displacements by minimising them away. In their proofs, they drop the first term in the lower bound and build the recovery sequence with no u term in h-1 . Note that it is by keeping the membrane term that our model is able to take into account and respond to the pre-stressing (internal misfit) Bh , e.g. compressive or tensile stresses in wafers. Theorem 5.5 (Upper bound, linearised von Karman regime). Let > 3 and consider displacements (u, v) X := W1,2 (; R2 ) × W1,2 (; R). There exists a sequence (yh )h>0 Y which Ph -converges to (u, v) such that lim h0 Ih (yh ) = IlvK(u, v), 24 M. DE BENITO DELGADO AND B. SCHMIDT with IlvK defined as in (3.3) by IlvK(u, v) := 1 2 Z Q2 su, -2 v dx0 , on X0 and by + elsewhere. Proof. We follow closely the notation and path of proof of Theorem 5.4. By a standard density argument it is enough to consider (u, v) X C (). Define yh (x0 , x3) := x0 hx3 + h-1 u(x0 ) h-2 v(x0 ) - h-1 x3 v(x0 ) 0 + h d(x0 , x3), with d W1, (1; R3 ). Then hyh = I + h-2 (e3 ^ v - ^ v e3) | {z } =:E +h-1 ( ^ u - x3 ^ 2 v + 3d e3) | {z } =:F +O(h ), and, using that Es = 0: > h yh hyh = (I + h-2 E> + h-1 F> )(I + h-2 E + h-1 F) + O(h ) = I + 2h-1 Fs + o(h-1 ). Define now Mh := hyh (I + h-1 Bh ). A few computations lead to 1 2 [(Mh )> Mh - I] = h-1 (Fs + Bs) + o(h-1 ), from which follows, after a Taylor approximation (recall from the proof of Theorem 5.4, that this can be done for sufficiently small h): 1 h2-2 Wh (x3, hyh ) = 1 2h2-2 [Q3(x3, [(Mh )> Mh - I]/2) +o(|(Mh )> Mh - I|2 ) = 1 2 Q3(x3, Fs + Bs) + o(1). Picking d such that: ((B(x3)·3 + 3d) e3)s = (L(x3, u - x32 v + Bs(x3)) e3)s, e.g. d(x0 , x3) := Z x3 0 L t, su - t2 v + Bs(t) - B(t)·3 dt, the term with L in Q2 cancels out and we obtain Q3(x3, Fs + Bs) = Q2 x3, su - x32 v + Bs(x3) . A HIERARCHY OF MULTILAYERED PLATE MODELS 25 Note that as proved in Theorem 5.4, the properties of L imply that the function d W1, (1; R3 ) so the previous computations are justified. We have therefore 1 h2-2 W0(x3, hyh ) 1 2 Q2 x3, su - x32 v + Bs(x3) a.e. in , and also Q2(x3, A) . |A|2 by Assumption 2.1.b. Because ui, v C () and Bs L , all arguments of Q2 are uniformly bounded and we can apply dominated convergence to conclude: 1 h2-2 Z 1 W0(x3, hyh ) - h0 1 2 Z 1 Q2 x3, su - x32 v + Bs(x3) dx = 1 2 Z Q2 su, -2 v dx0 . Set now R = I SO(3), c = 0 R3 for the rigid transformation in Definition 2.2. It remains to note that indeed Ph (yh ) (u, v) in X: 1 h-2 Z 1/2 -1/2 yh 3 (·, x3) dx3 - v in W1,2 (; R), 1 h-1 Z 1/2 -1/2 (yh0 (·, x3) - x0 ) dx3 - u in W1,2 (; R2 ), and the proof is complete. 6. -convergence of the interpolating theory Notation. Throughout this section we write A := su + 1 2 v v for the strain induced by a pair of displacements (u, v). As before, > 0. We now set to prove Theorem 3.3, which states that the functional of generalised von Karman type that we found in the preceding section, I vK(u, v) := 1 2 Z Z 1/2 -1/2 Q2 x3, A - x32 v + B(x3) dx3 dx0 , interpolates between the two adjacent regimes as or 0. As approaches infinity, we expect the optimal energy configurations to approach those of the linearised Kirchhoff model, whereas with tending to zero they should approach the linearised von Karman model. For this section we restrict ourselves to spaces where Korn-Poincare type inequalities hold. Definition 6.1. Let Xu := u W1,2 (; R2 ) : Z au = 0 and Z u = 0 , and Xv := v W2,2 (; R) : Z v = 0 and Z v = 0 . 26 M. DE BENITO DELGADO AND B. SCHMIDT We set Xw := Xu × Xv with the weak topologies. Additionally, from now on we assume without loss that the barycenter of be the origin so that R x0 dx0 = 0. Finally, for the -limit we require that be convex or simply connected with satisfying the condition in Remark 3.2.4 and recall the definition of the space of maps with singular Hessian W2,2 sh () := v W2,2 (; R) : det 2 v = 0 a.e. . Remark 6.2. There is no loss of generality in reducing to the space Xu × Xv: First we can always add an infinitesimal rigid motion to u and any affine function to v without changing su or 2 v. Second, although the nonlinear term v v does change after adding an affine function, the extra terms appearing happen to be a symmetric gradient which can be absorbed into su with a little help: For any g(x) = a · x + b for a, b R2 , we have (v + g) (v + g) = v v + a a + a v + v a = v v + sz (6.1) where we set z(x) := (2v(x) + a · x)a W2,2 (; R2 ). Therefore, for any fixed u W1,2 (; R2 ), v W2,2 () one can choose g(x) = -[(v) · x + (v)] and define u = u + z + r, v = v + g, with r(x) = Rx + c, for constants R := -1 || R au + az dx R2×2 ant and c := -1 || R u(x) + z(x) + Rx dx. For u, v we then have on the one hand R u = 0, R au = 0 and R v = 0, R v = 0 and on the other (note that sr = 0): I vK(u, v) = I vK(u - z - r, v - g) (6.1) = I vK(u - r, v) = I vK(u, v) as desired. Our first theorem identifies the types of convergence required in order to obtain precompactness of sequences of bounded energy. We use these definitions of convergence for the computation of the -limits. Theorem 6.3 (Compactness). Let (u, v)>0 be a sequence in Xw with finite energy sup >0 I vK(u, v) 6 C. Then: 1. The sequence (v) is weakly precompact in W2,2 () and the weak limit is in Xv W2,2 sh (). Additionally (u) is weakly precompact in W1,2 (; R2 ). 2. The sequence (1/2 u, v)0 is weakly precompact in W1,2 (; R2 ) × W2,2 () and the weak limit is in Xu × Xv. Proof. By assumption: C > Z Z 1/2 -1/2 Q2 x3, A - x32 v + B(x3) dx3 dx0 , and the uniform lower bound on Q2 in (2.4) yields Q2(x3, F) & |F|2 for all symmetric F and x3 (-1/2, 1/2) , A HIERARCHY OF MULTILAYERED PLATE MODELS 27 so that R 1/2 -1/2 Q2(x3, F(x3)) & R 1/2 -1/2 |F(x3)|2 . Now split the inner integral in half, and normalise to use Jensen's inequality. In the upper half: C & Z 2 Z 1/2 0 A - x32 v + Bs(x3) 2 dx3 dx0 & Z 2 Z 1/2 0 A - x32 v + Bs(x3) dx3 2 dx0 = Z A - 1 4 2 v + c 2 dx0 & A - 1 4 2 v 2 0,2 - c2 ||. An analogous computation for the lower half of the interval results in C > A + 1 4 2 v 0,2 , and bringing both bounds together we obtain: A 0,2 6 C and k2 vk0,2 6 C. (6.2) Two applications of Poincare's inequality to the second bound yield: kvk2,2 6 C for all > 0. Therefore a subsequence (not relabelled) v * v for some v Xv. Now consider (6.2) again and observe that with the Sobolev embedding W1,2 () , L4 () we know that kv vk0,2 = kvk2 0,4 . kvk2 1,2 6 kvk2 2,2 6 C. Together with (6.2) this implies su 0,2 6 C + C , (6.3) so, by the Korn-Poincare inequality, the sequence (u)>0 is bounded in W1,2 when and there exists a subsequence (not relabelled) u * u for some u Xu. Now if z * z in W1,2 (; R2 ), by the compact Sobolev embedding W1,2 , L4 we have z z in L4 and Z |z z - z z|2 dx - 0 0. So v v v v in L2 and from (6.2) and lower semicontinuity of the norm we deduce su + 1 2 v v 0,2 6 linf kAk0,2 = 0. ([23], Prop. 9) (applied to every ball contained in ) shows det 2 v = 0 a.e., and this concludes the proof of the first statement. 28 M. DE BENITO DELGADO AND B. SCHMIDT For the second statement we take 0. It only remains to prove precompactness for u since the pre- vious computation for (v)>0 applies for all . But it follows directly from (6.3) above: again with the Korn-Poincare inequality, the sequence (1/2 u)>0 is bounded in W1,2 , so it contains a weakly convergent subsequence 1/2 u * u Xu. We begin the proof of -convergence in Theorem 3.3 with the lower and upper bound and a few technical lemmas for the passage from = 3 to < 3. Theorem 6.4 (Lower bound, von Karman to linearised Kirchhoff). Assume is convex and let (u, v)>0 be a sequence in Xw such that v * v in Xv as . Then linf I vK(u, v) > IlKi(v). Proof. By Theorem 6.3 we only need to consider v X0 v := Xv W2,2 sh (), hence IlKi(v) < . We can minimise the inner integral pointwise and obtain a lower bound: I vK(u, v) = 1 2 Z Z 1/2 -1/2 Q2 x3, A - x32 v + B(x3) dx3 dx0 > 1 2 Z min AR2×2 Z 1/2 -1/2 Q2(x3, A - x32 v + B(x3)) dx3 dx0 = IlKi(v). As Q ? 2 is a convex quadratic form, we have by the convergence 2 v * 2 v in L2 : linf I vK(u, v) > linf IlKi(v) > IlKi(v). Theorem 6.5 (Upper bound, von Karman to linearised Kirchhoff). Assume is convex14 and fix some v Xv. There exists a sequence (u, v) Xw such that v v in W2,2 () and I vK(u, v) IlKi(v) as . Proof. Without loss of generality we may assume that v X0 v := Xv W2,2 sh (). By Theorem 7.1 we can work with functions v V0, see (7.2), which are smooth with singular Hessian, since they are dense in the restriction to X0 v . By ([23], Prop. 9) there exists a displacement u : R2 in W2,2 (; R2 ) such that su + 1 2 vv = 0. (6.4) Fix > 0 and, using Corollary 7.4, choose smooth functions C (), g C (; R2 ) such that sg + 2 v - Amin 2 0,2 < , where Amin L (; R2×2 sym) is defined as Amin := argmin AR2×2 sym Z 1/2 -1/2 Q2(t, A - t2 v + B(t)) dt. 14Alternatively we may assume that be simply connected and satisfy the assumption detailed in Remark 3.2.4, because Theorem 7.1, Corollary 7.4 and ([23], Prop. 9) also apply in this situation. A HIERARCHY OF MULTILAYERED PLATE MODELS 29 Define now the recovery sequence (u, v)>0 with u := u + 1 (v + g), v := v - 1 . Clearly v = v - -1/2 v as in W2,2 (). Furthermore su = su + sg + ( v)s + 2 v 2 v v = 2 v v + 1 2 - ( v)s, and -t2 v = -t2 v + t 2 , so that, using (6.4) and the fact that the product k k0,2 = kk2 0,4 is bounded we have I vK(u, v) = 1 2 Z Z 1/2 -1/2 Q2 t, 1/2 A - t2 v + B(t) dt dx0 = 1 2 Z Z 1/2 -1/2 Q2 t, sg + ( - t)2 v + B(t) dt dx0 + O(-1/2 ). Now subtract and add Amin inside Q2 and use Cauchy's inequality to get Z 1/2 -1/2 Q2 t, sg + 2 v - t2 v + B dt 6 1 + Z 1/2 -1/2 Q2(t, Amin - t2 v + B) dt + 1 4 Z 1/2 -1/2 Q2 t, sg + 2 v - Amin dt | {z } .ksg+2v-Amink2 0,2< = Z 1/2 -1/2 Q2(t, Amin - t2 v + B) dt + O0(1/2 ). We plug this in and obtain: I vK(u, v) 6 1 2 Z Z 1/2 -1/2 Q2(t, Amin - t2 v + B(t)) dt dx0 +O(-1/2 ) + O0(1/2 ) - 1 2 Z Z 1/2 -1/2 Q2(t, Amin - t2 v + B(t)) dt dx0 + O0(1/2 ). The proof is concluded by letting 0 and passing to a diagonal sequence. 30 M. DE BENITO DELGADO AND B. SCHMIDT We finish the proof of Theorem 3.3 with the lower and upper bounds for the transition from = 3 to > 3. The lack of constraints in the limit functional makes the proofs straightforward. Theorem 6.6 (Lower bound, von Karman to linearised von Karman). Let (u, v)>0 be a sequence in Xw such that (1/2 u, v) * (u, v) in Xw as 0. Then linf 0 I vK(u, v) > IlvK(u, v). Proof. We may assume that sup>0 I vK(u, v) 6 C. Then by Theorem 6.3 (v)>0 is bounded in W1,2 and by the Sobolev embedding W1,2 , L4 we have as before kv vk0,2 = kvk2 0,4 6 C. Consequently A = su + 2 v v * su in L2 as 0. By convexity of the quadratic form Q2 we conclude linf 0 I vK(u, v) > 1 2 Z Z 1/2 -1/2 Q2 x3, su - x32 v + B(x3) dx3 dx0 = IlvK(u, v). Theorem 6.7 (Upper bound, von Karman to linearised von Karman). Let (u, v) Xw. There exists a sequence (u, v)>0 Xw such that (1/2 u, v) (u, v) in Xw and I vK(u, v) IlvK(u, v) as 0. Proof. Define u := -1/2 u and v := v. Clearly (1/2 u, v) (u, v) and using again W1,2 , L4 we have: A = su + 1 2 1/2 v v - 0 su in L2 . Consequently: I vK(u, v) = 1 2 Z Z 1/2 -1/2 Q2 x3, A - x32 v + B(x3) dx3 dx0 - 0 1 2 Z Z 1/2 -1/2 Q2 x3, su - x32 v + B(x3) dx3 dx0 = IlvK(u, v), as stated. 7. Approximation and representation theorems A key ingredient in the proofs of the upper bounds is the density of certain smooth functions in the space where the energy is minimised. In particular, for the case (2, 3) we obtain a result proving that W2,2 maps with singular Hessian can be approximated by a specific set of smooth functions with the same property. In order to apply the results of [52] we may restrict ourselves to isometries which partition into finitely many A HIERARCHY OF MULTILAYERED PLATE MODELS 31 so-called bodies and arms. More precisely, suppose y : R3 is a W2,2 isometric immersion and denote by II = II(y) its second fundamental form, i.e., IIij = y,i ·(y,1 y,2),j. Then II is singular, and there exists fy W1,2 such that fy = II. We call : [0, l] , parameterised by arclength, a leading curve if it is orthogonal to the inverse images of fy on regions where fy is not constant. We denote by and the curvature and unit normal, respectively, i.e., 00 = . In fact, must be bounded, hence W2, . A subdomain 0 is said to be covered by a curve if 0 {(t) + s(t) : s R, t [0, l]}. As shown in [46], if R2 is a bounded convex domain, it can be partitioned into so-called bodies and arms. Here a body is a connected maximal subdomain on which y is affine and whose boundary contains more than two segments inside . An arm is a maximal subdomain () covered by some leading curve . Such a covering is possible also for general bounded Lipschitz domains, see ([26], Thms. 3 and 4). In [52] (built on [46]) it is shown that for convex the set A0 := y C (; R3 ) : y is an isometry finitely partitioning , (7.1) i.e., the set of C -smooth isometries with only a finite number of bodies and arms, is dense in the W2,2 - isometries. For Lipschitz domains, a direct application of the results in [26] only leads to approximations with possibly a countable number of arms within portions near the boundary. However, for domains for which there exists = with H1 () = 0 such that on its complement \ the outer unit normal to exists and is continuous, cf. Remark 3.2.4, Hornung constructs in [25] C -smooth approximations to a given W2,2 -isometry, which are non-affine only on a finite number of arms. In this sense, the set A0 in (7.1) is still dense in the W2,2 -isometries if it is understood as the set of C -smooth isometries y with only a finite number of bodies and arms on which y is non-affine. Here we show that, additionally, V0 := {v C () : > 0 s.t. v = y3 for some y A0}, (7.2) is W2,2 -dense in W2,2 sh .15 Theorem 7.1. Let R2 be a bounded, convex16 domain. Then the set V0 is W2,2 -dense in W2,2 sh (). In particular det 2 v = 0 for all v V0. Proof. Step 1: Approximation. Let v W2,2 sh () and > 0. By ([23], Thm. 10), we can find some v W2,2 sh () W1, () s.t. kv - vk2,2 < /2 and, for = () > 0 sufficiently small, kvk < 1/2. One can now apply ([23], Thm. 7) to construct an isometry y W2,2 (; R3 ) whose out-of-plane component y3 = v. By the density of A0 we find a smooth y A0 such that ky - yk2,2 < /2 and in particular ky3 - y3k2,2 < /2. Setting := y3/ V0 we conclude kv - k2,2 6 kv - vk2,2 + kv - k2,2 < . 15The density of C2() W2,2 sh () in W2,2 sh () was first announced in [46] to follow along the same lines as the density of smooth isometric immersions in the class of W2,2 isometric immersions. As this seems not to be straightforward, we follow a different route reducing the density of V0 in W2,2 sh to the density of A0 in the set of W2,2 isometric immersions. We are grateful to Peter Hornung for the help provided with this proof. 16Again the assumption of convexity may be dropped if one requires that still be simply connected and satisfy the assumption detailed in Remark 3.2.4, because A0 is still dense in the set of W2,2 isometric immersions and ([23], Thm. 7) holds true also in this situation. 32 M. DE BENITO DELGADO AND B. SCHMIDT Step 2: Inclusion. Let v V0 with v = y3, > 0 for some smooth isometry y A0. Recall that the second fundamental form II(y) of any smooth isometric immersion y is singular and the identity 2 yj = -II(y)nj holds for all j {1, 2, 3}, where n = y,1 y,2.17 Therefore det(2 v) = det(- II(y) n3) = 0 and the proof is complete. As it appears to be of independent interest we state here the following condensed version of our previous considerations. Remark 7.2. Let R2 be a bounded, simply connected, Lipschitz domain whose boundary contains a set = with H1 () = 0 such that on its complement \ the outer unit normal to exists and is continuous. Then the set W2,2 sh () C () is W2,2 -dense in W2,2 sh (). Once one can work with smooth functions, the essential tool for the construction of the recovery sequences for (2, 3) is the following representation theorem for maps with singular Hessian and its corollary, both inspired by [52]: In ([52], Lem. 3.3) it was shown that if y A0 and A C (; R2×2 sym) vanishes over a neighbourhood of N = {II(y) = 0}, then there exist , g1, g2 C () vanishing on N such that A = sg + II(y). While was assumed to be convex in [52], the same proof applies to the situation of domains where satisfies the assumption detailed in Remark 3.2.4. This is because , g1, g2 are chosen to vanish on bodies and those arms where y is affine and the construction in ([52], Lem. 3.3) is done over the finitely many covered domains over which y is non-affine only. Theorem 7.3. Let R2 be a bounded convex18 domain and v V0 and A C (; R2×2 sym) such that A 0 in a neighbourhood of {2 v = 0}. There exist maps , g1, g2 C () such that = gi = 0 on {2 v = 0} and A = sg + 2 v. Proof. Let > 0, y A0 s.t. v = y3. Using that 2 y3 = -II(y)n3 holds by virtue of y being an isometry, with n = y,1 y,2 being the unit normal vector, we have that A 0 in a neighbourhood of {II(y) = 0} {n3 = 0}, and {2 v = 0} = {II(y) = 0} {n3 = 0}. We can apply the above stated ([52], Lem. 3.3) to y in order to obtain functions , g1, g2 C () s.t. , g1, g2 = 0 on {II(y) = 0} and A = sg + II(y). By examining the proof of this Lemma one can see that , g 0 in a neighbourhood of {n3 = 0}: since over bodies one has , g1, g2 = 0 by construction, we need only consider arms. On these sets, if n3 vanishes at a point then it vanishes at a whole line perpendicular to the leading curve, because the latter is orthogonal to the level sets of the gradient. Now, because A = 0 in a neighbourhood of this line, when solving the equations in the proof of the Lemma which determine g then , one obtains u2,s = 0 and u2,t = 0, and with the boundary conditions u2 = 0 then u1 = 0 is a solution to the remaining equation. Hence g = 0 and = 0 on these lines. Since the functions so obtained are C , we can define := -/n3 if n3 6= 0 and = 0 otherwise, and this is a smooth function such that A = sg + 2 v. 17See ([42], Prop. 3) for a proof for W2,2 isometries on Lipschitz domains. 18Again instead being convex one may assume that it be simply connected and satisfy the assumption detailed in Remark 3.2.4. Theorem 7.1, Corollary 7.4, ([23], Thm. 7) and ([52], Lem. 3.3) (and its proof) are valid in this situation as well. A HIERARCHY OF MULTILAYERED PLATE MODELS 33 Corollary 7.4. Let be as in Theorem 7.3, v V0 and define for every x0 Amin(x0 ) = argmin AR2×2 sym Z 1/2 -1/2 Q2(t, A - t2 v(x0 ) + Bs) dt. Then Amin L2 (; R2×2 sym) and there exist sequences of functions k C (), gk C (; R2 ) such that sgk + k2 v - Amin L2(;R2×2) - 0 as k . Proof. Let k N be arbitrary. First, on the set {2 v = 0} we trivially have Amin A0 a constant matrix. Now let Ak C (; R2×2 ) with support in {2 v 6= 0} such that kAk - (Amin - A0)kL2(;R2×2) < 1 k , and use Theorem 7.3 to pick smooth k, gk on with Ak = sgk + k2 v. Set gk(x0 ) = gk(x0 ) + A0x0 . Then: sgk + k2 v - Amin L2 = sgk + k2 v - (Amin - A0) L2 < 1 k . Acknowledgements. We are grateful to Peter Hornung for the help provided with the proof of Theorem 7.1. Also the valu- able suggestions of the unknown referees are appreciated. 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Math. Anal. Appl. 458 (2018) 1236–1273. [29] C. S. Kim and S. J. Lombardo, Curvature and bifurcation of MgO-Al₂O₃ bilayer ceramic structures. J. Ceram. Process. Res. 9 (2008) 93–96. [30] G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40 (1850) 51–88. [31] Y. Klein, E. Efrati and E. Sharon, Shaping of elastic sheets by prescription of non-euclidean metrics. Science 315 (2007) 1116–1120. [32] R. Kupferman and J. P. Solomon, A Riemannian approach to reduced plate, shell, and rod theories. J. Funct. Anal. 266 (2014) 2989–3039. [33] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549–578. [34] M. Lewicka and D. Lučić, Dimension reduction for thin films with transversally varying prestrian: the oscillatory and the non-oscillatory case. Preprint (2018). [35] M. Lewicka, L. Mahadevan and M. R. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains. Proc. Roy. Soc. London Ser. A. Math. Phys. Eng. Sci. 467 (2011) 402–426. [36] M. Lewicka, M. G. Mora and M. R. Pakzad, Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity. Annali della Scuola normale superiore di Pisa, Classe di scienze 9 (2008) 253–295. [37] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover Publications, New York (1944). [38] C. Maor and A. Shachar, On the role of curvature in the elastic energy of non-Euclidean thin bodies. J. Elast. 134 (2019) 149–173. [39] C. B. Masters and N. Salamon, Geometrically nonlinear stress-deflection relations for thin film/substrate systems. Int. J. Eng. Sci. 31 (1993) 915–925. [40] M. G. Mora, S. Müller and M. G. Schultz, Convergence of equilibria of planar thin elastic beams. Indiana Univ. Math. J. 56 (2007) 2413–2438. [41] S. Müller, Mathematical problems in thin elastic sheets: scaling limits, packing, crumpling and singularities, in Vector-Valued Partial Differential Equations and Applications, Vol. 2179. Lecture Notes Math. Springer, Cham (2017) 125–193. [42] S. Müller and M. R. Pakzad, Regularity properties of isometric immersions. Mathematische Zeitschrift 251 (2005) 313–331. [43] S. Müller and M. R. Pakzad, Convergence of equilibria of thin elastic plates—the von Kármán case. Comm. Partial Differ. Equ. 33 (2008) 1018–1032. [44] S. Neukamm and I. Velčić, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity. Math. Models Methods Appl. Sci. 23 (2013) 2701–2748. [45] H. Paetzelt, V. Gottschalch, J. Bauer, H. Herrnberger and G. Wagner. Fabrication of III–V nano- and microtubes using MOVPE grown materials. Physica Status Solidi (A) 203 (2006) 817–824. [46] M. R. Pakzad, On the Sobolev space of isometric immersions. J. Differ. Geom. 66 (2004) 47–69. [47] V. Y. Prinz, D. Grützmacher, A. Beyer, C. David, B. Ketterer and E. Deckardt, A new technique for fabricating three-dimensional micro- and nanostructures of various shapes. Nanotechnology 12 (2001) 399–402. [48] J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press (2003). [49] N. Salamon and C. B. Masters, Bifurcation in isotropic thinfilm/substrate plates. Special topics in the theory of elastic: A volume in honour of Professor John Dundurs. Int. J. Solids Struct. 32 (1995) 473–481. [50] B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Model. Simul. 5 (2006) 664–694. [51] B. Schmidt, Minimal energy configurations of strained multi-layers. Cal. Var. Partial Diff. Equ. 30 (2007) 477–497. [52] B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy. J. Math. Pures Appl. 88 (2007) 107–122. [53] B. Schmidt, A Griffith-Euler-Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics. Math. Models Methods Appl. Sci. 27 (2017) 1685–1726. [54] O. G. Schmidt and K. Eberl, Thin solid films roll up into nanotubes. Nature 410 (2001) 168. [55] T. Von Kármán Festigkeitsprobleme im Maschinenbau, in Encyclopädie der Mathematischen Wissenschaften, Vol. IV/4. Teubner, Leipzig (1910) 311–385. COCV_2021__27_S1_A18_082e7c061-0ca1-4040-9965-9103aa33211e cocv190126 10.1051/cocv/202007010.1051/cocv/2020070 Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations* Li Juan 1 Li Wenqiang 2 0000-0001-8564-9105 Wei Qingmeng 3** 1 School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, P.R. China. 2 School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China. 3 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China. **Corresponding author: [e-mail] 01 03 2021 01 03 2021 2021 cocv/2021/01 Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science OuverteSupplementS17 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF) Full (DJVU)

By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function initially defined only for deterministic times t and states x to stopping times τ and random variables $$. The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case.

Strong dynamic programming principle coupled FBSDEs with jumps stochastic differential games HJBI equations 49N70 49L25 60H10 93E20 idline ESAIM: COCV 27 (2021) S17 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S17 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020070 www.esaim-cocv.org PROBABILISTIC INTERPRETATION OF A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS Juan Li1 , Wenqiang Li2 and Qingmeng Wei3,** Abstract. By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function initially defined only for deterministic times t and states x to stopping times and random variables L2 (, F , P; R). The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case. Mathematics Subject Classification. 49N70, 49L25, 60H10, 93E20. Received July 28, 2019. Accepted October 19, 2020. 1. Introduction This paper is devoted to the study of a probabilistic interpretation of the following multi-dimensional coupled system of Hamilton-Jacobi-Bellman-Isaacs (HJBI, for short) equations: Wi t (t, x) + sup uU inf vV n bi t, x, Wi(t, x), DWi(t, x)i(t, x, Wi(t, x), u, v), u, v DWi(t, x) +1 2 tr i i (t, x, Wi(t, x), u, v)D2 Wi(t, x) +fi t, x, W(t, x), DWi(t, x)i(t, x, Wi(t, x), u, v), u, v o = 0, (t, x) [0, T) × R, i K, Wi(T, x) = gi(x), x R, i K, (1.1) The work has been supported in part on one hand by the NSF of P.R. China (No. 11871037, 11971099), National Key R and D Program of China (No. 2018YFA0703900), NSFC-RS (No. 11661130148, NA150344), on the other hand by the Natural Science Foundation of Shandong Province (ZR2017MA015), Doctoral Scientific Research Fund of YantaiUniversity (No. SX17B09), and by the Science and Technology Development Plan Project of Jilin Province (20190103026JH). Keywords and phrases: Strong dynamic programming principle, coupled FBSDEs with jumps, stochastic differential games, HJBI equations. 1 School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, P.R. China. 2 School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China. 3 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China. ** Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 J. LI ET AL. where T > 0 is an arbitrarily fixed finite time horizon, U and V are two compact metric spaces, k 2 is an integer, K = {1, 2, · · · k} and W(t, x) = (W1(t, x), W2(t, x), . . . , Wk(t, x)), (t, x) [0, T] × R. The precise assumptions on the functions bi, i, fi, gi, 1 i K, will be given in Section 3. Note that the system of k HJBI equations (1.1) is coupled through the k-dimensional solution W(t, x). The probabilistic interpretation for partial differential equations (PDEs, for short) has been investigated by many authors, but for such coupled system of PDEs has been studied by few authors. When both U and V are single point sets, Pardoux, Pradeilles and Rao [8] obtained a probabilistic interpretation for the system (1.1) with the help of backward stochastic differential equations (BSDEs, for short) associated to a diffusion jump process. When either U or V is a singleton, the stochastic representation for (1.1) was studied by Buckdahn, Hu [2] by a stochastic control problem. We emphasize that the functions bi, i, i K, in (1.1) also do not depend on the variables y and z in [2, 8], and PDE (1.1) is totally new. On the other hand, for the case k = 1, the system (1.1) is reduced to a generalized HJBI equation whose probabilistic interpretation has been obtained by Li, Wei [7] using the lower value function of a stochastic differential game problem. The reader is also referred to [3­5, 9, 10] for the probabilistic interpretation of a HJBI equation and the references therein. In this paper, we introduce a stochastic differential game problem on a Wiener-Poisson space in order to give the probabilistic interpretation for (1.1) in the general case. Let us be more precise: The dynamics of our stochastic differential games is given by the following coupled FBSDE with jumps dXt,,i;u,v s = bNt,i s (s, Xt,,i;u,v s , Y t,,i;u,v s , Zt,,i;u,v s , us, vs)ds + Nt,i s (s, Xt,,i;u,v s , Y t,,i;u,v s , us, vs)dBs, Xt,,i;u,v t = , dY t,,i;u,v s = - ~ fNt,i s (s, Xt,,i;u,v s , Y t,,i;u,v s , Ht,,i;u,v s , Zt,,i;u,v s , us, vs)ds + k-1 P l=1 Ht,,i;u,v s (l)ds +Zt,,i;u,v s dBs + k-1 P l=1 Ht,,i;u,v s (l)dNs(l), s [t, T], Y t,,i;u,v T = gNt,i T (Xt,,i;u,v T ), (1.2) where Nt,i is a K-valued Markov process which will be specified in the next section, and ~ fi is introduced to overcome the difficulties related with the coupling in (1.1) (i = 1, . . . , k). The relationship between ~ fi and fi, i K, will be given in Section 3. We study the game of the type "strategy against control", so the lower value function is defined as follows: Wi(t, x) := essinf Bt,T esssup uUt,T Y t,x,i;u,(u) t , i K, (t, x) [0, T] × R, (1.3) where Ut,T is the set of admissible controls of player I and Bt,T is the set of nonanticipative strategies for player II; for the precise definitions of Ut,T and Bt,T , see the Definitions 3.4 and 3.5, respectively. We first prove that Wi(t, x), 1 i K, satisfy some regularity properties: They are deterministic, Lipschitz continuous in x and 1 2 -Holder continuous in t. In order to establish the relation between the lower value function (1.3) and the coupled system (1.1), the crucial step is to get the dynamic programming principle (DPP, for short) for Wi(t, x), i K. However, the classical DPP is not sufficient anymore in our framework, since stopping times are involved in the transformation between ~ fNt,i s and ~ fi, i K, in the proof of the existence of a viscosity solution. Then we consider the DPP (Thm. 3.17) in strong sense, with the help of the notion of the backward stochastic semigroup introduced by Peng in [10]. It is worth mentioning that this strong DPP is far from being obvious and its proof is not a standard generalization of the classical DPP. The key step is to deduce an important equality WNt,i (, ) = essinf B,T esssup uU,T Y ,,Nt,i ;u,(u) , i K, A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 3 (see Prop. 3.16) which is the extension of the definition for the lower value function (which is originally defined only for deterministic times t and states x) to stopping times and random variables L2 (, F , P; R). This generalization needs a series of auxiliary results, for which we provide the details in Section 3. The DPP will allow to prove that the lower value function W(t, x) = (W1(t, x), W2(t, x), . . . , Wk(t, x)) defined by (1.3) is a viscosity solution of the system of HJBI equations (1.1). Moreover, the uniqueness of the viscosity solution is also proved when the coefficients i, i K, in (1.1)­(1.2) are independent of y. In addition, our stochastic differential game problem has the upper value function Ui(t, x) := esssup At,T essinf vVt,T Y t,x,i;(v),v t , i K (for the notations, see the Defs. 3.4 and 3.5) and it has properties similar to Wi(t, x), i K. To avoid repetitions, we only state the main results for Ui(t, x): The upper value function Ui(t, x), i K, of the stochastic differential game is the (unique) viscosity solution of the following coupled HJBI equation: Ui t (t, x) + inf vV sup uU n bi(t, x, Ui(t, x), DUi(t, x)i(t, x, Ui(t, x), u, v), u, v)DUi(t, x) +1 2 tr i i (t, x, Ui(t, x), u, v)D2 Ui(t, x) +fi(t, x, U(t, x), DUi(t, x)i(t, x, Ui(t, x), u, v), u, v) o = 0, Ui(T, x) = gi(x), (1.4) where U(t, x) = (U1(t, x), U2(t, x), . . . , Uk(t, x)), (t, x) [0, T] × R, i K = {1, 2, . . . k}, k 2. As a byproduct, we obtain that the stochastic differential game has a value under the well-known Isaacs' condition. Our paper is organized as follows. In Section 2, we introduce the underlying probability space and some notations, and we recall the preliminaries of BSDEs with jumps. Section 3 is devoted to the formulation of the stochastic differential games and the study of the properties of the lower value functions Wi, i K. Moreover, we show that they satisfy the strong dynamic programming principle. In Section 4, we present the details of the relationship between the lower value functions of the stochastic differential games and the coupled HJBI equations. In the appendix we give the detailed illustration on the comparison theorem for FBSDEs with jumps and the proof of Theorem 3.17, respectively. 2. Preliminaries 2.1. Some notations Let K = {1, 2, . . . , k}, where k 2 is a given integer. Let (, F, P) be the completed product of the probability spaces (1, F1, P1) and (2, F2, P2), where ­ (1, F1, P1) is a classical Wiener space, namely 1 = C0(R; Rd ) is the set of all continuous functions from R to Rd with value 0 at time 0, F1 is the completed Borel -field on 1, P1 is the Wiener measure such that the canonical processes Bs() = (s), s R+, 1 and B-s() = (-s), s R+, 1, are two independent d-dimensional Brownian motions. We denote by FB = (FB s )s0 the completed filtration generated by the Brownian motion B, i.e., FB s = {Br, r (-, s]} NP1 , where NP1 is the collection of null-sets of P1. ­ (2, F2, P2) is a Poisson space: A point function p : Dp R L is a map defined on a countable subset Dp of the real line R, where L = K - {k} is equipped with the -field L of all subsets of L; 2 is the set of all point functions p on L. The counting measure N on R × L at p 2 is defined as follows N(p, (s, t] × ) = ]{r Dp (s, t] : p(r) }, L, s, t R, s < t, where ] denotes the cardinal number of elements of the set. We identify the point function p with N(p, ·). The -field F2 is defined as the smallest one on 2 with respect to which the coordinate mapping p N(p, (s, t] × ), L, s, t R, s < t, is measurable. For fixed > 0, we consider the probability 4 J. LI ET AL. measure P2 on (2, F2) such that the coordinate measure N becomes a Poisson random measure with the compensator N((s, t] × {l}) = (t - s) k-1 P n=1 n(l) = (t - s), l L. Denoting by N((s, t] × {l}) st the Poisson martingale measure defined as (N - N)((s, t] × {l}) st = N((s, t] × {l}) - (t - s) st , for all l L, we recall that the processes N((s, t] × {l}) 0stT , 1 l k - 1, are independent. Above n(·) is the Dirac measure over L, that is, n(l) = 1, if l = n, and n(l) = 0, otherwise. By defining FN t := {N((s, r] × ) : - < s r t, L}, t 0, we get the filtration generated by the Poisson random measure N is (FN t )t0: FN t = ( T s>t FN s ) NP2 , t 0. Finally, put = 1 × 2, P = P1 P2, F = (F1 F2) NP , where F is completed with respect to P. We denote by F = (Ft)t0 the filtration generated by Brownian motion B and Poisson random measure N, and augmented by all P-null sets, i.e., Ft = (FB t FN t ) NP , t 0. For 0 t s T, i K and l L, we put Ns = N((0, s]×L), Ns(l) = N((0, s]×{l}) and Ns(l) = Ns(l)-s. We introduce a K-valued Markov process Nt,i s as Nt,i s = i + k-1 P l=1 lN((t, s] × {l}) mod(k), where (j)mod(k) is identified with j0 K such that j - j0 is a multiple of k, for any given natural j 1. Let T > 0 be a finite time horizon. We introduce the following spaces of processes: ­ S2 (t, T; R) := n | : × [t, T] R is an (Ft)t0-adapted cadlag process, and E[ sup s[t,T ] |s|2 ] < + o ; ­ M2 (t, T; Rd ) := n | : × [t, T] Rd is an (Ft)t0-predictable process, and E h Z T t |t|2 dt i < + o ; ­ [L2 (P L)]k-1 = n H|H = (H(1), · · · , H(k - 1)) : × [t, T] × L Rk-1 is P L-measurable1 and k H k[L2(PL)]k-1 = E h Z T t k-1 X l=1 H2 s (l)ds i1 2 < + o ; ­ [L2 (L; R)]k-1 = n H(·)|H(·) : L R o . Moreover, for a map H(·) : L R we introduce the norm k H k[L2(L;R)]k-1 = [ k-1 P l=1 H2 (l)] 1 2 . We set B2 [t, T] = S2 (t, T; R) × S2 (t, T; R) × [L2 (P L)]k-1 × M2 (t, T; Rd ). 2.2. BSDEs with jumps We consider the following BSDE with jumps: dYt = -g(t, Yt, Ht, Zt)ds + ZtdBt + k-1 P l=1 Ht(l)dNt(l), t [0, T], YT = , (2.1) where T > 0 is an arbitrary but fixed time horizon, and the coefficient g : ×[0, T]×R×[L2 (L; R)]k-1 ×Rd R is P-measurable, for every fixed (y, h, z) R × [L2 (L; R)]k-1 × Rd , and satisfies: (H1) (i) There exists a constant C 0 such that, P-a.s., for all t [0, T], y1, y2 R, z1, z2 Rd , h1, h2 [L2 (L; R)]k-1 , |g(t, y1, h1, z1) - g(t, y2, h2, z2)| C(|y1 - y2|+ k h1 - h2 k[L2(L;R)]k-1 +|z1 - z2|); 1P denotes the -field of (Ft)t0-predictable subsets of × [t, T]. A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 5 (ii) E h Z T 0 |g(s, 0, 0, 0)|ds 2i < +. Let us recall some well-known results. Lemma 2.1. Under the condition (H1), for any random variable L2 (, FT , P; R), the BSDE with jumps (2.1) has a unique solution (Yt, Ht, Zt)t[0,T ] S2 (0, T; R) × [L2 (P L)]k-1 × M2 (0, T; Rd ). For some g : × [0, T] × R × [L2 (L; R)]k-1 × Rd R satisfying (H1), and for i {1, 2}, suppose the drivers gi are of the form gi(s, Y i s , Hi s, Zi s) = g(s, Y i s , Hi s, Zi s) + i(s), dsdP-a.e. Denote by (Y 1 , H1 , Z1 ), (Y 2 , H2 , Z2 ) the solution of the BSDE with jumps with the data (1, g1) and (2, g2), respectively. We have the following classical estimate: Lemma 2.2. Under (H1), the difference of (Y 1 , H1 , Z1 ), (Y 2 , H2 , Z2 ) satisfies: |Y 1 t - Y 2 t |2 + 1 2 E h Z T t e(s-t) |Y 1 s - Y 2 s |2 + |Z1 s - Z2 s |2 + k-1 X l=1 |H1 s (l) - H2 s (l)|2 ds | Ft i E h e(T -t) |1 - 2|2 | Ft i + E h Z T t e(s-t) |1(s) - 2(s)|2 ds | Ft i , P-a.s., t [0, T], where 2 + 2C + 4C2 . For the details on the above results, please refer to Lemma 2.3 in Li, Wei [7]; see also Barles, Buckdahn and Pardoux [1]. 3. Stochastic differential games 3.1. Formulation First we introduce the definitions of the sets of admissible control processes. Define U := n u : [0, T] × U is (Ft)t0-progressively measurable process o , V := n v : [0, T] × V is (Ft)t0-progressively measurable process o , where the control state spaces U and V are supposed to be compact metric spaces. U (resp., V) is the set of admissible control processes for player I (resp., II). For every i K, we consider continuous maps bi : [0, T] × R × R × Rd × U × V R, i : [0, T] × R × R × U × V Rd , fi : [0, T] × R × Rk × Rd × U × V R, gi : R R. 6 J. LI ET AL. Here we adopt the definition ~ fp as [2, 8]. For every p K, we define ~ fp : [0, T]×R×R×Rk-1 ×Rd ×U ×V R by ~ fp(t, x, y, h, z, u, v) = fp(t, x, ap , z, u, v), where h = (h(1), . . . , h(k - 1)), and ap = (ap 1, . . . , ap k) are related by ap j = y + h(k - p + j), j < p, ap p = y, j = p, ap j = y + h(j - p), j > p. (3.1) The vector ap = (ap 1, · · · , ap k) given by the right-hand side of (3.1) will be denoted by ap [y, h]. It is easy to check that, for all (t, x) [0, T] × R, z Rd , u U, v V, f1(t, x, a, z, u, v) = ~ f1(t, x, a1, h1 , z, u, v), f2(t, x, a, z, u, v) = ~ f2(t, x, a2, h2 , z, u, v), . . . . . . fk(t, x, a, z, u, v) = ~ fk(t, x, ak, hk , z, u, v), where hp = (hp (1), . . . , hp (k - 1)) with hp (j) = a(p+j)mod(k) - ap = ap+j - ap, 1 j k - p, ap+j-k - ap, k - p + 1 j k - 1. We assume that for every i K, bi, i, fi, gi satisfy the following conditions: (B1) bi, i, gi are Lipschitz continuous in (x, y, z), uniformly with respect to (t, u, v); fi is Lipschitz continuous in (x, a, z), uniformly with respect to (t, u, v). (B2) For all t [0, T], u U, v V , (x, y, z), (x0 , y0 , z0 ) R × R × Rd , h, h0 [L2 (L; R)]k-1 , (i) (bi(t, x, y, z, u, v) - bi(t, x0 , y0 , z0 , u, v))(y - y0 ) + (i(t, x, y, u, v) - i(t, x0 , y0 , u, v))(z - z0 ) -(fi(t, x, ai [y, h], z, u, v) - fi(t, x0 , ai [y0 , h0 ], z0 , u, v))(x - x0 ) + k-1 P l=1 (h(l) - h0 (l))(x - x0 ) 6 -1|x - x0 |2 - 2|y - y0 |2 - 3|z - z0 |2 - 4 k-1 P l=1 |h(l) - h0 (l)|2 , (ii) (gi(x) - gi(x0 ))(x - x0 ) µ1|x - x0 |2 , where the relationships between h, y and ai [y, h], are presented in (3.1), similar to h0 , y0 and ai [y0 , h0 ]; and µ1, 1, 2, 3, 4 are nonnegative constants with 1 + 2 > 0, 1 + 3 > 0, 1 + 4 > 0, 2 + µ1 > 0, 3 + µ1 > 0, and 4 + µ1 > 0. (B3) For fixed > 0, there exists a constant K1 0 such that for all i, j K, j 6= i, (t, x, z) [0, T] × R × Rd , and u U, v V , a, a0 Rk with ap = a0 p, p 6= j, and aj a0 j, fi(t, x, a, z, u, v) - fi(t, x, a0 , z, u, v) K1(aj - a0 j). A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 7 For given admissible controls u U, v V, i K, and initial data (t, ) [0, T] × L2 (, Ft, P; R), let us consider the dynamics of our stochastic coupled controlled system dXt,,i;u,v s = bNt,i s (s, Xt,,i;u,v s , Y t,,i;u,v s , Zt,,i;u,v s , us, vs)ds +Nt,i s (s, Xt,,i;u,v s , Y t,,i;u,v s , us, vs)dBs, s [t, T], Xt,,i;u,v t = , dY t,,i;u,v s = - ~ fNt,i s (s, Xt,,i;u,v s , Y t,,i;u,v s , Ht,,i;u,v s , Zt,,i;u,v s , us, vs)ds + k-1 P l=1 Ht,,i;u,v s (l)ds + Zt,,i;u,v s dBs + k-1 P l=1 Ht,,i;u,v s (l)dNs(l), Y t,,i;u,v T = gNt,i T (Xt,,i;u,v T ). (3.2) Remark 3.1. Note that, the involved coefficient in BSDE (3.2) is ~ fp, not fp. From the definition of ~ fp and assumptions (B1)-(B3), it is easy to check that for every i K, (i) ~ fi is Lipschitz with respect to (x, y, h, z), uniformly with respect to (t, u, v); (ii) For all t [0, T], u U, v V , (x, y, h, z), (x0 , y0 , h0 , z0 ) R × R × [L2 (L; R)]k-1 × Rd , (bi(t, x, y, z, u, v) - bi(t, x0 , y0 , z0 , u, v))(y - y0 ) + (i(t, x, y, u, v) - i(t, x0 , y0 , u, v))(z - z0 ) -( ~ fi(t, x, y, h, z, u, v) - ~ fi(t, x0 , y0 , h0 , z0 , u, v))(x - x0 ) + k-1 P l=1 (h(l) - h0 (l))(x - x0 ) 6 -1|x - x0 |2 - 2|y - y0 |2 - 3|z - z0 |2 - 4 k-1 P l=1 |h(l) - h0 (l)|2 , where 1, 2, 3, 4 are the same constants as in (B2). (iii) (B3) is equivalent with the condition that, for any i, j K, j 6= i, t [0, T], (x, y, z) R × R × Rd , and u U, v V , h, h0 Rk-1 with hp = h0 p, p 6= j, and hj h0 j, ~ fi(t, x, y, h, z, u, v) - ~ fi(t, x, y, h0 , z, u, v) K1(hj - h0 j). Remark 3.2. The authors of [2] used the assumption K1 > -1. However, if K1 < 0, examples show that the comparison theorem does, in general, not hold, while for K1 0 the comparison result follows from the comparison theorem in [11] and a passage to the limit. We give more details in the Appendix A.1. Remark 3.3. In fact, FBSDE (3.2) is a special case of fully coupled FBSDEs with jumps considered in [6, 7]. Thus it follows from Lemma 2.1 in [6] that for every u U, v V, (3.2) has a unique solution (Xt,,i;u,v , Y t,,i;u,v , Ht,,i;u,v , Zt,,i;u,v ) B2 [t, T] under the assumptions (B1) and (B2). Moreover, standard estimates for coupled FBSDEs with jumps (see, Prop. 3.1 in [6]) show that, there exists some constant C > 0 such that, for all t [0, T], i K, u U, v V and , 0 L2 (, Ft, P; R), (i) E h sup tsT |Xt,,i;u,v s - Xt,0 ,i;u,v s |2 + sup tsT |Y t,,i;u,v s - Y t,02 ,i;u,v s |2 + Z T t |Zt,,i;u,v s - Zt,0 ,i;u,v s |2 ds + Z T t k-1 X l=1 |Ht,,i;u,v s (l) - Ht,0 ,i;u,v s (l)|2 ds | Ft i C| - 0 |2 , (ii) E h sup tsT |Xt,,i;u,v s |2 + sup tsT |Y t,,i;u,v s |2 + Z T t |Zt,,i;u,v s |2 ds + Z T t k-1 X l=1 |Ht,,i;u,v s (l)|2 ds | Ft i C(1 + ||2 ). (3.3) 8 J. LI ET AL. Given control processes u U, v V, and the initial data (t, x, i) [0, T] × R × K, the cost functional of our stochastic differential game is defined as J(t, x, i; u, v) := Y t,x,i;u,v t , (3.4) where the process Y t,x,i;u,v is the first component of the solution of FBSDE (3.2) with the initial condition (t, x, i). The relations (3.3) and (3.4) imply that, there exists a constant C > 0 such that for any x, x0 R, (i) |J(t, x, i; u, v) - J(t, x0 , i; u, v)| C|x - x0 |; (ii) |J(t, x, i; u, v)| C(1 + |x|), for all i K, u U, v V. (3.5) As a consequence, for all i K, t [0, T], L2 (, Ft, P; R), we also have J(t, , i; u, v) = J(t, x, i; u, v)|x= = Y t,,i;u,v t , P-a.s., (3.6) which is a classical result in stochastic control theory, see, for example, Theorem 3.1 in [6]. Now we introduce the following subspaces of admissible controls. Let 1, 2 be two stopping times such that t 1 < 2 T, P-a.s. We define the random interval [[1, 2]] := {(t, ) [0, T] × , 1() t 2()}. Definition 3.4. Let u0 U, v0 V . A process u = {ur(), (r, ) [[1, 2]]} (resp., v = {vr(), (r, ) [[1, 2]]}) is an admissible control for player I (resp., II) on [[1, 2]], if uI[[1,2]] + u0 I[[0,T ]]\[[1,2]] (resp. vI[[1,2]] + v0 I[[0,T ]]\[[1,2]]) is (Fr)-progressively measurable and with values in U (resp., V ). The set of all admissible controls for player I (resp., II) on [[1, 2]] is denoted by U1,2 (resp., V1,2 ) . We identify two pro- cesses u and u in U1,2 and write u u on [[1, 2]], if P{u = u a.e. in [[1, 2]]} = 1. Similarly we interpret v = v on [[1, 2]] in V1,2 . The nonanticipative strategies for the game are defined as follows: Definition 3.5. A nonanticipative strategy for player I on [[1, 2]] is a mapping : V1,2 U1,2 such that, for all F-stopping time S : [[1, 2]] and all v1, v2 V1,2 , with v1 v2 on [[1, S]], it holds that (v1) (v2) on [[1, S]]. Nonanticipative strategies for player II on [[1, 2]], : U1,2 V1,2 , are defined similarly. The set of all nonanticipative strategies : V1,2 U1,2 for player I on [[1, 2]] is denoted by A1,2 , while the set of all nonanticipative strategies : U1,2 V1,2 for player II on [[1, 2]] is denoted by B1,2 . The lower and upper value functions of our stochastic differential game are defined respectively as follows: (lower) Wi(t, x) := essinf Bt,T esssup uUt,T J(t, x, i; u, (u)), (upper) Ui(t, x) := esssup At,T essinf vVt,T J(t, x, i; (v), v), (t, x) [0, T] × R, i K. (3.7) Now we intend to establish the relationship between FBSDEs (3.2) and PDEs (1.1) via the lower value function Wi, i K. In the last section , we will explain which system of HJBI equations the upper value function Ui, i K will be related with. As the essential infimum and the essential supremum of the Ft-measurable cost functional J(t, x, i; u, (u)) over a family of control processes and strategies, Wi(t, x) is an Ft-measurable random variable. But it turns out to be deterministic. From now on, we only present the properties of the lower value function, the upper value function Ui can be analyzed in the same manner. Proposition 3.6. For any (t, x, i) [0, T]×R×K, the lower value function Wi(t, x) is a deterministic function: E[Wi(t, x)] = Wi(t, x), P-a.s. Proof. This result is a direct consequence of Proposition 3.1 in Li and Wei [7]; see also Buckdahn, Hu and Li [3]. A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 9 From the definition of the lower value function combined with (3.5), we see that the following results hold true. Lemma 3.7. There exists a constant C > 0 such that, for all i K, x, x0 R, t [0, T], (i) |Wi(t, x) - Wi(t, x0 )| C|x - x0 |; (ii) |Wi(t, x)| C(1 + |x|). (3.8) Lemma 3.8. Suppose that (B1) and (B2) hold true. Then, for all u U, v V, the cost functional J(t, x, i; u, v) is monotonic in the following sense: for every i K, x, x R, t [0, T], J(t, x, i; u, v) - J(t, x, i; u, v) x - x 0, P-a.s. Furthermore, the lower value function Wi(t, x) is monotonic: Wi(t, x) - Wi(t, x) x - x 0, i K, t [0, T], x, x R. For the proof, please refer to Lemma 3.4 in Li and Wei [7]. 3.2. Dynamic Programming Principle To get the dynamic programming principle (DPP, for short), we adapt the notion of the stochastic backward semigroup which was first introduced by Peng [10] to our problem. Given (t, x, i) [0, T] × R × K, u U, v V, for all stopping times , such that t T, P-a.s., and L2 (, F , P; R), we define Gt,x,i;u,v , [] := Y t,x,i;u,v , (3.9) where (Xt,x,i;u,v s , Y t,x,i;u,v s , Ht,x,i;u,v s , Zt,x,i;u,v s )s[t,] is the solution of the following FBSDE with the random time horizon : dXt,x,i;u,v s = bNt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , Zt,x,i;u,v s , us, vs)ds +Nt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , us, vs)dBs, Xt,x,i;u,v t = x, dY t,x,i;u,v s = - ~ fNt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , Ht,x,i;u,v s , Zt,x,i;u,v s , us, vs)ds + k-1 P l=1 Ht,x,i;u,v s (l)ds + Zt,x,i;u,v s dBs + k-1 P l=1 Ht,x,i;u,v s (l)dNs(l), s [t, ], Y t,x,i;u,v = . (3.10) Remark 3.9. (i) By Lemma 2.5 in [7] equation (3.10) has a unique solution (Xt,x,i;u,v s , Y t,x,i;u,v s , Ht,x,i;u,v s , Zt,x,i;u,v s )s[t,] under the assumptions (B1) and (B2)-(i). (ii) When the terminal Y t,x,i;u,v also depends on the solution of the forward SDE, i.e., Y t,x,i;u,v = (, Xt,x,i;u,v ), where : × R R is F B(R)-measurable, linear growth and Lipschitz in x, then, by Theorem 3.2 in [6], there exists some 0 > 0 independent of (t, x) and (u, v), such that for any 0 0, (3.10) has a unique solution on the small interval [t, t + ]. Then we will consider t t + , and = (, Xt,x,i;u,v ), and have Y t,x,i;u,v = Gt,x,i;u,v , [(, Xt,x,i;u,v )], where Y t,x,i;u,v s is the first component of the solution of (3.10) with the terminal (, Xt,x,i;u,v ) at the random time horizon . 10 J. LI ET AL. For the solution (Xt,x,i;u,v , Y t,x,i;u,v , Ht,x,i;u,v , Zt,x,i;u,v ) of FBSDEs (3.2) with = x, it is easy to check that Gt,x,i;u,v t, [Y t,x,i;u,v ] = Gt,x,i;u,v t,T [gNt,i T (Xt,x,i;u,v T )] = Y t,x,i;u,v t = J(t, x, i; u, v). (3.11) With the help of the notion of the backward semigroup, we have the following classical DPP which is the weak one in our paper. Theorem 3.10. (Weak-DPP) Suppose (B1), (B2) and (B3). There exists some 0 > 0 small enough such that, for any 0 0, t [0, T - ], x R, i K, we have Wi(t, x) = essinf Bt,t+ esssup uUt,t+ G t,x,i;u,(u) t,t+ [WNt,i t+ (t + , X t,x,i;u,(u) t+ )], (3.12) where (Xt,x,i;u,v , Y t,x,i;u,v , Zt,x,i;u,v , Ht,x,i;u,v ) is the solution of the following coupled FBSDE with jumps on [t, t + ]: dXt,x,i;u,v s = bNt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , Zt,x,i;u,v s , us, vs)ds +Nt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , us, vs)dBs, s [t, t + ], Xt,x,i;u,v t = x, dY t,x,i;u,v s = - ~ fNt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , Ht,x,i;u,v s , Zt,x,i;u,v s , us, vs)ds + k-1 P l=1 Ht,x,i;u,v s (l)ds + Zt,x,i;u,v s dBs + k-1 P l=1 Ht,x,i;u,v s (l)dNs(l), Y t,x,i;u,v t+ = WNt,i t+ (t + , Xt,x,i;u,v t+ ). (3.13) Its proof uses the same arguments as those in the proof of Theorem 3.1 in [7] and therefore is omitted. We will provide the proof of the strong DPP, which generalizes Theorem 3.10. Remark 3.11. From Lemma 3.8 it follows that, for all i K, t [0, T], x, x R, Wi(t, x)-Wi(t, x) x-x 0, which doesn't mean (B2)-(ii). However, Lemma 3.7 tells us that, for all i K, t [0, T], Wi(t, x) is Lipschitz in x. Therefore, from Remark 3.9-(ii), the stochastic backward semigroup in (3.12) makes sense. Next we recall the following continuity property of the lower value function Wi(t, x) in t, which is also a consequence of Theorem 3.2 in Li, Wei [7]. Proposition 3.12. Assume (B1), (B2) and (B3). Then, there exists a constant C such that, for every x R, i K, t, t0 [0, T], |Wi(t, x) - Wi(t0 , x)| C(1 + |x|)|t - t0 | 1 2 . That means, Wi(t, x) is 1 2 -Holder continuous in t. To prepare the strong DPP, we need the following auxiliary results. Lemma 3.13. For any stopping time : t T, we have essinf Bt,T esssup uUt,T Y ,x,i;u,(u) = essinf B,T esssup uU,T Y ,x,i;u,(u) , P-a.s. (3.14) Proof. Here, we briefly present the details for (3.14) in two steps. A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 11 (i) The first one is to prove essinf B,T esssup uU,T Y ,x,i;u,(u) essinf Bt,T esssup uUt,T Y ,x,i;u,(u) , P-a.s. (3.15) For any arbitrarily fixed Bt,T , given any u0 Ut, , we define the restriction u0 1 of to B,T : u0 1 (u1) = (u0 u1)|[,T ], u1 U,T , where u := u0 u1 = u0I[t,) + u1I[,T ] Ut,T . It is easy to check that, according to Definition 3.5, u0 1 B,T . Note that Y ,x,i;u,(u) denotes the state of the solution Y of (3.2) with initial data (, x, i) at . Combining this with the definitions of u and u0 1 , we get Y ,x,i;u1, u0 1 (u1) = Y ,x,i;u,(u) . (3.16) Thus, for all Bt,T , essinf 0B,T esssup u1U,T Y ,x,i;u1,0 (u1) esssup u1U,T Y ,x,i;u1, u0 1 (u1) = esssup u1U,T Y ,x,i;u0u1,(u0u1) esssup uUt,T Y ,x,i;u,(u) , which yields essinf 0B,T esssup u1U,T Y ,x,i;u1,0 (u1) essinf Bt,T esssup uUt,T Y ,x,i;u,(u) . Hence, (3.15) holds true. (ii) We prove now the converse relation essinf B,T esssup uU,T Y ,x,i;u,(u) essinf Bt,T esssup uUt,T Y ,x,i;u,(u) , P-a.s. (3.17) For any > 0, there exists a 1 B,T such that essinf 1B,T esssup u1U,T Y ,x,i;u1,1(u1) esssup u1U,T Y ,x,i;u1, 1 (u1) - , P-a.s., for the proof it is standard now, see, for instance, Remark 3.7 in [7]. Given an arbitrary 0 Bt, , for any u Ut,T , define u1 as the restriction of u to [, T], i.e., u1 = u|[,T ], and as the extension of 1 to [t, T]: (u) := 0(u|[t,]) 1(u1). Obviously, Bt,T . Similar to (3.16), we also have Y ,x,i,u1, 1 (u1) = Y ,x,i;u, (u) . Therefore, essinf 1B,T esssup u1U,T Y ,x,i;u1,1(u1) esssup u1U,T Y ,x,i;u1, 1 (u1) - = esssup uUt,T Y ,x,i;u, (u) - essinf Bt,T esssup uUt,T Y ,x,i;u,(u) - , P-a.s. From the arbitrariness of , we get (3.17). Combining the above two steps, we complete the proof. 12 J. LI ET AL. For any t [0, T], let be a stopping time such that t T. For any positive integer N, let tN j = t+ (T -t)j N , 0 j N, tN -1 := 0. We define N := N X j=0 tN j I{tN j-1<tN j }, (3.18) then N is {}-measurable stopping time, and N , as N . Lemma 3.14. For the stopping time N in (3.18) and L2 (, F , P; R), we have Wi(N , ) = essinf Bt,T esssup uUt,T E[Y N ,,i;u,(u) N | F ], P-a.s. Proof. For L2 (, F , P; R), (u, v) Ut,T × Vt,T , we know (X,,i;u,v , Y ,,i;u,v , H,,i;u,v , Z,,i;u,v ) is the unique solution of FBSDE (3.2) with instead of t. From the uniqueness of solution (XN ,x,i;u,v , Y N ,x,i;u,v , HN ,x,i;u,v , ZN ,x,i;u,v ) and the fact that also N P j=0 (XtN j ,x,i;u,v , Y tN j ,x,i;u,v , HtN j ,x,i;u,v , ZtN j ,x,i;u,v )I{N =tN j } solves FBSDE (3.2) with N instead of t, it follows that (XN ,x,i;u,v , Y N ,x,i;u,v , HN ,x,i;u,v , ZN ,x,i;u,v ) = N X j=0 (XtN j ,x,i;u,v , Y tN j ,x,i;u,v , HtN j ,x,i;u,v , ZtN j ,x,i;u,v )I{N =tN j }, and in particular, Y N ,x,i;u,v N = N X j=0 Y tN j ,x,i;u,v tN j I{N =tN j } = N X j=0 J(tN j , x, i; u, v)I{N =tN j }, P-a.s. Hence, Wi(N , x) = N X j=0 Wi(tN j , x)I{N =tN j } = N X j=0 I{N =tN j } essinf BtN j ,T esssup uUtN j ,T J(tN j , x, i; u, (u)) = essinf Bt,T esssup uUt,T Y N ,x,i;u,(u) N , P-a.s., where the latter equality follows from standard arguments for the essential infimum and the essential supermum of families of FN -measurable random variables as well as Lemma 3.13. Again, from the above relation and from standard arguments (see, for example, Rem. 3.7 in [7]) it follows that for fixed i: (i) For any > 0, any Bt,T , there exists some u Ut,T , such that Wi(N , x) Y N ,x,i;u ,(u ) N + , P-a.s. (ii) For any > 0, there exists some Bt,T , such that for all u Ut,T , Wi(N , x) Y N ,x,i;u, (u) N - , P-a.s. A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 13 As Wi(N , x) is F -measurable (Wi is continuous, and hence Borel measurable, and N is {}-measurable), we have (i) For any > 0, and any Bt,T , there exists some u Ut,T , such that Wi(N , x) E[Y N ,x,i;u ,(u ) N | F ] + , P-a.s. (ii) For any > 0, there exists some Bt,T , such that for all u Ut,T , Wi(N , x) E[Y N ,x,i;u, (u) N | F ] - , P-a.s. Hence, Wi(N , x) = essinf Bt,T esssup uUt,T E[Y N ,x,i;u,(u) N | F ], P-a.s. Finally, (3.5) and (3.8) combined with a standard approximation argument for L2 (, F , P; R) allow to show Wi(N , ) = Wi(N , x)|x= = essinf Bt,T esssup uUt,T E[Y N ,,i;u,(u) N | F ], P-a.s. Next, we have the following estimate. Lemma 3.15. For all stopping time (t T), L2 (, F , P; R), i K, (u, v) Ut,T × Vt,T , we have E[Y N ,,i;u,v N | F ] - Y ,,i;u,v 2 C(N - )(1 + ||2 ), P-a.s., N 1. Proof. First of all, we have E[Y N ,,i;u,v N | F ] - Y ,,i;u,v E[Y N ,,i;u,v N - Y ,,i;u,v N | F ] + E[Y ,,i;u,v N | F ] - Y ,,i;u,v . (3.19) Now we work on the two terms at the right side of the above inequality. To simplify the notations, we omit the indication of the controls u, v in the following proof. For the forward equation, from (3.3)­(ii) (extended to stopping times), we get E[| - X,,i N |2 | F ] = E h Z N bN,i s (s, X,,i s , Y ,,i s , Z,,i s )ds + Z N N,i s (s, X,,i s , Y ,,i s )dBs 2 | F i C(1 + ||2 )|N - |. (3.20) For the backward equations with the different data (, , i) and (N , , i), we apply Lemma 2.2 by setting 1 := gN,i T (X,,i T ), 2 := gN N ,i T (XN ,,i T ), 1(s) := 0, 2(s) := ~ fN N ,i s (s, XN ,,i s , Y N ,,i s , HN ,,i s , ZN ,,i s ) - ~ fN,i s (s, X,,i s , Y N ,,i s , HN ,,i s , ZN ,,i s ), s [N , T], 14 J. LI ET AL. then we get E h |Y N ,,i N - Y ,,i N |2 | F i CE h |gN N ,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i + CE h Z T N |2(s)|2 ds | F i CE h |gN N ,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i + CE h Z T N |XN ,,i s - X,,i s |2 ds | F i + CE h Z T N |If (s)|2 ds | F i , (3.21) where If (s) = ~ fN N ,i s (s, XN ,,i s , Y N ,,i s , HN ,,i s , ZN ,,i s ) - ~ fN,i s (s, XN ,,i s , Y N ,,i s , HN ,,i s , ZN ,,i s ). Firstly, we consider E h |gN N ,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i . Obviously, E h |gN N ,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i CE h 1{N,i N =i}|gN N ,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i + CE h 1{N,i N 6=i}|gN N ,i T (XN ,,i T ) - gN,i T (XN ,,i T )|2 | F i + CE h 1{N,i N 6=i}|gN,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i = I1 + I2 + I3, where I1 = CE h 1{N,i N =i}|gN N ,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i , I2 = CE h 1{N,i N 6=i}|gN N ,i T (XN ,,i T ) - gN,i T (XN ,,i T )|2 | F i , I3 = CE h 1{N,i N 6=i}|gN,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i . For I1, on {N,i N = i}, we know, NN ,i T = N N ,N,i N T = N,i T , and from the uniqueness of the solution we get X,,i T = X N ,X,,i N ,N,i N T = X N ,X,,i N ,i T , P-a.s. Hence, from (3.3)­(i) and (3.20), we know I1 = CE h 1{N,i N =i}E |gN N ,i T (XN ,,i T ) - gN,i T (X N ,X,,i N ,i T )|2 | FN | F i CE h 1{N,i N =i}| - X,,i N |2 | F i C(1 + ||2 )|N - |. A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 15 For I2, from the linear growth property of gi, i K and (3.3)­(ii), I2 = CE h 1{N,i N 6=i}E |gN N ,i T (XN ,,i T ) - gN,i T (XN ,,i T )|2 | FN | F i CE h 1{N,i N 6=i}E (1 + |XN ,,i T |2 ) | FN | F i CE h 1{N,i N 6=i} | F i (1 + ||2 ) = CP{N,i N 6= i}(1 + ||2 ) = C(1 - e-(k-1)(N -) )(1 + ||2 ) C(1 + ||2 )|N - |. For I3, from the Lipschitz property of gi, i K, (3.3)­(ii), and (3.20), I3 = CE h 1{N,i N 6=i}E |gN,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | FN | F i CE h 1{N,i N 6=i}E |XN ,,i T - X N ,X,,i N ,N,i N T |2 | FN | F i CE h 1{N,i N 6=i}(1 + ||2 + |X,,i N |2 ) | F i C(1 + ||2 )|N - | + CE h 1{N,i N 6=i}|X,,i N |2 | F i C(1 + ||2 )|N - | + CE h 1{N,i N 6=i}|X,,i N - |2 | F i + CE h 1{N,i N 6=i}||2 | F i C(1 + ||2 )|N - |. Therefore, we get E h |gN N ,i T (XN ,,i T ) - gN,i T (X,,i T )|2 | F i C(1 + ||2 )|N - |. (3.22) Now we focus on E h R T N |XN ,,i s - X,,i s |2 ds | F i . Similarly, from (3.3), (3.20) and the proof for I3, we have E h Z T N |XN ,,i s - X,,i s |2 ds | F i TE h sup s[N ,T ] |XN ,,i s - X,,i s |2 | F i =TE h 1{N,i N =i}E sup s[N ,T ] |XN ,,i s - X N ,X,,i N ,i s |2 | FN | F i + TE h 1{N,i N 6=i}E sup s[N ,T ] |XN ,,i s - X N ,X,,i N ,N,i N s |2 | FN | F i TE h 1{N,i N =i}|X,,i N - |2 | F i + TE h 1{N,i N 6=i} 1 + ||2 + |X,,i N |2 | F i C(N - )(1 + ||2 ). (3.23) 16 J. LI ET AL. Combined with (3.21), (3.22) and (3.23), using (3.3)­(ii) and the proof for I2, we have E h |Y N ,,i N - Y ,,i N |2 | F i C(1 + ||2 )|N - | + CE h Z T N |If (s)|2 ds | F i C(1 + ||2 )|N - | + CE h 1{N,i N 6=i}E Z T N 1 + |XN ,,i s |2 + |Y N ,,i s |2 + |ZN ,,i s |2 + k-1 X l=1 |HN ,,i s (l)|2 ds | FN | F i C(1 + ||2 )|N - | + C(1 + ||2 )E h 1{N,i N 6=i} | F i C(1 + ||2 )|N - |, (3.24) where we have used the linear growth properties of ~ fi, i K. On the other hand, as N is {}-measurable and still from (3.3)­(ii) we know |E[Y ,,i N | F ] - Y ,,i | = E h Z N ~ fN,i s (s, X,,i s , Y ,,i s , H,,i s , Z,,i s )ds - Z N k-1 X l=1 H,,i s (l)ds F i C(N - ) 1 2 E h Z N 1 + |X,,i s |2 + |Y ,,i s |2 + k-1 X l=1 |H,,i s (l)|2 + |Z,,i s |2 ds | F i1 2 C(N - ) 1 2 (1 + ||), P-a.s. (3.25) Therefore, from (3.19), (3.24) and (3.25), we get |E[Y N ,,i N | F ] - Y ,,i |2 C(N - )(1 + ||2 ), P-a.s., N 1. Based on the above results, we generalize the results for the lower value function Wi(t, x), (t, x) [0, T] × R, i K to the case for stopping times and random variables L2 (, F , P; R) Proposition 3.16. For every fixed initial data (t, i) [0, T] × K, every stopping time with values in [t, T], and all L2 (, F , P; R), WNt,i (, ) = essinf B,T esssup uU,T Y ,,Nt,i ;u,(u) , P-a.s. Proof. As Lemma 3.15 implies that essinf Bt,T esssup uUt,T E[Y N ,,i;u,(u) N | F ] - essinf Bt,T esssup uUt,T Y ,,i;u,(u) 2 C(N - )(1 + ||2 ) C T - t N (1 + ||2 ) 0, as N , P-a.s., A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 17 and Wi is continuous, we get from Lemma 3.14 that Wi(, ) = essinf Bt,T esssup uUt,T Y ,,i;u,(u) , P-a.s., i K. As K is finite, and Nt,i is F -measurable, it follows that from Lemma 3.13 WNt,i (, ) = essinf Bt,T esssup uUt,T Y ,,Nt,i ;u,(u) = essinf B,T esssup uU,T Y ,,Nt,i ;u,(u) , P-a.s., i K. The following result is the strong DPP which plays a crucial role in the proof of the existence of a viscosity solution. Theorem 3.17. (Strong-DPP) Suppose (B1), (B2) and (B3). There exists some 0 > 0 small enough such that, for all t [0, T - 0], and all stopping time with 0 t t + 0 T, x R, i K, we have Wi(t, x) = essinf Bt, esssup uUt, G t,x,i;u,(u) t, [WNt,i (, Xt,x,i;u,(u) )], P-a.s., (3.26) where (Xt,x,i;u,v , Y t,x,i;u,v , Zt,x,i;u,v , Ht,x,i;u,v ) is the solution of the following FBSDE with jumps on [t, ]: dXt,x,i;u,v s = bNt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , Zt,x,i;u,v s , us, vs)ds +Nt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , us, vs)dBs, s [t, ], Xt,x,i;u,v t = x, dY t,x,i;u,v s = - ~ fNt,i s (s, Xt,x,i;u,v s , Y t,x,i;u,v s , Ht,x,i;u,v s , Zt,x,i;u,v s , us, vs)ds + k-1 P l=1 Ht,x,i;u,v s (l)ds + Zt,x,i;u,v s dBs + k-1 P l=1 Ht,x,i;u,v s (l)dNs(l), Y t,x,i;u,v = WNt,i (, Xt,x,i;u,v ). (3.27) We postpone the proof to the appendix. 4. Viscosity solution 4.1. Existence In this subsection, we take advantage of the above results to prove that the value function W(t, x) = (W1(t, x), W2(t, x), · · · , Wk(t, x)), (t, x) [0, T] × R, is a viscosity solution of the following system of coupled HJBI equations: ( Wi t (t, x) + sup uU inf vV {Li u,vWi(t, x) + fi(t, x, W(t, x), DWi(t, x)i(t, x, Wi(t, x), u, v), u, v)} = 0, Wi(T, x) = gi(x), (t, x, i) [0, T) × R × K, (4.1) where for each t [0, T), i K, C2 ([0, T] × R), the differential operator is Li u,v(t, x) = 1 2 tr i i (t, x, (t, x), u, v)D2 (t, x) +bi(t, x, (t, x), D(t, x)i(t, x, (t, x), u, v), u, v)D(t, x). First we present the definition of viscosity solution for HJBI equation (4.1). 18 J. LI ET AL. Definition 4.1. Let W = (W1, W2, · · · , Wk) belong to C([0, T] × R; Rk ). The function W is said to be (i) a viscosity subsolution of the system (4.1) if Wi(T, x) gi(x), i K, x R, and for all i K, C3 l,b([0, T] × R), for all (t, x) [0, T) × R such that (t, x) is a local maximum point of Wi - , we have t (t, x) + sup uU inf vV n Li u,v(t, x) + fi(t, x, W(t, x), D(t, x)i(t, x, Wi(t, x), u, v), u, v) o 0; (ii) a viscosity supersolution of the system (4.1) if Wi(T, x) gi(x), i K, x R, and for all i K, C3 l,b([0, T] × R), for all (t, x) [0, T) × R such that (t, x) is a local minimum point of Wi - , we have t (t, x) + sup uU inf vV n Li u,v(t, x) + fi(t, x, W(t, x), D(t, x)i(t, x, Wi(t, x), u, v), u, v) o 0; (iii) a viscosity solution of the system (4.1) if W is both a viscosity subsolution and a viscosity supersolution of (4.1). Here C3 l,b([0, T] × R) denotes the set of the real-valued functions that are continuously differentiable up to the third order and whose derivatives of order 1 to 3 are bounded. The following is the main result. Theorem 4.2. Assume that (B1)-(B3) hold. Moreover, suppose that 0 < < 1 (k-1)T . Then the function W is a viscosity solution of the system (4.1) of coupled HJBI equations. We begin with proving some auxiliary results first in order to prepare for the proof of Theorem 4.2. Let (t, x, i) [0, T) × R × K. For any given > 0 with t + T, the stopping time is defined as = (t + ) inf{s t : Nt,i s 6= i}. We consider the following system: dXu,v s = bi(s, Xu,v s , Y u,v s , Zu,v s , us, vs)ds + i(s, Xu,v s , Y u,v s , us, vs)dBs, s [t, ], Xu,v t = x, dY u,v s = - ~ fi(s, Xu,v s , Y u,v s , Hu,v s , Zu,v s , us, vs)ds + k-1 P l=1 Hu,v s (l)ds +Zu,v s dBs + k-1 P l=1 Hu,v s (l)dNs(l), s [t, ], Y u,v = ( , Xu,v )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , Xu,v )1{Nt,i =(l+i)mod(k)} , (4.2) where C3 l,b([0, T] × R). From the wellposedness and regularity results of FBSDEs (see the Thms. 3.2 and 3.4 in [6]), under the assumption (B1), we know there exists some 0 > 0 only depending on the Lipschitz constants of the coefficients, such that for all (0, 0], (4.2) has a unique solution (Xu,v , Y u,v , Hu,v , Zu,v ) B2 [t, ]; A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 19 and for all p 2, there is some constant Cp > 0 such that for all 0, 0 , (i) E h sup ts |Xu,v s |p + sup ts |Y u,v s |p + Z t |Zu,v s |2 ds p 2 + Z t k-1 X l=1 |Hu,v s (l)|2 ds p 2 | Ft i Cp(1 + |x|p ), P-a.s.; (ii) E h sup ts |Xu,v s - x|p | Ft i Cp(1 + |x|p ), P-a.s.; (iii) E h Z t |Zu,v s |2 ds p 2 + Z t k-1 X l=1 |Hu,v s (l)|2 ds p 2 | Ft i Cp p 2 (1 + |x|p ), P-a.s. (4.3) Now we define Y 1,i,u,v s := (s, Xu,v s ) + k-1 X l=1 Z s t W(l+i)mod(k) (r, Xu,v r ) - (r, Xu,v r ) dNr(l), s [t, ]. (4.4) Lemma 4.3. Let the process (Y 1,i,u,v s )s[t,] be defined as the first component of the solution (Y 1,i,u,v , Z1,i,u,v , H1,i,u,v ) of the following BSDE with jumps: dY 1,i,u,v s = -F(s, Xu,v s , Y 1,i,u,v s , H1,i,u,v s , Z1,i,u,v s , us, vs)ds + k-1 P l=1 H1,i,u,v s (l)ds +Z1,i,u,v s dBs + k-1 P l=1 H1,i,u,v s (l)dNs(l), s [t, ], Y 1,i,u,v = 0, (4.5) where we have used the notations L(s, x, y, h, z, u, v) = s (s, x) + 1 2 tr i i (s, x, y, u, v)D2 (s, x) + bi(s, x, y, z, u, v)D(s, x) + ~ fi(s, x, y, h, z, u, v), F(s, x, y, h, z, u, v) = L(s, x, y + (s, x), h + W(i+·)mod(k)(s, x) - (s, x)1, z + D(s, x)i(s, x, y + (s, x), u, v), u, v), and h + W(i+·)mod(k) (s, x) - (s, x)1 = (hj + W(i+j)mod(k) (s, x) - (s, x))1jk-1. Then, for all s [t, T], Y 1,i,u,v s = Gt,x,i;u,v s, h ( , Xu,v )1{Nt,i =i} + k-1 X l=1 W(l+i)mod(k) ( , Xu,v )1{Nt,i =(l+i)mod(k)} i - Y 1,i,u,v s . Remark 4.4. There exists some constant C > 0 such that for all (x, y, h, z), (x0 , y0 , h0 , z0 ) R × R × Rk-1 × Rd , for any s [t, T], u U, v V, (i) |L(s, x, y, h, z, u, v) - L(s, x0 , y0 , h0 , z0 , u, v)| C(1 + |x| + |x0 | + |y| + |y0 |)(|x - x0 | + |y - y0 |) + C(|h - h0 | + |z - z0 |), (ii) |L(s, x, y, h, z, u, v)| C(1 + |x|2 + |y|2 + |h| + |z|), (iii) |F(s, x, y, h, z, u, v)| C(1 + |x|2 + |y|2 + |h| + |z|). 20 J. LI ET AL. It is easy to check that F(s, x, y, h, z, u, v) is Lipschitz in (h, z), uniformly with respect to (s, x, y, u, v), and |F(s, x, 0, 0, 0, u, v)| C(1 + |x|2 ), (s, x, u, v) [0, T] × R × U × V for some constant C 0. Proof of Lemma 4.3 Let (Y u,v , Hu,v , Zu,v ) denote the solution of the BSDE in (4.2). From the definition of the backward semigroup we have, Y u,v s = Gt,x,i;u,v s, h ( , Xu,v )1{Nt,i =i} + k-1 X l=1 W(l+i)mod(k) ( , Xu,v )1{Nt,i =(l+i)mod(k)} i . On the other hand, Y 1,i,u,v = ( , Xu,v ) + k-1 P l=1 W(l+i)mod(k) ( , Xu,v ) - ( , Xu,v ) N (l) = ( , Xu,v )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , Xu,v )1{Nt,i =(l+i)mod(k)} = Y u,v , (see (4.2)). (4.6) Moreover, applying Ito's formula to Y 1,i,u,v s - Y u,v s and using (4.2), we get d(Y 1,i,u,v s - Y u,v s ) = h s (s, Xu,v s ) + 1 2 tr i i (s, Xu,v s , Y u,v s , us, vs)D2 (s, Xu,v s ) + bi(s, Xu,v s , Y u,v s , Zu,v s , us, vs)D(s, Xu,v s ) + ~ fi(s, Xu,v s , Y u,v s , Hu,v s , Zu,v s , us, vs) i ds - k-1 X l=1 Hu,v s (l) - W(l+i)mod(k) (s, Xu,v s ) + (s, Xu,v s ) ds - Zu,v s - D(s, Xu,v s )i(s, Xu,v s , Y u,v s , us, vs) dBs - k-1 X l=1 Hu,v s (l) - W(l+i)mod(k) (s, Xu,v s ) + (s, Xu,v s ) dNs(l). By setting Y 1,i,u,v s = Y u,v s - Y 1,i,u,v s , Z1,i,u,v s = Zu,v s - D(s, Xu,v s )i(s, Xu,v s , Y u,v s , us, vs), H1,i,u,v s (l) = Hu,v s (l) - W(l+i)mod(k) (s, Xu,v s ) + (s, Xu,v s ), 1 l k - 1, s [t, ], (4.7) we get BSDE (4.5). The uniqueness of the solution (Xu,v , Y u,v , Hu,v , Zu,v ) of (4.2) gives the stated result. Lemma 4.5. For some C R+ independent of > 0, it holds E h Z t |Y 1,i,u,v s | + k-1 X l=1 |H1,i,u,v s (l)| + |Z1,i,u,v s | ds | Ft i C 5 4 , P-a.s., (4.8) for all (0, 0), and all u Ut, , v Vt, , where 0 > 0 is small enough . A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 21 Proof. From (4.7), (4.6) and (4.4), we know Y 1,i,u,v s = Y u,v s - Y 1,i,u,v s = -E[Y u,v - Y u,v s | Fs ] + E[Y 1,i,u,v - Y 1,i,u,v s | Fs ] = E h Z s ~ fi(r, Xu,v r , Y u,v r , Hu,v r , Zu,v r , ur, vr)dr - Z s k-1 X l=1 Hu,v r (l)dr | Fs i +E h ( , Xu,v ) - (s , Xu,v s ) | Fs i +E h k-1 X l=1 Z s W(l+i)mod(k) (r, Xu,v r ) - (r, Xu,v r ) dNr(l) | Fs i , s [t, t + ]. Therefore, from (4.3) |Y 1,i,u,v s | CE h Z s 1 + |Xu,v r | + |Y u,v r | + k-1 X l=1 |Hu,v r (l)| + |Zu,v r | dr | Fs i + CE h |( , Xu,v ) - (s , Xu,v s )| | Fs i + E h k-1 X l=1 Z s (W(l+i)mod(k)(r, Xu,v r ) - (r, Xu,v r ))dNr(l) | Fs i CE h Z s 1 + |Xu,v r | + |Y u,v r | + k-1 X l=1 |Hu,v r (l)| + |Zu,v r | dr | Fs i + C + CE |Xu,v - Xu,v s | | Fs + E h k-1 X l=1 Z s W(l+i)mod(k)(r, Xu,v r ) - (r, Xu,v r ) dr | Fs i C 1 2 E h Z t+ s (1 + |Xu,v r |2 + |Y u,v r |2 + k-1 X l=1 |Hu,v r (l)|2 + |Zu,v r |2 )dr | Fs i1 2 + C + C 1 2 (1 + |Xu,v s |) C 1 2 (1 + |Xu,v s |), P-a.s., s [t, t + ]. (4.9) From (4.7), we get |Z1,i,u,v s | C(1 + |Xu,v s | + |Y u,v s | + |Zu,v s |), k-1 P l=1 |H1,i,u,v s (l)| C(1 + |Xu,v s | + k-1 P l=1 |Hu,v s (l)|), P-a.s., s [t, ]. (4.10) 22 J. LI ET AL. Applying Ito's formula to |Y 1,i,u,v s |2 , from (4.5) we obtain |Y 1,i,u,v t |2 + E h Z t |Z1,i,u,v r |2 dr + Z t k-1 X l=1 |H1,i,u,v r (l)|2 dr | Ft i = 2E h Z t Y 1,i,u,v r F(r, Xu,v r , Y 1,i,u,v r , H1,i,u,v r , Z1,i,u,v r , ur, vr)dr | Ft i -E h Z t 2Y 1,i,u,v r k-1 X l=1 H1,i,u,v r (l)dr | Ft i 2E h Z t Y 1,i,u,v r F(r, Xu,v r , Y 1,i,u,v r , H1,i,u,v r , Z1,i,u,v r , ur, vr)dr | Ft i +CE h Z t |Y 1,i,u,v r |2 dr | Ft i + 2 E h Z t k-1 X l=1 |H1,i,u,v r (l)|2 dr | Ft i . (4.11) Thus, combing this estimate with Remark 4.1, (4.9), (4.10) as well as (4.3) yields |Y 1,i,u,v t |2 + E h Z t |Z1,i,u,v r |2 dr + Z t k-1 X l=1 |H1,i,u,v r (l)|2 dr | Ft i 4E h Z t Y 1,i,u,v r F(r, Xu,v r , Y 1,i,u,v r , H1,i,u,v r , Z1,i,u,v r , ur, vr)dr | Ft i + CE h Z t |Y 1,i,u,v r |2 dr | Ft i CE h Z t |Y 1,i,u,v r |(1 + |Xu,v r |2 + |Y 1,i,u,v r |2 + |Z1,i,u,v r | + k-1 X l=1 |H1,i,u,v r (l)|)dr | Ft i +CE h Z t |Y 1,i,u,v r |2 dr | Ft i C 1 2 E h Z t (1 + |Xu,v r |2 + |Xu,v r |3 )dr | Ft i + CE h Z t |Y 1,i,u,v r |3 dr | Ft i +C 1 2 E h Z t (1 + |Xu,v r |)|Y u,v r |dr | Ft i + C 1 2 E h Z t (1 + |Xu,v r |)|Zu,v r |dr | Ft i +C 1 2 E h Z t (1 + |Xu,v r |) k-1 X l=1 |Hu,v r (l)|dr | Ft i + CE h Z t (1 + |Xu,v r |2 )dr | Ft i C 3 2 . (4.12) Therefore, from (4.9), (4.12) and (4.3)­(i) we have E h Z t |Y 1,i,u,v r | + |Z1,i,u,v r | + k-1 X l=1 |H1,i,u,v r (l)| dr | Ft i C 1 2 E h Z t (1 + |Xu,v r |)dr | Ft i + C 1 2 E h Z t |Z1,i,u,v r |2 dr | Ft i1 2 +C 1 2 E h Z t k-1 X l=1 |H1,i,u,v r (l)|2 dr | Ft i1 2 C 5 4 , P-a.s. A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 23 Let us focus now on the following BSDE with jumps: dY 2,i,u,v s = -F(s, x, 0, H2,i,u,v s , Z2,i,u,v s , us, vs)ds + k-1 P l=1 H2,i,u,v s (l)ds +Z2,i,u,v s dBs + k-1 P l=1 H2,i,u,v s (l)dNs(l), s [t, ], Y 2,i,u,v = 0. (4.13) Lemma 4.6. There is some C R+ such that, for all (0, 0) and for all u Ut, , v Vt, , where 0 is small enough, it holds (i) |Y 1,i,u,v t - Y 2,i,u,v t | C 4 3 , P-a.s. (ii) E h Z t k-1 X l=1 |H1,i,u,v s (l) - H2,i,u,v s (l)|2 + |Z1,i,u,v s - Z2,i,u,v s |2 ds|Ft i C 5 3 , P-a.s. (4.14) Proof. As F(s, x, y, h, z, u, v) is Lipschitz in (h, z), uniformly with respect to (s, x, y, u, v), a classical BSDE estimate yields |Y 1,i,u,v t - Y 2,i,u,v t |2 + E h Z t k-1 X l=1 |H1,i,u,v s (l) - H2,i,u,v s (l)|2 + |Z1,i,u,v s - Z2,i,u,v s |2 ds|Ft i E h Z t |F(s, Xu,v s , Y 1,i,u,v s , H1,i,u,v s , Z1,i,u,v s , us, vs) - F(s, x, 0, H1,i,u,v s , Z1,i,u,v s , us, vs)|2 ds|Ft i CE h Z t (1 + |x| + |Xu,v s | + |Y 1,i,u,v s |)2 (|Xu,v s - x| + |Y 1,i,u,v s |)2 ds | Ft i . (4.15) Consequently, from (4.3) (i)­(ii) and (4.9), |Y 1,i,u,v t - Y 2,i,u,v t |2 + E h Z t k-1 X l=1 |H1,i,u,v s (l) - H2,i,u,v s (l)|2 + |Z1,i,u,v s - Z2,i,u,v s |2 ds | Ft i CE h sup s[t,] (1 + |Xu,v s |)2 |Xu,v s - x|2 + (1 + |Xu,v s |)2 | Ft i C E h sup s[t,] (1 + |Xu,v s |)6 | Ft i1 3 E h sup s[t,] (|Xu,v s - x|3 + 3 2 (1 + |Xu,v s |)3 ) | Ft i2 3 C( 2 3 + ) C 5 3 , (4.16) 24 J. LI ET AL. for all (u, v) and all (0, 0), for some 0 > 0 sufficiently small. On the other hand, |Y 1,i,u,v t - Y 2,i,u,v t | = E h Z t (F(s, Xu,v s , Y 1,i,u,v s , H1,i,u,v s , Z1,i,u,v s , us, vs) - F(s, x, 0, H2,i,u,v s , Z2,i,u,v s , us, vs))ds|Ft i -E h Z t k-1 X l=1 (H1,i,u,v s (l) - H2,i,u,v s (l))ds|Ft i CE h Z t (1 + |x| + |Xu,v s | + |Y 1,i,u,v s |)(|Xu,v s - x| + |Y 1,i,u,v s |) + k-1 X l=1 |H1,i,u,v s (l) - H2,i,u,v s (l)| + |Z1,i,u,v s - Z2,i,u,v s | ds|Ft i CE h Z t |Xu,v s - x| + |Y 1,i,u,v s | + |Xu,v s - x|2 + |Y 1,i,u,v s |2 ds | Ft i +C 1 2 E h Z t k-1 X l=1 |H1,i,u,v s (l) - H2,i,u,v s (l)|2 + |Z1,i,u,v s - Z2,i,u,v s |2 ds|Ft i1 2 , and from (4.9) as well as (4.3) and (4.16) we obtain |Y 1,i,u,v t - Y 2,i,u,v t | C E h sup s[t,] |Xu,v s - x|2 + |Xu,v s - x|4 | Ft i1 2 + C 3 2 E h sup s[t,] (1 + |Xu,v s |2 )|Ft i +C 1 2 E h Z t k-1 X l=1 |H1,i,u,v s (l) - H2,i,u,v s (l)|2 + |Z1,i,u,v s - Z2,i,u,v s |2 ds|Ft i1 2 C 3 2 + C 4 3 C 4 3 , P-a.s. Lemma 4.7. Let Y 3,i be the solution of the following equation: dY 3,i s = -F0(s, x, 0, H3,i s , 0)ds + k-1 P l=1 H3,i s (l)[dNs(l) + ds], s [t, ], Y 3,i = 0, (4.17) where the function F0 is defined by F0(s, x, y, h, z) = sup uU inf vV F(s, x, y, h, z, u, v). (4.18) Then, there is some 0 > 0 small enough such that, for all (0, 0), P-a.s., esssup uUt, essinf vVt, Y 2,i,u,v t = Y 3,i t . (4.19) A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 25 Proof. Let us define F(s, x, y, h, z, u) = inf vV F(s, x, y, h, z, u, v), (s, x, y, h, z, u) [0, T]×R×R×Rk-1 ×Rd ×U, and consider, for u Ut, , the following BSDE: dY 3,i,u s = -F(s, x, 0, H3,i,u s , Z3,i,u s , us)ds + k-1 P l=1 H3,i,u s (l)(dNs(l) + ds) + Z3,i,u s dBs, s [t, ], Y 3,i,u = 0, (4.20) for all (0, 0]. It is clear that there exists a unique solution (Y 3,i,u s , H3,i,u s , Z3,i,u s ) to (4.20). Moreover, Y 3,i,u t = essinf vVt, Y 2,i,u,v t , P-a.s., for all u Ut, . Indeed, from the definition of F and the comparison theorem for BSDEs with jumps (see Thm. 2.5 in [11]), we have Y 3,i,u t essinf vVt, Y 2,i,u,v t , P-a.s., for all u Ut, . On the other hand, there exists a Borel measurable function v2 : [0, T] × R × Rk-1 × Rd × U V such that F(s, x, 0, h, z, u) = F(s, x, 0, h, z, u, v2 (s, x, h, z, u)), for any (s, x, h, z, u). By setting e v2 s = v2 (s, x, H3,i,u s , Z3,i,u s , us), s [t, ], we see e v2 Vt, , and F(s, x, 0, H3,i,u s , Z3,i,u s , us) = F(s, x, 0, H3,i,u s , Z3,i,u s , us, e v2 s ), s [t, ]. From the uniqueness of the solution of the BSDE with jumps, we conclude that (Y 3,i,u , H3,i,u , Z3,i,u ) = (Y 2,i,u,e v2 , H2,i,u,e v2 , Z2,i,u,e v2 ), and in particular, Y 3,i,u t = Y 2,i,u,e v2 t , P-a.s., for all u Ut, . Consequently, Y 3,i,u t = essinf vVt, Y 2,i,u,v t , P-a.s., for all u Ut, . Finally, as F0(s, x, y, h, z) = sup uU F(s, x, y, h, z, u), by a similar argument we get the desired result: Y 3,i t = esssup uUt, Y 3,i,u t = esssup uUt, essinf vVt, Y 2,i,u,v t , P-a.s. Remark 4.8. Notice that the solution (Y 3,i , H3,i ) of (4.17) is independent of the Brownian motion B but still depends on the Poisson random measure N. Now we extend equation (4.17) to the interval [t, t + ] as follows: dY 0, s = -1[t,](s)F0(s, x, 0, H0, s , 0)ds + k-1 P l=1 H0, s (l)[dNs(l) + ds], s [t, t + ], Y 0, t+ = 0. (4.21) 26 J. LI ET AL. Remark 4.9. It is easy to check that Y 0, s = Y 3,i s , s [t, ], 0, s [ , t + ], H0, s = H3,i s , s [t, ], 0, s [ , t + ], So we have Y 0, t = Y 3,i t = esssup uUt, essinf vVt, Y 2,i,u,v t , P-a.s. Furthermore, we consider dY 0, s = -F0(s, x, 0, 0, 0)ds, s [t, t + ], Y 0, t+ = 0, (4.22) and we can estimate the difference between Y 0, t and Y 0, t as follows: Lemma 4.10. There exists a constant C R+ such that for all (0, 0), for some 0 > 0, |Y 0, t - Y 0, t | C 3 2 , P-a.s. Proof. We notice that dY 0, s = -F0(s, x, 0, H0, s , 0)ds + k-1 P l=1 H0, s (l)[dNs(l) + ds], s [t, t + ], Y 0, t+ = 0, (4.23) where H0, s = 0, s [t, t + ]. On the other hand, from (4.22), Y 0, = Z t+ F0(s, x, 0, 0, 0)ds, and as |F0(s, x, 0, 0, 0)| C(1 + |x|2 ) = C (indeed, x is fixed), we have from the definition of , |Y 0, | C(t + - ) k-1 X l=1 C1{N((t,t+]×{l})1}, that is, E[|Y 0, |2 | Ft] Ck2 2 1 - P N((t, t + ] × {l}) = 0 = Ck2 2 (1 - e- ) Ck2 3 . As we have dY 0, s = -F0(s, x, 0, H0, s , 0)ds + k-1 P l=1 H0, s (l)[dNs(l) + ds], s [t, ], Y 0, = R t+ F0(s, x, 0, 0, 0)ds, (4.24) and dY 0, s = -F0(s, x, 0, H0, s , 0)ds + k-1 P l=1 H0, s (l)[dNs(l) + ds], s [t, ] Y 0, = 0, (4.25) A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 27 we get from a BSDE standard estimate, |Y 0, t - Y 0, t |2 + E h Z t k-1 X l=1 |H0, s (l) - H0, s (l)|2 ds | Ft i CE |Y 0, - Y 0, |2 | Ft C3 , (0, 0), P-a.s. (4.26) Consequently, |Y 0, t - Y 0, t | C 3 2 , (0, 0), P-a.s. Now we give the proof of Theorem 4.2. Proof. (1) Let us show that W is a viscosity supersolution. For this we suppose that i K, C3 l,b([0, T] × R) and (t, x) [0, T) × R are such that Wi - attains its minimum at (t, x). Without loss of generality we may also suppose that (t, x) = Wi(t, x). Then ( , Xu,v )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , Xu,v )1{Nt,i =(l+i)mod(k)} Wi( , Xu,v )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , Xu,v )1{Nt,i =(l+i)mod(k)} = WNt,i ( , Xu,v ), (u, v) Ut, × Vt, . Recall from (4.4) that Y 1,i,u,v t = (t, x), and recall also Lemma 4.3. Then, from the comparison theorem (the proof is similar to Theorem 3.3 in [6]) for fully coupled FBSDEs with jumps we have, for all (0, 0) and for all u Ut, , Bt, , Y 1,i,u,(u) t = Y u,(u) t - (t, x) = G t,x,i;u,(u) t, h ( , X u,(u) )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , X u,(u) )1{Nt,i =(l+i)mod(k)} i - (t, x) G t,x,i;u,(u) t, h WNt,i ( , X t,x,i;u,(u) ) i - Wi(t, x), (4.27) where G t,x,i;u,(u) t, [·] is defined in (3.9), and (Xt,x,i;u,(u) , Y t,x,i;u,(u) , Ht,x,i;u,(u) , Zt,x,i;u,(u) ) is the unique solution of the following FBSDE with jumps on [t, ]: dX t,x,i;u,(u) s = bNt,i s (s, X t,x,i;u,(u) s , Y t,x,i;u,(u) s , Z t,x,i;u,(u) s , us, (u)s)ds +Nt,i s (s, X t,x,i;u,(u) s , Y t,x,i;u,(u) s , us, (u)s)dBs, s [t, ], X t,x,i;u,(u) t = x, dY t,x,i;u,(u) s = - ~ fNt,i s (s, X t,x,i;u,(u) s , Y t,x,i;u,(u) s , H t,x,i;u,(u) s , Z t,x,i;u,(u) s , us, (u)s)ds + k-1 P l=1 H t,x,i;u,(u) s (l)ds + Z t,x,i;u,(u) s dBs + k-1 P l=1 H t,x,i;u,(u) s (l)dNs(l), Y t,x,i;u,(u) = WNt,i ( , X t,x,i;u,(u) ). Combined with the strong DPP (Thm. 3.17), this yields essinf Bt, esssup uUt, Y 1,i,u,(u) t essinf Bt, esssup uUt, G t,x,i;u,(u) t, [WNt,i ( , X t,x,i;u,(u) )] - Wi(t, x) = 0, P-a.s. 28 J. LI ET AL. Furthermore, Lemma 4.6 implies that essinf Bt, esssup uUt, Y 2,i,u,(u) t essinf Bt, esssup uUt, Y 1,i,u,(u) t + C 4 3 C 4 3 , P-a.s. Consequently, from essinf vVt, Y 2,i,u,v t Y 2,i,u,(u) t , Bt, , we get esssup uUt, essinf vVt, Y 2,i,u,v t essinf Bt, esssup uUt, Y 2,i,u,(u) t C 4 3 , P-a.s. On the other hand, according to Lemma 4.7, Y 3,i t = esssup uUt, essinf vVt, Y 2,i,u,v t C 4 3 , P-a.s. It then follows from Remark 4.9 and Lemma 4.10, Y 0, t C 4 3 . Therefore, from equation (4.22), sup uU inf vV F(t, x, 0, 0, 0, u, v) = F0(t, x, 0, 0, 0) 0, that is, t (t, x) + sup uU inf vV n Li u,v(t, x) + ~ fi(t, x, (t, x), W(i+·)mod(k) (t, x) - (t, x)1, D(t, x)i(t, x, (t, x), u, v), u, v) o = t (t, x) + sup uU inf vV n Li u,v(t, x) + fi(t, x, W(t, x), D(t, x)i(t, x, (t, x), u, v), u, v) o 0. Hence, W is a viscosity supersolution of (4.1). (2) We now show that W is a viscosity subsolution of (4.1). Suppose i K, C3 l,b([0, T] × R) and (t, x) [0, T) × R are such that Wi - attains its maximum at (t, x). Without loss of generality, we assume again that (t, x) = Wi(t, x). For this, we only need to prove that sup uU inf vV F(t, x, 0, 0, 0, u, v) = F0(t, x, 0, 0, 0) 0. Let us suppose that this is not true. Then there exists some > 0 such that sup uU inf vV F(t, x, 0, 0, 0, u, v) = F0(t, x, 0, 0, 0) - < 0, (4.28) and we can find a measurable function g : U V such that F(t, x, 0, 0, 0, u, g(u)) - 3 4 , for all u U. Moreover, since F(·, x, 0, 0, 0, ·, ·) is uniformly continuous on [0, T] × U × V, there exists some T - t R > 0 such that F(s, x, 0, 0, 0, u, g(u)) - 1 2 , for all u U and |s - t| R. (4.29) A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 29 On the other hand, notice that now we have ( , X u,(u) )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , X u,(u) )1{Nt,i =(l+i)mod(k)} Wi( , X u,(u) )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , X u,(u) )1{Nt,i =(l+i)mod(k)} = WNt,i ( , X u,(u) ). Similar to (4.27), from the comparison theorem for fully coupled FBSDEs with jumps (see, Thm. 3.3 in [6]) and Lemma 4.1 we have Y 1,i,u,(u) t = Y u,(u) t - (t, x) = G t,x,i;u,(u) t, [( , X u,(u) )1{Nt,i =i} + k-1 P l=1 W(l+i)mod(k) ( , X u,(u) )1{Nt,i =(l+i)mod(k)} ] - (t, x) G t,x,i;u,(u) t, [WNt,i ( , X t,x,i;u,(u) )] - Wi(t, x), where G t,x,i;u,(u) t, [·] is defined in (3.9). Then, the strong DPP gives essinf Bt, esssup uUt, Y 1,i,u,(u) t 0, P-a.s., and in particular, esssup uUt, Y 1,i,u,g(u) t 0, P-a.s. Here, by putting gs(u)() = g(us()), (s, ) [t, T] × , we identify g as an element of Bt, . Given an arbitrary > 0 we can choose u Ut, (depending on > 0) such that Y 1,i,u ,g(u ) t -. From Lemma 4.6 we have Y 2,i,u ,g(u ) t -C 4 3 - , P-a.s. (4.30) Recall that Y 2,i,u ,g(u ) t = Z t F(s, x, 0, H2,i,u ,g(u ) s , Z2,i,u ,g(u ) s , u s, g(u s))ds - Z t k-1 X l=1 H2,i,u ,g(u ) s (l)ds - Z t Z2,i,u ,g(u ) s dBs - Z t k-1 X l=1 H2,i,u ,g(u ) s (l)dNs(l). Using that (h, z) F(s, x, 0, h, z, u, v) is Lipschitz, uniformly in (s, x, u, v), we have F(s, x, 0, H2,i,u ,g(u ) s , Z2,i,u ,g(u ) s , u s, g(u s)) F(s, x, 0, 0, 0, u s, g(u s)) + C k-1 X l=1 |H2,i,u ,g(u ) s (l)| + |Z2,i,u ,g(u ) s | . Hence, Y 2,i,u ,g(u ) t E h Z t F(s, x, 0, 0, 0, u s, g(u s))ds | Ft i +CE h Z t k-1 X l=1 |H2,i,u ,g(u ) s (l)| + |Z2,i,u ,g(u ) s | ds | Ft i . (4.31) 30 J. LI ET AL. Notice that, from (4.8) and (4.14)­(ii) we have E h Z t k-1 X l=1 |H2,i,u ,g(u ) s (l)| + |Z2,i,u ,g(u ) s | ds|Ft i E h Z t k-1 X l=1 |H1,i,u ,g(u ) s (l)| + |Z1,i,u ,g(u ) s | ds|Ft i +E h Z t k-1 X l=1 |H1,i,u ,g(u ) s (l) - H2,i,u ,g(u ) s (l)| + |Z1,i,u ,g(u ) s - Z2,i,u ,g(u ) s | ds|Ft i C 5 4 + C 1 2 E h Z t k-1 X l=1 |H1,i,u ,g(u ) s (l) - H2,i,u ,g(u ) s (l)|2 + |Z1,i,u ,g(u ) s - Z2,i,u ,g(u ) s |2 ds|Ft i1 2 C 5 4 + C 4 3 C 5 4 . Substituting this in (4.31) yields Y 2,i,u ,g(u ) t E h Z t F(s, x, 0, 0, 0, u s, g(u s))ds | Ft i + C 5 4 . (4.32) Consequently, from (4.29), for all (0, 0 R), Y 2,i,u ,g(u ) t - 1 2 E h - t | Ft i + C 5 4 . (4.33) From the definition of , = t + on {Nt,i t+ = i}. Thus, E h | Ft i (t + )P{Nt,i t+ = i | Ft} = (t + )P{Nt,i t+ = i} = (t + )P{N((t, t + ] × {l}) = 0, 1 l k - 1} = (t + )e-(k-1) =: (). As 0 () = e-(k-1) (1 - (k - 1)(t + )) e-(k-1) (1 - (k - 1)T) > 0, the function : [0, T] R is increasing, and there exists some [0, ], such that () - (0) = 0 (), i.e., E h - t | Ft i () - (0) = 0 () = e-(k-1) (1 - (k - 1)(t + )) e-(k-1) (1 - (k - 1)T)(> 0). Hence, from (4.33), Y 2,i,u ,g(u ) t - 1 2 E h - t | Ft i + C 5 4 - 1 2 e-(k-1) (1 - (k - 1)T) + C 5 4 . (4.34) From (4.30) and (4.34), -C 1 3 - -1 2 e-(k-1) (1 - (k - 1)T) + C 1 4 , P-a.s. Letting 0, and then 0, we deduce that 0, which induces a contradiction. Therefore, sup uU inf vV F(t, x, 0, 0, 0, u, v) = F0(t, x, 0, 0, 0) 0, and from the definition of F we see that W is a viscosity subsolution of (4.1). Therefore, we obtain W is a viscosity solution of (4.1). A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 31 4.2. Uniqueness First we define the following space of continuous functions: = { C([0, T] × R) : A > 0 such that lim |x| (t, x) exp{-A[log((|x|2 + 1) 1 2 )]2 } = 0, uniformly in t [0, T]}. The uniqueness result of viscosity solution of the following HJBI equation (which is a special case of (4.1) when the diffusion coefficients i, i K, are independent of y) will be presented in : Wi t (t, x) + sup uU inf vV n 1 2 tr (i i (t, x, u, v)D2 Wi(t, x)) +bi(t, x, Wi(t, x), DWi(t, x)i(t, x, u, v), u, v)DWi(t, x) +fi(t, x, W(t, x), DWi(t, x)i(t, x, u, v), u, v) o = 0, Wi(T, x) = gi(x), (t, x, i) [0, T] × R × K, (4.35) where W(t, x) = (W1(t, x), W2(t, x), · · · , Wk(t, x)), (t, x) [0, T] × R. Theorem 4.11. Under the assumptions (B1)-(B3), the lower value function W is the unique viscosity solution in of the coupled HJBI equation (4.35). Proving the uniqueness of the viscosity solution similar techniques as those in [7] can be applied in our case. So the details are omitted here. We only study the lower value function W(t, x). For the upper value function U(t, x) = (U1(t, x), U2(t, x), · · · , Uk(t, x)), we have the same result. Theorem 4.12. Under our assumptions (B1)-(B3), the upper value function U(t, x) = (U1(t, x), U2(t, x), · · · , Uk(t, x)) is a viscosity solution of the following system of HJBI equations: Ui t (t, x) + inf vV sup uU n bi(t, x, Ui(t, x), DUi(t, x)i(t, x, Ui(t, x), u, v), u, v)DUi(t, x) +1 2 tr (i i (t, x, Ui(t, x), u, v)D2 Ui(t, x)) +fi(t, x, U(t, x), DUi(t, x)i(t, x, Ui(t, x), u, v), u, v) o = 0, Ui(T, x) = gi(x), (t, x) [0, T] × R, i K = {1, 2, · · · k}. (4.36) When i, i K, do not depend on y, we also get the uniqueness of the viscosity solution U(t, x) in . Remark 4.13. If for all (t, x, i) [0, T] × R × K, (y, p, A) R × R × R, a Rk , the following Isaacs' condition holds true: sup uU inf vV n 1 2 tr i i (t, x, u, v)A + bi(t, x, y, pi(t, x, u, v), u, v) · p + fi(t, x, a, pi(t, x, u, v), u, v) o = inf vV sup uU n 1 2 tr i i (t, x, u, v)A + bi(t, x, y, pi(t, x, u, v), u, v) · p + fi(t, x, a, pi(t, x, u, v), u, v) o , then the equations (4.35) and (4.36) coincide (when i, i K, do not depend on y). The uniqueness of the viscosity solution implies that W(t, x) = U(t, x). That is, there exists a value for our stochastic differential games under the Isaacs' condition. 32 J. LI ET AL. Appendix A. A.1 Comparison theorem for FBSDEs with jumps In this section, we show that the comparison theorem for the dynamics (3.2) holds true only when K1 0, where K1 is the constant in the condition (B3). For simplicity, we suppress the controls (u, v) in equation (3.2), i.e., the following FBSDE with jumps, dXt,,i s = bNt,i s (s, Xt,,i s , Y t,,i s , Zt,,i s )ds + Nt,i s (s, Xt,,i s , Y t,,i s )dBs, s [t, T], Xt,,i t = , dY t,,i s = - ~ fNt,i s (s, Xt,,i s , Y t,,i s , Ht,,i s , Zt,,i s ) - k-1 P l=1 Ht,,i s (l) ds +Zt,,i s dBs + k-1 P l=1 Ht,,i s (l)dNs(l), s [t, T], Y t,,i T = gNt,i T (Xt,,i T ), (A.1) where i K, and initial data (t, ) [0, T] × L2 (, Ft, P; R). From Remark 3.3, we know that, under the assumptions (B1) and (B2), equation (A.1) has a unique solution (Xt,,i , Y t,,i , Ht,,i , Zt,,i ) B2 [t, T]. We now show that under the additional condition (B3) the comparison theorem for this equation holds true (or, equivalently, if condition (iii) in Rem. 3.1 is satisfied). In fact, when the constant K1 > 0 we get the comparison theorem by arguments similar to those in Theorem 2.5 in [11]. For the situation K1 = 0, we consider an increasing positive sequence {n} n=1 R+ with n < , n , n , and the FBSDE with jumps (A.1) with replaced by n (but we keep Ns(l) defined in Subsection 2.1 with ), that is, dXt,,i,n s = bNt,i s (s, Xt,,i,n s , Y t,,i,n s , Zt,,i,n s )ds + Nt,i s (s, Xt,,i,n s , Y t,,i,n s )dBs, s [t, T], Xt,,i,n t = , dY t,,i,n s = - ~ fNt,i s (s, Xt,,i,n s , Y t,,i,n s , Ht,,i,n s , Zt,,i,n s ) - n k-1 P l=1 Ht,,i,n s (l) ds +Zt,,i,n s dBs + k-1 P l=1 Ht,,i,n s (l)dNs(l), s [t, T], Y t,,i,n T = gNt,i T (Xt,,i,n T ). (A.2) It is easy to check that, for every n N, this equation has a unique solution (Xt,,i,n , Y t,,i,n , Ht,,i,n , Zt,,i,n ) B2 [t, T]. For n N and i K, we introduce ~ fn i (s, x, y, h, z) = ~ fi(s, x, y, h, z) + ( - n) k-1 X l=1 h(l), (s, x, y, z, h) [t, T] × R × R × Rk-1 × Rd . Then the equation (A.2) can be rewritten as dXt,,i,n s = bNt,i s (s, Xt,,i,n s , Y t,,i,n s , Zt,,i,n s )ds + Nt,i s (s, Xt,,i,n s , Y t,,i,n s )dBs, s [t, T], Xt,,i,n t = , dY t,,i,n s = - ~ fn Nt,i s (s, Xt,,i,n s , Y t,,i,n s , Ht,,i,n s , Zt,,i,n s ) - k-1 P l=1 Ht,,i,n s (l) ds +Zt,,i,n s dBs + k-1 P l=1 Ht,,i,n s (l)dNs(l), s [t, T], Y t,,i,n T = gNt,i T (Xt,,i,n T ). (A.3) A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 33 Since the coefficient ~ fn i (s, x, y, h, z) satisfies the condition (iii) in Remark 3.1 with K1 + ( - n) > K1 = 0, we get the comparison results for the FBSDE with jumps (A.3) for all n N. Note that it follows from standard arguments that E h sup s[t,T ] |Xt,,i,n s - Xt,,i s |2 + sup s[t,T ] |Y t,,i,n s - Y t,,i s |2 + Z T t |Zt,,i,n s - Zt,,i s |2 ds + k-1 X l=1 Z T t |Ht,,i,n s (l) - Ht,,i s (l)|2 ds i 0, as n . Thus, also for the limit equation (A.1) with K1 = 0, we have the comparison theorem. This proves that we obtain the comparison theorem for FBSDE with jumps (A.1) when the constant K1 0. It is worth to point out that, in general, there is no comparison result for FBSDEs with jumps of the type (A.1) when K1 < 0. Let us give a counter-example. Example A.1. We choose K = {1, 2}, i.e., k = 2. Referring to the notations introduced in Subsection 2.1, we put N (s, t] := N (s, t] × {1} , N (s, t] := (N - N) (s, t] × {1} , 0 s t T. Let the coefficients b = 0, = 0, and ~ fi(s, x, y, h, z) = - k-1 P l=1 h(l) = -h(1), for some > 0. Moreover, for K1 = - < 0, we have ~ fi(s, x, y, h, z) - ~ fi(s, x, y, h0 , z) = K1 h(1) - h0 (1) , but as K1 < 0, (B3) or condition (iii) in Remark 3.1 is not satisfied. Notice that now (A.1) takes the form dYs = ( + )Hsds + HsdNs = Hsds + HsdNs, s [0, T]. (A.4) Endowed with a terminal condition L2 , {N (0, T] }, P , BSDE (A.4) has a unique square integrable adapted solution (Y, H). Let us consider the both terminal conditions 1 = 0, 2 = N (0, T] , and denote by (Y i , Hi ) the unique solution of (A.4) with Y i T = i , i = 1, 2, respectively. Obviously, (Y 1 s , H1 s ) (0, 0), (Y 2 s , H2 s ) = N (0, s] - (T - s), 1 , s [0, T]. We observe that, although 1 2 with P{2 > 1 } > 0, we have Y 2 0 = -T < 0 = Y 1 0 , for any > 0 (and, hence, for any K1 < 0). This shows that, if K1 < 0, the comparison theorem fails. If we consider i = i + N (0, T] , i = 1, 2, and denote by (Y i , Hi ) the unique solution of (A.4) with Y i T = i , i = 1, 2, respectively. It is easy to check that, for i = 1, 2, (Y i s , Hi s) = i + N (0, s] - (T - s), 1 , s [0, T]. Then, although 1 < i , i = 1, 2, Y i 0 = i - T 2 - T < 0 = Y 1 0 , if T > 2 , 34 J. LI ET AL. which also implies that the comparison result fails for K1 = - < - 2 T . We see, in particular that, choosing T > 0 large enough, we can have K1 < 0 arbitrarily near to zero. Remark A.2. We consider i in order to get the explicit solution, instead of considering the terminal condition Y i T = Ni T = i + N (0, T] mod(2) {1, 2}. But, as i + N (0, T] (i + N((0, T]))mod(2) > 0, the comparison theorem should also hold for i + N((0, T]) if we had the comparison theorem. A.2 The proof of Theorem 3.17 (Strong-DPP) For convenience, we set Wi(t, x) = essinf Bt, esssup uUt, G t,x,i;u,(u) t, [WNt,i (, Xt,x,i;u,(u) )]. We want to prove that Wi(t, x) and Wi(t, x) coincide. For this we only need to prove the following three lemmas. Lemma A.3. For all i K, (t, x) [0, T] × R, Wi(t, x) is deterministic. The proof of this lemma is similar to the proof of Proposition 3.6; so we omit it here. Lemma A.4. For all i K, (t, x) [0, T] × R, it holds Wi(t, x) Wi(t, x), P-a.s. Proof. Let Bt,T be arbitrarily fixed. For any given u2 U,T , we define, for u1 Ut, , 1(u1) := (u1 u2)|[t,], where u1 u2 := u11[t,] + u21(,T ], belongs to Ut,T . It is easy to check that 1 Bt, and 1 is independent of the special choice of u2 U,T due to the nonanticipativity property of . Consequently, from the definition of Wi(t, x), Lemma 5.1 and Remark 3.7 in [7], we know for any > 0, there exists u 1 Ut, such that Wi(t, x) G t,x,i;u 1,1(u 1) t, [WNt,i (, X t,x,i;u 1,1(u 1) )] + , P-a.s. (A.5) We now estimate WNt,i (, X t,x,i;u 1,1(u 1) ). For u2 U,T , we define u 1 2 (u2) := (u 1 u2)|[,T ]. Then u 1 2 B,T . From Proposition 3.16 with choosing = X t,x,i;u 1,1(u 1) , we have WNt,i (, X t,x,i;u 1,1(u 1) ) = essinf 2B,T esssup u2U,T J , X t,x,i;u 1,1(u 1) , Nt,i ; u2, 2(u2) esssup u2U,T J , X t,x,i;u 1,1(u 1) , Nt,i ; u2, u 1 2 (u2) , P-a.s. By similar arguments to (A.5) it follows that there exists u 2 U,T such that WNt,i (, X t,x,i;u 1,1(u 1) ) J , X t,x,i;u 1,1(u 1) , Nt,i ; u 2, u 1 2 (u 2) + , P-a.s. A SYSTEM OF COUPLED HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS 35 It is easy to check that (u ) = 1(u 1) u 1 2 (u 2), where u = u 1 u 2 belongs to Ut,T . Then we have WNt,i (, X t,x,i;u 1,1(u 1) ) J , X t,x,i;u 1,1(u 1) , Nt,i ; u 2, u 1 2 (u 2) + = J , Xt,x,i;u ,(u ) , Nt,i ; u , (u ) + , P-a.s. (A.6) Therefore, from (A.5), (A.6) and the comparison theorem for FBSDE with jumps (see Thm. 3.3 in [6]) we obtain Wi(t, x) G t,x,i;u 1,1(u 1) t, [WNt,i (, X t,x,i;u 1,1(u 1) )] + G t,x,i;u 1,1(u 1) t, [J , Xt,x,i;u ,(u ) , Nt,i ; u , (u ) + ] + G t,x,i;u 1,1(u 1) t, [J , Xt,x,i;u ,(u ) , Nt,i ; u , (u ) ] + C, P-a.s. (A.7) where the last inequality follows from Corollary 3.1 in [6]. Finally, from (A.7) we get Wi(t, x) G t,x,i;u ,(u ) t, [J , Xt,x,i;u ,(u ) , Nt,i ; u , (u ) ] + C = G t,x,i;u ,(u ) t, [Y t,x,i;u ,(u ) ] + C = Y t,x,i;u ,(u ) t + C = J(t, x, i; u , (u )) + C esssup uUt,T J(t, x, i; u, (u)) + C, P-a.s. From the arbitrariness of Bt,T , we conclude that Wi(t, x) essinf Bt,T esssup uUt,T J(t, x, i; u, (u)) + C = Wi(t, x) + C, (A.8) which yields the desired result by letting 0. Lemma A.5. For all i K, (t, x) [0, T] × R, it holds Wi(t, x) Wi(t, x), P-a.s. Proof. From the definition of Wi(t, x) and the standard arguments, we know that for any > 0, there exists 1 Bt, such that for all u1 Ut, , Wi(t, x) G t,x,i;u1, 1 (u1) t, [WNt,i (, X t,x,i;u1, 1 (u1) )] - , P-a.s. (A.9) We now give the estimate of WNt,i (, X t,x,i;u1, 1 (u1) ). It follows from Proposition 3.16 with choosing = X t,x,i;u1, 1 (u1) that WNt,i (, X t,x,i;u1, 1 (u1) ) = essinf 2B,T esssup u2U,T J , X t,x,i;u1, 1 (u1) , Nt,i ; u2, 2(u2) , P-a.s. Using a similar method, we deduce that there exists 2 B,T (depending on u1) such that for all u2 U,T , WNt,i (, X t,x,i;u1, 1 (u1) ) J , X t,x,i;u1, 1 (u1) , Nt,i ; u2, 2(u2) - , P-a.s. (A.10) For u Ut,T , we define (u) := 1(u1) 2(u2), 36 J. LI ET AL. where u1 = u|[t,], u2 = u|(,T ]. Then Bt,T . Indeed, let S : [t, T] be an F-stopping time and u, u Ut,T be such that u u on [[t, S]]. Decomposing u, u into u1, u1 U[t,], u2, u2 U(,T ] such that u = u1 u2, u = u1 u2. Then we have u1 = u1 on [[t, S ]] which yields that 1(u1) = 1(u1) on [[t, S ]] since 1 is nonanticipating. On the other hand, we have u2 = u2 on ]], S ]] which gives that 2(u2) = 2(u2) on ]], S ]]. Therefore, we get (u) := 1(u1) 2(u2) = 1(u1) 2(u2) = (u) on [[t, S]], from which it follows that Bt,T . Let u Ut,T be arbitrarily given and decomposed into u1 = u|[t,] Ut, and u2 = u|(,T ] U,T . Then from (A.9), (A.10) and Corollary 3.1 in [6], we get Wi(t, x) G t,x,i;u1, 1 (u1) t, [J , X t,x,i;u1, 1 (u1) , Nt,i ; u2, 2(u2) - ] - G t,x,i;u1, 1 (u1) t, [J , X t,x,i;u1, 1 (u1) , Nt,i ; u2, 2(u2) ] - C = G t,x,i;u, (u) t, [Y t,x,i;u, (u) ] - C = Y t,x,i;u, (u) t - C = J(t, x, i; u, (u)) - C, P-a.s. Consequently, it follows from the arbitrariness of u Ut,T that Wi(t, x) esssup uUt,T J(t, x, i; u, (u)) - C essinf Bt,T esssup uUt,T J(t, x, i; u, (u)) - C = Wi(t, x) - C, P-a.s. (A.11) Finally, letting 0 we get Wi(t, x) Wi(t, x). References [1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60 (1997) 57­83. [2] R. Buckdahn and Y. Hu, Probabilistic interpretation of a coupled system of Hamilton-Jacobi-Bellman equations. J. Evol. Equ. 10 (2010) 529­549. [3] R. Buckdahn, Y. Hu and J. Li, Stochastic representation for solutions of Isaacs' type integral-partial differential equations. Stochastic Proc. Appl. 121 (2011) 2715­2750. [4] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47 (2008) 444­475. [5] J. Li and Q.M. Wei, Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 52 (2014) 1622­1662. [6] J. Li and Q.M. Wei, Lp-estimates for fully coupled FBSDEs with jumps. Stoch. Process. Appl. 124 (2014) 1582­1611. [7] J. Li and Q.M. Wei, Stochastic differential games for fully coupled FBSDEs with jumps. Appl Math Optim. 71 (2015) 411­448. [8] E. Pardoux, F. Pradeilles and Z. Rao, Probabilistic interpretation of a system of semi-linear parabolic partial differential equations. Ann. Inst. Henri Poincare Probab. Statist. 33 (1997) 467­490. [9] S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stoch. Stoch. Rep. 38 (1992) 119­134. [10] S. Peng, BSDE and stochastic optimizations, in Topics in Stochastic Analysis, edited by J. Yan, S. Peng, S. Fand and L. Wu. Science Press, Beijing (1997). [11] M. Royer, Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116 (2006) 1358­1376. [1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60 (1997) 57–83. [2] R. Buckdahn and Y. Hu, Probabilistic interpretation of a coupled system of Hamilton-Jacobi-Bellman equations. J. Evol. Equ. 10 (2010) 529–549. [3] R. Buckdahn, Y. Hu and J. Li, Stochastic representation for solutions of Isaacs’ type integral-partial differential equations. Stochastic Proc. Appl. 121 (2011) 2715–2750. [4] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47 (2008) 444–475. [5] J. Li and Q. M. Wei, Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 52 (2014) 1622–1662. [6] J. Li and Q. M. Wei, L$$-estimates for fully coupled FBSDEs with jumps. Stoch. Process. Appl. 124 (2014) 1582–1611. [7] J. Li and Q. M. Wei, Stochastic differential games for fully coupled FBSDEs with jumps. Appl Math Optim. 71 (2015) 411–448. [8] E. Pardoux, F. Pradeilles and Z. Rao, Probabilistic interpretation of a system of semi-linear parabolic partial differential equations. Ann. Inst. Henri Poincaré Probab. Statist. 33 (1997) 467–490. [9] S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stoch. Stoch. Rep. 38 (1992) 119–134. [10] S. Peng, BSDE and stochastic optimizations, in Topics in Stochastic Analysis, edited by J. Yan, S. Peng, S. Fand and L. Wu. Science Press, Beijing (1997). [11] M. Royer, Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116 (2006) 1358–1376. COCV_2021__27_S1_A19_02e5bb060-7d9d-443a-8b21-b740172dfc9acocv20003410.1051/cocv/202006510.1051/cocv/2020065 Observability and unique continuation of the adjoint of a linearized simplified compressible fluid-structure model in a 2d channel 0000-0001-8986-9869 Mitra Sourav * Institute of Mathematics, University of Würzburg, 97074, Germany. *Corresponding author: [e-mail] SupplementS18 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF)Full (DJVU)

We consider a compressible fluid structure interaction model in a 2D channel with a simplified expression of the net force acting on the structure appearing at the fluid boundary. Concerning the structure we will consider a damped Euler-Bernoulli beam located on a portion of the boundary. In the present article we establish an observability inequality for the adjoint of the linearized fluid structure interaction problem under consideration which in principle is equivalent with the null controllability of the linearized system. As a corollary of the derived observability inequality we also obtain a unique continuation property for the adjoint problem.

Observability unique continuation adjoint compressible Navier-Stokes damped beam fluid-structure Carleman estimate 76N25 76N10 93B05 93B07 93B18 ANR project IFSMACS ANR-15-CE40-0010 idline ESAIM: COCV 27 (2021) S18 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S18 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020065 www.esaim-cocv.org OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT OF A LINEARIZED SIMPLIFIED COMPRESSIBLE FLUID-STRUCTURE MODEL IN A 2D CHANNEL Sourav Mitra* Abstract. We consider a compressible fluid structure interaction model in a 2D channel with a simpli- fied expression of the net force acting on the structure appearing at the fluid boundary. Concerning the structure we will consider a damped Euler-Bernoulli beam located on a portion of the boundary. In the present article we establish an observability inequality for the adjoint of the linearized fluid structure interaction problem under consideration which in principle is equivalent with the null controllability of the linearized system. As a corollary of the derived observability inequality we also obtain a unique continuation property for the adjoint problem. Mathematics Subject Classification. 76N25, 76N10, 93B05, 93B07, 93B18. Received February 17, 2020. Accepted October 2, 2020. 1. Introduction This article deals with the observability and unique continuation properties of the adjoint of a linearized compressible fluid structure interaction problem. In order to introduce our model in a fixed domain it is first important to present the non linear fluid structure interaction dynamics and obtain the linear model via a suitable linearization procedure. We remark that this linearization process is not unique and depend on the structure of the map which we will use to bring the time dependent domain to a fixed reference configuration. 1.1. Motivation In this section, we introduce the full non-linear compressible fluid structure interaction model which we aim at studying from the controllability point of view, even though our work is only a preliminary work in this direction. Our goal here is to explain how, starting from a control problem for a compressible fluid-structure interaction model, we derive a linear model (cf. Sect. 1.1.4) which should, in principle, contain some of the main difficulties related to the non-linear model. Keywords and phrases: Observability, unique continuation, adjoint, compressible Navier-Stokes, damped beam, fluid-structure, Carleman estimate. Institute of Mathematics, University of Wurzburg, 97074, Germany. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 S. MITRA Figure 1. Domain t. 1.1.1. The Non-Linear Model We first define a few notations corresponding to the fluid and the structural domain. Let d > 0 be a constant and = (0, d) × (0, 1). We set s = (0, d) × {1}, ` = (0, d) × {0}, = s `. For a given function : s × (0, ) (-1, ), which will correspond to the displacement of the one dimensional beam, let us denote by t and s,t the following sets t = {(x, y) | x (0, d), 0 < y < 1 + (x, t)} = domain of the fluid at time t (Fig. 1), s,t = {(x, y) | x (0, d), y = 1 + (x, t)} = the beam at time t. The reference configuration of the beam is s and we set T = × (0, T), s T = s × (0, T), f s T = t(0,T )s,t × {t}, ` T = ` × (0, T), QT = × (0, T), f QT = t(0,T )t × {t}. (1.1) We consider a fluid with density and velocity u. The fluid structure interaction system coupling the compressible Navier-Stokes and the damped Euler-Bernoulli beam equation is modeled by the following equations t + div(u) = 0 in f QT , (tu + (u.)u) - µu - (µ + µ0 )divu + p() = 0 in f QT , tt - txx + xxxx = (Tf )2 on s T . (1.2) We assume that at the fluid structure interface the following impermeability condition holds u(·, t) · nt = (0, t) · nt on f s T , (1.3) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 3 where nt is the outward unit normal to s,t given by nt = - x p 1 + (x)2 ~ e1 + 1 p 1 + (x)2 ~ e2, (~ e1 = (1, 0) and ~ e2 = (0, 1)). The fixed boundary ` T is assumed to be impermeable and here the impermeability condition is given as follows u(·, t) · n = u2 = 0 on ` T , (1.4) where n is the unit outward normal to `. The fluid boundary is supplemented with the following slip condition curl(u) = 0 on f s T l T , (1.5) where curlu = (u1 y - u2 x ), denotes the vorticity of the vector field u. In the system (1.2), the real constants µ, µ0 are the Lame coefficients which are supposed to satisfy µ > 0, (µ0 + 2µ) > 0. In our case the fluid is isentropic i.e. the pressure p() is only a function of the fluid density and is given by p() = a , where a > 0 and > 1 are positive constants. We assume that there exists a constant external force Pext > 0 which acts on the beam. We then introduce the positive constant defined by the relation Pext = a . To incorporate this external forcing term Pext into the system of equations (1.2), we introduce the following P() = p() - Pext = a - a . (1.6) The non-homogeneous source term of the beam equation (Tf )2 is the net surface force on the structure which is the resultant of force exerted by the fluid on the structure and the external force Pext and it is assumed to be of the following simplified form (Tf )2 = (-(µ0 + 2µ)divu nt + Pnt) |s,t p 1 + (x)2 · ~ e2 on s T . (1.7) Remark 1.1 (The physical model and simplification.). The stress tensor corresponding to a Newtonian fluid with velocity u and pressure p is of the following form: S(u, p) = (2µD(u) + µ0 div uId) - pId, (1.8) where where Id is the identity matrix and D(u) is the symmetric gradient given by D(u) = 1 2 (u + T u). 4 S. MITRA In view of the expression (1.8) of the stress tensor, the net force acting on the beam should be given as follows: (Tf )ph 2 = ([-2µD(u) - µ0 divu Id] · nt + Pnt) |s,t p 1 + (x)2 · ~ e2 on s T . (1.9) Instead of using the force (1.9), we assume, for technical reasons (see Rem. 2.1), that the net force acting on the beam is given by (1.7). Although this might seem physically irrelevant, let us point out that the resulting simplified model (1.2)­(1.7) admits an energy equality which is explained in the following. d Assuming the data and the unknowns , u and are periodic in the x direction, we can formally derive the following energy dissipation law for the system (1.2)­(1.7) (the detailed computation can be found in Section ([30], p. 211, App.)) 1 2 d dt Z (t) |u|2 dx + d dt Z (t) a ( - 1) dx + 1 2 d dt L Z 0 |t|2 dx + 1 2 d dt L Z 0 |xx|2 dx + µ Z (t) |curlu|2 dx + (µ0 + 2µ) Z (t) |divu|2 dx + L Z 0 |tx|2 dx = -Pext Z s t. (1.10) Let us also point out that a similar simplified expression of stress tensor was considered in [21] and [20]. We would like to refer the readers to the Remark 2.3 for the technical details behind considering the simplified model (1.2)­(1.7). 1.1.2. Control Problem and Extension Arguments Our goal will be to discuss a control problem with controls acting from the boundary in the x-variable. So far, we did not make precise the boundary conditions in the x-variable, as the controls we shall consider will precisely act on these boundaries. But in fact, the boundary control functions will never appear explicitly, as we will first do an extension argument in the direction of the channel and then study the distributed controllability for (1.2)­(1.7) in the extended domain with controls localized in the extension of the domain. We thus take L > 0 and embed s into TL × {1} and into TL × (0, 1) where TL is the one dimensional torus identified with (-L, d + L) with periodic conditions. Then we consider the controls ve t (for the density), vue t (for the velocity) and ve b,t (for the beam), where e t and e b,t are the characteristics functions of the sets e t and e b,t which are defined as follows e t = {(x, y) | x [-L, 0), 0 < y < 1 + (x, t)} {(x, y) | x (d, d + L], 0 < y < 1 + (x, t)}, e b,t = {(x, y) | x [-L, 0), y = 1 + (x, t)} {(x, y) | x (d, d + L], y = 1 + (x, t)}. (1.11) Notice that e b,t is the boundary of e t (Fig. 2). To write the control system we further introduce the following notations ex t = {(x, y) | x TL, 0 < y < 1 + (x, t)} = extended domain of the fluid at time t, ex s,t = {(x, y) | x TL, y = 1 + (x, t)} = the extended beam at time t. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 5 Figure 2. TL = (-L, d + L), e t = e 1 t e 2 t . Our control system then reads as follows t + div(u) = ve t in t(0,T ) ex t × {t}, (tu + (u.)u) - µu - (µ + µ0 )divu + P() = vue t in t(0,T ) ex t × {t}, u2 = t + xu1 on t(0,T ) ex s,t × {t}, u2 = 0 on (TL × {0}) × (0, T), curl(u) = 0 on t(0,T ) ex s,t × {t}, curl(u) = 0 on (TL × {0}) × (0, T), u(·, 0) = u0 in TL × (0, 1), (·, 0) = 0 in TL × (0, 1), tt - txx + xxxx = (Tf )2 + ve b,t on (TL × {1}) × (0, T), (·, 0) = 0 and t(·, 0) = 1 in TL × {1}. (1.12) It is standard to deduce a boundary controllability result for the system (1.2)­(1.7) from a controllability result of the system (1.12) by restricting the data at the boundaries in the x-variable. 1.1.3. Transformation of the problem to a fixed domain To transform the system (1.12) in the reference configuration, for satisfying 1 + (x, t) > 0 for all (x, t) TL × (0, T), we introduce the following changes of variables (t) : ex t - TL × (0, 1) defined by (t)(x, y) = (x, z) = x, y 1 + (x, t) , : t(0,T )ex t × {t} - (TL × (0, 1)) × (0, T) defined by (x, y, t) = (x, z, t) = x, y 1 + (x, t) , t . (1.13) Remark 1.2. It is easy to prove that for each t [0, T), the map (t) is a C1 - diffeomorphism from ex t onto TL × (0, 1) provided that (1 + (x, t)) > 0, for all x TL and that (·, t) C1 (TL). 6 S. MITRA Observe that the map (t) can be uniquely extended to the boundary ex s,t with values in TL × {1}, by using the same formula (1.13)1. With the change of variable (t) (introduced in (1.13)), the control zones in the reference configuration are written as follows = ((-L, 0) × (0, 1)) ((d, d + L] × (0, 1)) = {(t)(x, y) | (x, y) e t}, b = ((-L, 0) × {1}) ((d, d + L] × {1}) = {(t)(x, y) | (x, y) e b,t}. (1.14) We set the following notations b (x, z, t) = (-1 (x, z, t)), b u(x, z, t) = (b u1, b u2) = u(-1 (x, z, t)), (x, z, t) (TL × (0, 1)) × (0, T), b 0(x, z) = 0(-1 (0)(x, z)), b u0(x, z) = u0(-1 (0)(x, z)), (x, z) TL × (0, 1), vb (x, z, t) = ve t (-1 (x, z, t)), vb u(x, z, t) = vue t (-1 (x, z, t)), (x, z, t) (TL × (0, 1)) × (0, T), vb (x, 1, t) = ve b,t (-1 (x, 1, t)), x TL and t (0, T). (1.15) After transformation the nonlinear control problem (1.12) is rewritten as following t b + b u1 1 (1+) (b u2 - zt - zb u1x) · b + b divb u = F1(b , b u, ) + vb in (TL × (0, 1)) × (0, T), b (tb u + (b u · )b u) - µb u - (µ0 + µ)(divb u) + P(b ) = F2(b , b u, ) + vb u in (TL × (0, 1)) × (0, T), b u · n = b u2 = t + xb u1 on (TL × {1}) × (0, T), b u(·, t) · n = 0 on (TL × {0}) × (0, T), curlb u = z b u1 - xb u2 = z b u1 (1 + ) - zxz b u2 (1 + ) on ((TL × {0, 1}) × (0, T), b u(·, 0) = b u0 in TL × (0, 1), b (·, 0) = b 0 in TL × (0, 1), tt - txx + xxxx = - (µ0 + 2µ)divb u + P(b ) + F3(b , b u, ) + vb on (TL × {1}) × (0, T), (0) = 0 and t(0) = 1 in TL × {1}, (1.16) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 7 where the non homogeneous terms F1(b , b u, ), F2(b , b u, ) and F3(b , b u, ), are non linear in its arguments and are given by F1(b , b u, ) = 1 (1 + ) zz b u1xb + b z b u2 , F2(b , b u, ) = -b tb u + zb z b ut - b b u1xb u + b u1z b uxb z + µ xxb u - zz b u (1 + ) - 2xzxz b u + zz b uz2 (x)2 (1 + ) + z b u (1 + )zxx - 2(x)2 z (1 + ) + (µ + µ0 )· b xxu1 - xz b u1zx - xz xz b u1 - zz b u1zx (1 + ) + z b u1 (1 + )zxx - 2(x)2 z (1 + ) - xz b u2 (1 + ) - xzzz b u2 (1 + ) - xz b u1 (1 + ) - xzzz b u1 (1 + ) - zz b u2 (1 + ) - (xP(b ) - zP(b )zx)~ e1, F3(b , b u, ) = (µ0 + 2µ) zz b u1x (1 + ) - z b u2 (1 + ) , (1.17) and n denotes the unit normal to the boundary (TL × {0, 1}) of the extended reference domain (TL × (0, 1)), i.e. n = (0, 1) on TL × {1}, (0, -1) on TL × {0}. In view of the expression above of the normal to the boundary at the fixed reference configuration, from now on we will just use the notation w2 and -w2 to denote w · n on TL × {1} and TL × {0} respectively for a vector field w. The interface boundary condition (1.16)5 is obtained by considering the trace of the following calculation (which makes use of chain rule of derivatives): curl u = y u1 - xu2 = z b u1yz - xb u2 - z b u2xz = curl b u - z b u1 (1 + ) + zxz b u2 (1 + ) , and finally using the relation curl u = 0 on t(0,T )ex s,t × {t}. Since = 0 on TL × {0}, (as we do not consider a structure on TL × {0}) (1.16)5 of course implies that curl b u = 0 on TL × {0}. 1.1.4. A control problem for the linearized model around the stationary state (, u, 0) We first pose the question of local exact controllability to the steady state (, u, 0), where > 0 is such that Pext = a and u = u1 0 , u1 > 0 is a constant, (1.18) which is obviously a stationary solution of the system (1.2)­(1.7) or (1.12) without the control functions. To be more precise, if (0, u0, 0, 1) is close in a suitable topology to (, u, 0, 0), then the question of local exact controllability of (1.12), aims in finding control functions (v, vu, v) such that the solution of (1.12) satisfies ((T), u(T), (T), t(T)) = (, u, 0, 0). 8 S. MITRA Since we give a sense to the solution of system (1.12) by posing it into a fixed domain, i.e. the system (1.16) we will rather talk about the controllability of (b , b u, , t) around (, u, 0, 0). Of course (, u, 0) is also a stationary solution to (1.16) without the control functions. The following change of unknowns e = b - , e u = b u - u, = - 0, (1.19) reduces the controllability problem to the state (, u, 0, 0) for (b , b u, , t) into a local null controllability problem for (e , e u, , t). We further introduce the following notations corresponding to the control functions which are consistent with the new unknowns defined in (1.19). ve = vb , ve u = vb u. (1.20) In fact, as we would like to obtain a local null-controllability problem for (e , e u, , t), it seems reasonable to start by considering the linearized problem around the state (0, 0, 0, 0). Starting from system (1.16) and the non-linear terms (1.17) and dropping all the non-linear terms in (e , e u, , t), we obtain te + u1xe + dive u = ve in (TL × (0, 1)) × (0, T), (te u + u1xe u) - µe u - (µ0 + µ)(dive u) +P0 ()e = ve u in (TL × (0, 1)) × (0, T), e u2 = t + u1x on (TL × {1}) × (0, T), e u2 = 0 on (TL × {0}) × (0, T), curle u = 0 on ((TL × {0, 1}) × (0, T), e u(·, 0) = b u0 - u = e u0 in TL × (0, 1), e (·, 0) = b 0 - = e 0 in TL × (0, 1), tt - txx + xxxx = - (µ0 + 2µ)dive u + P0 ()e ) + vb on (TL × {1}) × (0, T), (0) = 0 and t(0) = 1 in TL × {1}. (1.21) In the present article we will prove an observability inequality corresponding to the adjoint of the system (1.21). The observability inequality proved here implies by duality the null controllability of the linear system (1.21). But the observation will be obtained using strong norms of the unknowns which will only allow the attainment of the null controllability result for (1.21) in a very weak sense (i.e. in some negative order Sobolev spaces), which are not enough to pass from the null controllability of the linear system (1.21) to the local exact controllability of the non linear model (1.16). This is the reason why we will only present the observability result for the adjoint of (1.21) without stating the corresponding controllability of (1.21) in negative order Sobolev spaces. 1.2. Main result: Observability of the adjoint of (1.21) In order to study the null controllability of the linearized problem (1.21), the classical strategy is to prove the observability of the adjoint system of (1.21). Here, the adjoint system of (1.21) is computed with respect OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 9 to the scalar product L2 (TL × (0, 1)) × L2 (TL × (0, 1)) × L2 (TL × {1}) × L2 (TL × {1}) and reads as follows: -t - u1x - P0 ()divv = 0 in (TL × (0, 1)) × (0, T), -(tv + u1xv) - µv - (µ0 + µ)(divv) - = 0 in (TL × (0, 1)) × (0, T), v2 = on (TL × {1}) × (0, T), v2 = 0 on (TL × {0}) × (0, T), curlv = 0 on ((TL × {0, 1}) × (0, T), v(·, T) = vT in TL × (0, 1), (·, T) = T in TL × (0, 1), tt + txx + xxxx = (t + u1x)[(µ0 + 2µ)div v + ] on (TL × {1}) × (0, T), (T) = T and t(T) = 1 T in TL × {1}. (1.22) A formal derivation of the adjoint model (1.22) from (1.21) is given in the Appendix, Section A.1. The well- posedness of system (1.22) is stated as the following result, which is proved in Section 2.1: Theorem 1.3. Let (T , vT , T , 1 T ) H2 (TL × (0, 1)) × H3 (TL × (0, 1)) × H3 (TL × {1}) × H1 (TL × {1}), (1.23) and the following compatibility relations hold (i) (a) (vT )2 = T , on TL × {1}, (b) (vT )2 = 0, on TL × {0}, (ii) curl vT = 0, on TL × {0, 1} (iii) (a) -u1x(vT )2 + µ (vT )2 + (µ+µ0 ) z(div vT ) -zT = 1 T on TL × {1}, (b) -u1x(vT )2 + µ (vT )2 + (µ+µ0 ) z(div vT ) -zT = 0 on TL × {0}. (1.24) Then the system (1.22) admits a unique solution (, v, ) which satisfies the following regularity C0 ([0, T]; H2 (TL × (0, 1))) C1 ([0, T]; H1 (TL × (0, 1))), v L2 (0, T; H3 (TL × (0, 1))) H1 (0, T; H2 (TL × (0, 1))) H2 (0, T; L2 (TL × (0, 1))), L2 (0, T; H4 (TL × {1})) H1 (0, T; H2 (TL × {1})) H2 (0, T; L2 (TL × {1})). (1.25) Remark 1.4. A similar well posedness result can be proved for the linear primal problem (1.21) which we do not state here. For the statement and the proof of this result one can consult ([30], Thm. 4.1.3, p. 153) and ([30], Sect. 4.2.2, p. 172). The central result of the present article is the observability inequality of the adjoint system (1.22): Theorem 1.5. Let (, u, 0) be as in (1.18), T > 0 be such that T > d u1 , (1.26) 10 S. MITRA and L = 5u1T > 0. (1.27) There exists a positive constant C such that for all (T , vT , T , 1 T ) H2 (TL × (0, 1)) × H3 (TL × (0, 1)) × H3 (TL × {1}) × H1 (TL × {1}), (1.28) satisfying the compatibility conditions (1.24), then the solution (, v, ) of the problem (1.22) (in the sense of Theorem 1.3) satisfies the following observability inequality: k(·, 0)kH1(TL×(0,1)) + kv(·, 0)kH2(TL×(0,1)) + k((·, 0), t(·, 0))kH3(TL×{1}))×H1(TL×{1}) 6 CkkL2(T b ) + CkvkL2(0,T ;H2())H1(0,T ;H1()) + CkkL2(0,T ;H1()). (1.29) where the notations and b for the observation sets were introduced in (1.14) and T = × (0, T), T b = b × (0, T). Comments on the choice of T and L in Theorem 1.5: At a first glance the assumption (1.26) containing d might seem surprising since d does not explicitly appear in (1.22) and (1.29). The reason is that now we are dealing with the extended fluid domain TL × (0, 1) and the choice (1.26) asserts that L > 5d. We recall (1.26) and (1.27). The condition (1.26) means that the time of observability should be greater than the time taken to cross the channel length d by a particle moving with a velocity (u1, 0). This condition is imposed due to the hyperbolic nature of the transport equation satisfied by in the system (1.22). In fact this condition plays a key role in obtaining the observability estimate for a hyperbolic transport equation in Section 3.4. Our proof in Section 3.4 depends on a duality argument and a controllability result for the dual to the problem considered in Section 3.4 which is obtained in [14]. Hence one can consult [14] for a more explicit construction of the controlled trajectory where the restriction of the final time (1.26) comes into play. The role of (1.27) can be better explained after we define some Carleman weights in Section 3. Hence we refer the readers to the Remark 3.1 for the explanation behind choosing L as in (1.27). Still, let us mention that from the control point of view, the value of L > 0 does not play any role in the restriction argument presented in Section 1.1.2. The following is a corollary to Theorem 1.5 and corresponds to the unique continuation property for the system (1.22): Corollary 1.6. Let (, u, 0) be as in (1.18), T > 0 and L > 0 satisfies (1.26) and (1.27). Further let (T , vT , T , 1 T ) satisfies (1.28) along with the compatibility conditions (1.24). If the solution (, v, ) of the problem (1.22) (in the sense of Thm. 1.3) solves (, v) = (0, 0) in T and = 0 in T b , then (, v, ) = 0 in ((TL × (0, 1)) × (0, T))2 × ((TL × {1}) × (0, T)), where T = × (0, T), T b = b × (0, T). The proof of Corollary 1.6 will follow from an intermediate step in the proof of Theorem 1.5. The proof is included in Section 4.3. Remark 1.7. As a special case the results Theorem 1.5 and Corollary 1.6 imply the observability and unique continuation of the adjoint of linearized compressible Navier-Stokes equations in a 2D channel where the fluid OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 11 velocity satisfies the Navier-slip boundary condition without friction at the lateral boundaries i.e v2 = 0 and (2µD(v) + µ0 div vId)n · ~ = 0 on (TL × {0, 1}), where n and ~ respectively denotes the normal and tangent to the boundary. To the best of our knowledge this result is new in itself. The only articles so far dealing with the controllability and observability issues of compressible Navier-Stokes equations in dimension more than one are [14] and [33] where the problem is posed in a torus. 1.3. Ideas and strategy Now we will briefly discuss our ideas and strategy to prove Theorem 1.5. The underlying idea behind the proof of Theorem (1.5) is the identification of the suitable unknowns to track down the dynamics of (, v, ). It is well known that the coupling of and v is strong. When considering the primal problem (1.21), the dynamics between e , e u can be made simpler by introducing the effective viscous flux, see [29] and [17]. For the adjoint problem, a similar quantity, already used in [14], also simplifies the description of the dynamics: q = (µ0 + 2µ)div v + . (1.30) This can be termed as the dual version of the effective viscous flux. Now in our case it is important to identify the behavior of q at the boundaries and specially at the fluid solid interface. This way we obtain a closed loop system solved by (, q, ). For details we refer the readers to Section 2. One can in particular look into the system (2.3) to observe that unlike the coupling between and v in system (1.22), the coupling between and q is of lower order. Also it is easier to deal with (, q, ), since it has less degrees of freedom in comparison with (, v, ). We use this new set of unknowns (, q, ) both to prove the well posedness result stated in Theorem 1.3 and the observability Theorem 1.5. In Section 2 we prove Theorem 1.3. Next we focus in proving an observability inequality for the system satisfied by (, q, ). In that direction we first separately study the observability inequalities of some scalar equations i.e an adjoint damped beam equation, an adjoint heat equation and an adjoint transport equation. The observability estimates for the adjoint damped beam and the adjoint heat equations rely on Carleman estimates while for the adjoint transport equation we use a duality argument and some controllability estimates motivated from [14]. The main difficulty here is to obtain these separate observability estimates with a single goal of combining them suitably to obtain an observability for the coupled system solved by (, q, ). For the parabolic hyperbolic couplings this question is handled in the articles [1], [14] (for compressible Navier-Stokes equations), [33] (for compressible heat conducting fluid) and [8] (for damped viscoelasticity equations). The idea is to use compatible weight functions for the parabolic and hyperbolic equations so that the resulting observability estimates can be suitably combined. In our case along with a parabolic hyperbolic coupling there is a direct coupling between q and at the fluid boundary (see the system (2.3) for details). Hence to use the ideas from [1], [8] and [14] in our case there is not many options other than considering a one dimensional weight function, since the beam is one dimensional. We introduce such weight function in Section 3.1. Then using this weight function we state a Carleman estimate for the adjoint damped beam equation, see Section 3.2 for details. This Carleman estimate is taken from a very recent article [31] or more appropriately from ([30], p. 182, Sect. 4.3.2). Then using the same weight function for an adjoint heat equation with Neumann boundary condition we recover a Carleman estimate proved in [18]. Of course the weight functions considered in [18] and the present article are not the same. This Carleman estimate is included in Section 3.3. In the beginning of Section 3.3, we also point out the difference between our weight function used in proving the Carleman estimate for an adjoint heat equation with the one standard in the literature. Then the same weight function is used to obtain an observability estimate for an adjoint transport equation in Section 3.4. This is done first by obtaining some controllability estimates in the spirit of [14] followed by a duality argument. 12 S. MITRA Next in Section 4 we combine the Carleman estimates obtained in Section 3. First using suitably large values of the parameters we are able to prove a Carleman estimate (4.1) and an inequality corresponding to the unique continuation property for the system satisfied by (, q, ). This inequality is explicitly given by (4.9). It is not surprising to obtain an estimate of the form (4.9) by combining three different observability estimates. In fact as it is well known that the Carleman parameters quantify the compactness of a system hence our strategy to obtain the inequality (4.9) strongly relates on the proof of Step 1 in Theorem 1.3, where we prove the well posedness of (2.3) by gaining a time integrability of a suitable fixed point map. After this unique continuation estimate is used to show an observability estimate of (, q, ) at some intermediate time, see (4.10) for details. We further use a well posedness result for the system satisfied by (, q, ) to obtain an observability estimate over (, q, ) at initial time t = 0. Finally using this observability estimate over (, q, )(·, 0) we recover an observability estimate of (, div v, )(·, 0), which combined with an observability inequality of curl v(·, 0) furnishes the desired inequality (1.29). 1.4. Related bibliography Concerning the incompressible Navier-Stokes equations in a 2D domain one can find a result proving the local exact controllability to trajectories with localized boundary control in [23]. It is assumed in [23] that the fluid satisfies no vorticity boundary condition in the complement of the control part of the boundary. Local exact controllability to trajectories for incompressible Navier-Stokes equations in a 3D domain with distributed control and homogeneous Dirichlet boundary condition can be found in [26]. With less regularity assumption on the target trajectory the result in [26] was improved in [19]. We would also like to mention the article [24] for the local exact distributed controllability to trajectories for incompressible Navier-Stokes equations in a 3D domain with non linear Navier-slip boundary condition. In all of these articles the fluid is assumed to be homogeneous i.e the fluid density is constant. In a very recent article [2] the authors prove the local exact boundary controllability to smooth trajectories for a non homogeneous incompressible Navier-Stokes equation in a three dimensional domain. For global controllability results for incompressible Navier-Stokes equations we refer the readers to [12], [7] and the references therein. Now we quote a few articles dealing with the controllability issues of fluid structure interaction models. In fact to the best of our knowledge the only known results concerning the controllability issues of a fluid structure interaction problem in dimension greater than one deals with the motion of a rigid body inside a incompressible fluid modeled by Navier-Stokes equations where the structural motion are given by the balance of linear and angular momentum. Local null controllability of such an interaction problem in dimension two can be found in [5] and [25]. In dimension three a local null controllability for such a system is proved in [4]. The article [35] deals with the problem of feedback stabilization (in infinite time) for an incompressible fluid structure interaction problem in a 2D channel where the structure appears at the fluid boundary and is modeled by an Euler-Bernoulli damped beam, the one we consider in (1.21)8­(1.21)9. To our knowledge so far there does not exist any article dealing with the finite time controllability of a fluid structure interaction problem (neither for incompressible nor compressible fluids) in dimension more than one where the structure appears at the fluid boundary. We would also like to refer the readers to [22], [41] and [42] for observability estimates individually for the Euler-Bernoulli plate equations and Kirchoff plate systems without damping. We also like to quote a few articles from the literature dealing with the controllability issues of compressible Navier-Stokes equation. In fact our strategy to handle the coupling of the fluid velocity and density in the system (1.21) amounts in introducing a new unknown namely the effective viscous flux and this strategy is inspired from the article [14]. The article [15] concerns the motion of a fluid in dimension one whereas [14] and [33] deal with fluid flows in dimension two and three. For the controllability issues of one dimensional compressible Navier Stokes equations we also refer the readers to [10] and [11]. Concerning the unique continuation property of incompressible Stokes and Oseen equations we refer the readers to [16] and [38]. For the use of Carleman estimates in proving controllability and unique continuation properties of parabolic and elliptic PDE's one can also consult [37]. The article [14] dealing with the local exact controllability of the compressible Navier-Stokes equations set in a three dimensional torus of course OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 13 proves observability results for the adjoint of the corresponding linearized system which is stronger than unique continuation. But to the best of our knowledge there is no unique continuation result available even for linear compressible Navier-Stokes equations (without the structure) in dimension more than one where the domain is not entirely a torus. As a particular situation Corollary 1.6 implies the unique continuation property for the adjoint of linear compressible Navier-Stokes equations in a 2D channel TL × (0, 1) where the fluid velocity solves Navier-slip boundary conditions without friction at the lateral boundaries of the channel. 1.5. Outline Section 2 is devoted for the proof of the well posedness result Theorem 1.3. In Section 3 we prove several observability estimates. More precisely in Section 3.2 we state a Carleman estimate for the adjoint of a damped beam equation, in Section 3.3 we prove Carleman estimate for the adjoint of a heat equation and finally in Section 3.4 we prove some observability estimates for a transport equation. In Section 4, we suitably combine the Carleman estimates obtained in Section 3 to furnish the proof of the central result Theorem 1.5. We also include the proof of Corollary 1.6 in Section 4.3. The final Section A contains the formal derivation of the adjoint system, the proof of Lemma 2.2, which is a intermediate step in proving Theorem 1.5, and a result on parabolic regularization, Lemma A.1 which in turn is used during the proof of Lemma 4.2. 2. Well posedness result for the adjoint problem (1.22) This section is devoted for the proof of Theorem 1.3. As it turns out, this proof will also give some insights of the strength of the various coupling between the equations in (1.22), which will also help in the proof of Theorem 1.5. 2.1. Proof of Theorem 1.3 The proof is divided into two main steps. The first one consists in introducing the new unknown q = (µ0 + 2µ)div v + , (2.1) and looking at the system satisfied by (, q, ), which turns out to be easier to analyze than the full system (1.22). The second step will then consist in deducing from the regularity on (, q, ) suitable estimates for the function v in (1.22). Remark 2.1. The unknown q in (2.1) can be interpreted as the dual version of the effective viscous flux introduced for instance in [29], see also [17]. In fact, this quantity already appeared in [14] when studying the controllability properties of a compressible fluid (without structure and controls acting on the whole boundary), where it helps to weaken the coupling of the parabolic and hyperbolic effects of the system. Here, the interesting point is that this quantity is also suitable to deal with the coupling with the structure lying on the boundary. There, we strongly use that the force acting on the beam is given by (1.7) instead of the more natural one (1.9), which would not yield such a clean understanding of the coupling between the fluid and the structure. 2.1.1. Step 1. She system Satisfied by (, q, ) We first derive the system that (, q, ) should satisfy provided (, v, ) satisfies (1.22) and has the regularity given by (1.25). Indeed, these regularities allow to take the trace of 2 v and a.e on (TL × {0, 1}) × (0, T). Hence we can consider the trace of the equation (1.22)2 and use (1.22)3­(1.22)5 to have the following a.e on 14 S. MITRA ((TL × {1})) × (0, T) : 0 = -(tv2 + u1xv2) - µ(xxv2 + zzv2) - (µ + µ0 )(xzv1 + zzv2) - z = -(t + u1x) - (µ0 + 2µ)(xzv1 + zzv2) - z (using (1.22)3, (1.22)5) = -(t + u1x) - zq. (2.2) Similarly one can obtain that on the boundary (TL × {0}) × (0, T), q satisfies zq = 0 a.e on (TL × {0}) × (0, T). Hence with the formal calculations above and using (1.22) we obtain the following system satisfied by the unknowns (, q, ) : -t - u1x + P0 () = P0 () q in (TL × (0, 1)) × (0, T), -(tq + u1xq) - q - P0 () q = - P0 ()2 in (TL × (0, 1)) × (0, T), zq = -(t + u1x) on (TL × {1}) × (0, T), zq = 0 on (TL × {0}) × (0, T), q(·, T) = qT in TL × (0, 1), (·, T) = T in TL × (0, 1), tt + txx + xxxx = (t + u1x)q on (TL × {1}) × (0, T), (T) = T and t(T) = 1 T in TL × {1}, (2.3) where = (µ0 + 2µ), and qT = divvT + T . The well-posedness result for the system (2.3) is stated in form of the following lemma: Lemma 2.2. There exists a constant C > 0 such that for any (T , qT , T , 1 T ) H1 (TL × (0, 1)) × H2 (TL × (0, 1)) × H3 (TL × {1}) × H1 (TL × {1}) (2.4) satisfying the following compatibility conditions (i) zqT = -(1 T + u1xT ) on TL × {1}, (ii) zqT = 0 on TL × {0}, (2.5) the system (2.3) admits a unique solution (, q, ) which satisfies C0 ([0, T]; H1 (TL × (0, 1))) C1 ([0, T]; L2 (TL × (0, 1))), q L2 (0, T; H3 (TL × (0, 1))) H3/2 (0, T; L2 (TL × (0, 1))), L2 (0, T; H4 (TL × {1})) H1 (0, T; H2 (TL × {1})) H2 (0, T; L2 (TL × {1})). (2.6) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 15 and the following estimate kkC0([0,T ];H1(TL×(0,1)))C1([0,T ];L2(TL×(0,1))) + kqkC0([0,T ];H2(TL×(0,1))) + kkC0([0,T ];H3(TL×{1}))C1([0,T ];H1(TL×{1})) 6 C(kqT kH2(TL×(0,1)) + kT kH1(TL×(0,1)) + k(T , 1 T )kH3(TL×{1})×H1(TL×{1})). (2.7) The proof of Lemma 2.2 is included in Section A. Next using Lemma 2.2 we will obtain the regularities for v and further prove that the triplet (, v, ) solves the system 1.22. 2.1.2. Step 2: Constructing v In order to complete the proof of the existence of a solution (, v, ) of (1.22), we first set qT = divvT + T and solve the system (2.3) with initial data (T , qT , T , 1 T ). Note that the regularity and compatibility conditions of the initial data (T , vT , T , 1 T ) in Theorem 1.3 precisely imply the regularity and compatibility conditions of (T , qT , T , 1 T ) required by Lemma 2.2, so that Lemma 2.2 applies, yielding functions (, q, ) solving (2.3). Besides, if T H2 (TL × (0, 1)), it is clear that the solution of (2.3)(1) in fact satisfies C0 ([0, T]; H2 (TL × (0, 1))) C1 ([0, T]; H1 (TL × (0, 1))), since the source term of the transport equation belongs to C0 ([0, T]; H1 (TL × (0, 1))) (by using interpolation result ([27], p. 19, Thm. 3.1) and (2.6)2). For a detailed proof one can imitate the arguments used in proving ([32], Thm. 2.4). We then construct the function v by solving the equation -(tv + u1xv) - µv - (µ0 + µ)(divv) - = 0 in (TL × (0, 1)) × (0, T), v2 = on (TL × {1}) × (0, T), v2 = 0 on (TL × {0}) × (0, T), curlv = 0 on (TL × {0, 1}) × (0, T), v(·, T) = vT in TL × (0, 1). (2.8) Note that we already know the regularity of the source terms and from (2.6), so this step only consists in constructing a solution to a parabolic equation with source terms having given regularities. One can prove the following regularities of v : v L2 (0, T; H3 (TL × (0, 1))) H1 (0, T; H2 (TL × (0, 1))) H2 (0, T; L2 (TL × (0, 1))), (2.9) solving the system (2.8). The proof of such a parabolic regularity result for the system (2.8) with no vorticity boundary condition is classical in the literature. For the details of the proof we refer the readers to ([30], p. 168, Sect. 4.2.1.2). With the strong regularity framework (2.9) and with in the space (2.6) one can verify the formal argument (2.2) and conclude that the triplet (, v, ) solves the system (1.22) in the functional framework (1.25). Now in order to prove the uniqueness of the solution (, v, ) of (1.22) in the spaces (1.25), we proceed as follows. Let us assume that there exists two solutions (1, v1, 1) and (2, v2, 2) to the problem (1.22) in the functional spaces (1.25) with the same initial datum. The strong regularities of v1 (as well as v2) and 1 (as well as 2) allows us to verify that (1, q1, 1) and (2, q2, 2) solves (2.3), where q1 = (div v1 + 1) and q2 = (div v2 + 2). 16 S. MITRA But the solution of the system (2.3) is unique (thanks to the Banach fixed point argument used in Step 2) in the framework (2.6). Hence 1 = 2 and 1 = 2. Now one observes that v1 (respectively v2) solves (2.8) with (1, 1) (respectively (2, 2)). Since (1, 1) = (2, 2), from the uniqueness of the solution to the linear problem (2.8) we infer that v1 = v2. Hence the solution to the problem (1.22) is unique in the functional framework (1.25). This concludes the proof of Theorem 1.3. Remark 2.3. At this point the role of the dual version of effective viscous flux q is clear in order to prove the existence result Theorem 1.3. We now explain more clearly the reason behind considering the simplified model (1.2)­(1.7) instead of using the more physical expression (1.9) (recall Remark 1.1). In our analysis we use the dual version of effective viscous flux to reduce the strength of the parabolic hyperbolic coupling in the system (1.22). With the simplified expression (1.7) of the net surface force on the beam we are able to write the adjoint of the linearization of the system (1.2)­(1.7) as a closed loop system in terms of the unknowns: dual of fluid density, beam displacement and q. On the other hand it does not seem to be possible if one considers the expression (1.9) for the net force acting on the beam. In some sense the effective viscous flux does not prove to be a good unknown to weaken the parabolic hyperbolic coupling near the boundary for the adjoint to the system (1.2)­(1.9). 3. Carleman estimates for scalar equations From now onwards we fix our final time horizon T and the length L of our torus such that they satisfy (1.26) and (1.27) respectively. The goal of this section is to provide Carleman estimates for the various equations involved in the system (1.22), in particular: ­ a Carleman estimate for the beam equation set in the torus; ­ a Carleman estimate for the heat equation with non-homogeneous Neumann boundary conditions; ­ a Carleman estimate for the transport equation. One of the difficulties of our work is that, in order to prove Theorem 1.5, one should be able to couple all these Carleman estimates in a suitable way. In order to do that, we will consider one weight function which allows to derive Carleman estimates for the beam equation, the heat equation and the transport equation simultaneously. 3.1. Construction of the weight function 1. We first introduce a function on TL such that C6 (TL), (x) > 0 in TL, inf {|(x)| | x TL \ {(-5u1T, -3u1T) (d + u1T, d + 3u1T)}} > 0, (3.1) 2. Now we define 0 C6 (TL × [0, T]) as follows 0 (x, t) = (x - u1t) for all (x, t) TL × [0, T], (3.2) i.e 0 solves (t + u1x)0 = 0 in (TL × (0, 1)) × (0, T). (3.3) In view of (3.1), one can easily verify the following inf |0 (x, t)| (x, t) [-u1T, d + u1T] × [0, T] > 0, (3.4) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 17 3. Next we will define a weight function in the time variable. We now recall (1.26). Let T0 > 0, T1 > 0 and > 0, small enough such that 2T0 + 2T1 < T - d + 12 u1 . (3.5) Now we choose a weight function (t) C4 (0, T) such that (t) = 1 t2 , t [0, T0], is strictly decreasing t [T0, 2T0], 1 t [2T0, T - 2T1], is strictly increasing t [T - 2T1, T - T1], 1 (T - t)2 , t [T - T1, T]. (3.6) Observe that (t) blows up at the terminal points {0} and {T} of the interval (0, T). The choice of such a , specifically the fact that is constant in a sub interval [2T0, T - 2T1] plays a important role in proving the controllability of transport equation. 4. In view of 0 (x, t) and (t) we finally introduce the following weight functions in TL × [0, T], ( (x, t) = (t)(e6k0 k - e(0 (x,t)+4k0 k) ), (x, t) = (t)e(0 (x,t)+4k0 k) , (3.7) where > 1 is a positive parameter. Remark 3.1. Recall that we have fixed L = 3u1T. The reason lies in the choice (3.2) of 0 which travels along TL with a velocity u1. Of course this choice plays a very important role in Section 3.4 while obtaining an observability estimate for a hyperbolic transport equation. In that case the domain (0, d) × (0, 1) needs to be embedded in TL × (0, 1), for L large enough such that inf{|0 (·, t)|} is positive in a neighborhood of (0, d) for all t [0, T], since this is crucial to obtain parabolic Carleman estimates i.e. the Carleman estimates for the damped beam and heat equations (stated respectively in Thms. 3.2 and 3.3). Now the choice L = 3u1T serves this purpose and provides enough room so that (3.4) holds. From now on we will denote by c, a generic strictly positive small constant and by C, a large constant, where both of them are independent of the parameters s (> 1) and (> 1). In our computations afterwards we will frequently use the following estimates, valid on TL × (0, T): |(i) x | 6 Ci for all i {1, 2, 3, 4}, |t| 6 C3/2 , |tt| 6 C2 2 , |tx| 6 C2 3/2 , |txx| 6 C3 3/2 , |txxx| 6 C4 3/2 , |ttx| 6 C3 2 and |ttxx| 6 C4 2 , (3.8) and |(i) x | 6 Ci for all i {1, 2, 3, 4}, |t| 6 C3/2 , |tt| 6 C2 2 , |tx| 6 C2 3/2 , |txx| 6 C3 3/2 |txxx| 6 C4 3/2 , |ttx| 6 C3 2 and |ttxx| 6 C4 2 , (3.9) and, for large enough, for all (x, t) [-u1T, d + u1T] × (0, T) and i {2, 4}, - (i) x = (i) x > ci . (3.10) 18 S. MITRA We will need a few more properties of the weight functions for later analysis and we list them below. We will use the following inequality (t + u1x) (e2s ) 6 0 in (TL × (0, 1)) × (0, T - 2T1), (3.11) while proving Theorem 3.7 (specifically in obtaining (3.77) from (3.76)). The proof of (3.11) is easy and can be obtained in view of (3.3) and (3.6), especially the fact that (t) is constant in [2T0, T - 2T1]. The following inequalities will be important in obtaining an observability inequality in Lemma 4.2 from a Carleman estimate proved in Lemma 4.1: e-2s 6 C on T , i e-2s 6 C on T T b for i {-1, 1, 7}, c > 0, s.t. e-2s > c on (TL × (0, 1)) × (2T0, T - 2T1) (3.12) and i e-2s > c on (TL × {1}) × (2T0, T - 2T1) for i {-3, -1, 1, 3, 5, 7}. For convenience, we further introduce the following shorthand notations which will be used from now onwards mainly for writing the domain of integrals. Qex T = (TL × (0, 1)) × (0, T), T = × (0, T), T1 T = (TL × {1}) × (0, T), T0 T = (TL × {0}) × (0, T), T b = b × (0, T), (3.13) where and b was introduced in (1.14). 3.2. Carleman estimate for an adjoint damped beam equation In the following section we state a Carleman estimate for the adjoint of the damped beam equation: tt + txx + xxxx = f in T1 T , (., T) = T , t(., T) = 1 T in TL × {1}. (3.14) The main theorem of this section is stated as follows Theorem 3.2. There exist constants C > 0, s0 > 1, 0 > 1 such that for all solving (3.14) with initial datum T H3 (TL) and 1 T H1 (TL) and source term f L2 (T1 T ), for all s > s0, and > 0, s7 8 ZZ T1 T 7 ||2 e-2s + s5 6 ZZ T1 T 5 |x|2 e-2s + s3 4 ZZ T1 T 3 (|xx|2 + |t|2 )e-2s + s2 ZZ T1 T (|tx|2 + |xxx|2 )e-2s + 1 s ZZ T1 T 1 (|tt|2 + |txx|2 + |xxxx|2 )e-2s (3.15) 6 C ZZ T1 T |f|2 e-2s + Cs7 8 ZZ T b 7 ||2 e-2s , where the notations T1 T and T b were introduced in (3.13). We will not go into the details of the proof of Theorem 3.2 but only comment on it with suitable reference. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 19 Comment on the proof of Theorem 3.2. The Carleman estimate stated in Theorem 3.2 is already proved in the article [31] but with a difference in the weight functions 0 (x, t) and (t). To be precise unlike [31], in the present article we have used a weight function 0 (x, t) which depends on the time variable and further the definition (3.6) of (t) also varies from the one used in ([31], p. 2, (1.6)). This only leads to minor modifications in the estimates (3.8) and (3.9). The proof of Theorem 3.2 can be carried out essentially in a similar way as the proof of Carleman estimate ([31], Thm. 1.3). For a detailed proof of Theorem 3.2 with the weight functions defined in Section 3.1 we refer the readers to ([30], p. 182, Sect. 4.3.2). 3.3. Carleman estimate for an adjoint heat equation In this section we consider the following adjoint heat equation: - tq - q = f1 in Qex T , zq = f2 on T1 T , zq = 0 on T0 T , q(., T) = qT in TL × (0, 1), (3.16) where f1 L2 (Qex T ), f2 L2 (T1 T ). Theorem 3.3. There exist positive constants C, s1 > 1 and 1 > 1 such that for all f1 L2 (Qex T ), and f2 L2 (T1 T ), (3.17) for all qT L2 (TL × (0, 1)), for all s > s1 and > 1, the weak solution q of (3.16) satisfies the following inequality ZZ Qex T e-2s (s2 |q|2 + s3 4 3 |q|2 ) + s2 3 ZZ T1 T e-2s 2 |q|2 6 C ZZ Qex T e-2s |f1|2 + s ZZ T1 T e-2s |f2|2 + s3 4 ZZ T e-2s 3 |q|2 , (3.18) where the notations Qex T , T1 T and T are introduced in (3.13). Theorem 3.3 will be proved mainly by using the similar line of arguments as used to prove ([18], Thm. 1). The difference with [18] occurs in the construction of the weight functions. To be precise, the weight (t) in time is defined in [18] as (t) = 1 t(T -t) , whereas in our case (t) is as defined in (3.6). Further unlike [18], the function 0 (defined in (3.2)) travels in time with a constant velocity u1. Both of these differences can be classically handled just by using the estimates (3.8), (3.9) and (3.10) of and . One can also consult [2] for similar issues. Above all in [18] and in most other articles in the literature it is assumed that 0 vanishes at the boundary of the domain. Of course this assumption does not serve our purpose since we are working with a beam at the boundary. In our case, 0 just depends on (x, t) implying in particular z = z = 0 on T1 T T0 T . (3.19) Using this property, we still recover the same Carleman estimate for the heat equation with non homogeneous boundary condition obtained in [18]. Hence without going into the details we will just sketch the main steps of the proof of Theorem 3.3 and other supporting lemmas, we comment with references for their proofs. 20 S. MITRA For the proof of Theorem 3.3 we will need an auxiliary result: a Carleman inequality for heat equation with homogeneous Neumann boundary conditions, stated below: Lemma 3.4. There exist positive constants C, s2 > 1 and 2 > 1 such that for all s > s2, > 2, for all T L2 (TL × (0, 1)), and for all f3 L2 (Qex T ), the solution of the following problem - t - = f3 in Qex T , z = 0 on T1 T T0 T , (., T) = T in TL × (0, 1), (3.20) satisfies ZZ Qex T e-2s s2 ||2 + s3 4 3 ||2 6 C ZZ Qex T e-2s |f3|2 + s3 4 ZZ T e-2s 3 ||2 . (3.21) Lemma 3.4 will allow to construct by duality suitable solutions to a control problem: Lemma 3.5. There exist positive constants C > 0, s3 > 1 and 3 > 1, such that for all s > s3, > 3 and for all G satisfying ZZ Qex T -3 |G|2 e2s < , (3.22) there exists a solution (Y, H) of the following control problem tY - Y = G + H in Qex T , zY = 0 on T1 T T0 T , Y (·, 0) = 0 in TL × (0, 1), Y (·, T) = 0 in TL × (0, 1), (3.23) which satisfies Y L2 (0, T; H2 (TL × (0, 1))) H1 (0, T; L2 (TL × (0, 1))) C0 ([0, T]; H1 (TL × (0, 1))). and the following estimate: s3 4 ZZ Qex T |Y |2 e2s + s2 ZZ Qex T -2 |Y |2 e2s +s2 3 ZZ T1 T -1 |Y |2 e2s + ZZ T -3 |H|2 e2s 6 C ZZ Qex T -3 |G|2 e2s . (3.24) The proof of Theorem 3.3 then follows from Lemma 3.5 by duality. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 21 Details of the proof are given hereafter. Section 3.3.1 is dedicated to the proof of Lemma 3.4, Section 3.3.2 to the proof of Lemma 3.5, and Section 3.3.3 to the proof of Theorem 3.3. In Section 3.3.4, we prove an additional result, which is a corollary of Theorem 3.3: Corollary 3.6. There exist positive constants C, s2(> s1) and 2(> 1) (where s1 and 1 are the constants as in Theorem 3.3) such that if all the assumptions of Theorem 3.3 satisfied and if s > s2 and > 2, then the solution q of (3.16) satisfies the following inequality ZZ Qex T e-2s ( 1 s |q|2 + s2 |q|2 ) + ZZ T1 T e-2s |q|2 6 C 1 s22 ZZ Qex T e-2s 1 2 |f1|2 + 1 s ZZ T1 T e-2s 1 |f2|2 + s2 ZZ T e-2s |q|2 , (3.25) where the notations Qex T , T1 T are introduced in (3.13). 3.3.1. Proof of Lemma 3.4 Proof. We set 1 = e-s . (3.26) In view of (3.19) one observes that z1 = 0 on T1 T T0 T . (3.27) Besides, with f3 as in (3.20), 1 satisfies e-s f3 = e-s (- t - ) = e-s (- t(es 1) - (es 1)) = P1, where the operator P can be written as P = P11 + P21 + R1, where P11 = - t1 + 2s0 · 1 + 2s2 |0 |2 1, P21 = -1 - st1 - s2 2 2 |0 |2 1, R1 = -s0 1 + s2 |0 |2 1. (3.28) We then use that P11 + P21 = f3e-s - R1 and then ZZ Qex T |P11|2 + ZZ Qex T |P21|2 + 2 ZZ Qex T P11P21 = ZZ Qex T |f3e-s - R1|2 6 2 ZZ Qex T |f3|2 e-2s + 2 ZZ Qex T |R1|2 . (3.29) 22 S. MITRA We now compute the scalar product of P11 with P21. In fact, these computations are very similar to the classical ones, and one should only remark that the integrations by parts do not yield any bad terms on the boundaries of the domain. We shall write LO.T. to design lower order terms, that is terms which can be bounded as follows: |L.O.T| 6 C 1 s + 1 ZZ Qex T e-2s s2 ||2 + s3 4 3 ||2 . From now on we will be frequently using the estimates (3.8), (3.9) and (3.10) without mentioning them precisely all the time. This estimates will also be used to furnish the 0 L.O.T0 terms. Note in particular that we have ZZ Qex T |R1|2 = L.O.T. Computations. We write ZZ Qex T P11P21 = 3 X i,j=1 Jij, where Ji,j is the scalar product of the i-th term of P11 with the j-th term of P21. Computation of J11 : J11 = ZZ Qex T t11 = - ZZ Qex T t |1|2 2 = 0. (3.30) Computation of J12 : J12 = 2 s ZZ Qex T (t1 · 1) t = 1 2 2 s ZZ Qex T t|1|2 t = - 1 2 2 s ZZ Qex T |1|2 tt = L.O.T. (3.31) Computation of J13 : J13 = s2 2 ZZ Qex T 2 |0 |2 t1 · 1 = s2 2 2 ZZ Qex T t|1|2 2 |0 |2 = - s2 2 2 ZZ Qex T |1|2 t(2 |0 |2 ) = L.O.T. (3.32) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 23 Computation of J21 : J21 = -2s ZZ Qex T (0 · 1)1 = 2s ZZ Qex T ((0 · 1)) · 1 = 2s2 ZZ Qex T |0 · 1|2 + 2s ZZ Qex T (2 0 · 1) · 1 + s ZZ Qex T 0 · |1|2 = 2s2 ZZ Qex T |0 · 1|2 + 2s ZZ Qex T (2 0 · 1) · 1 - s2 ZZ Qex T |0 |2 |1|2 - s ZZ Qex T 0 |1|2 (3.33) = 2s2 ZZ Qex T |0 · 1|2 - s2 ZZ Qex T |0 |2 |1|2 + L.O.T. where the third line from the second in (3.33) follows because the boundary integral ZZ T1 T T0 T z0 |1|2 vanishes since 0 is only a function of (x, t). Computation of J22 : J22 = -2 s2 ZZ Qex T (0 · 1)t1 = s2 ZZ Qex T div t0 |1|2 = L.O.T. (3.34) Computation of J23 : J23 = -2s3 3 ZZ Qex T 3 (0 · 1)|0 |2 1 = s3 3 ZZ Qex T div(|0 |2 0 3 )|1|2 . (3.35) Computation of J31 : J31 = -2s2 ZZ Qex T |0 |2 11 = 2s2 ZZ Qex T |0 |2 |1|2 + 2s2 ZZ Qex T (|0 |2 )1 · 1 = 2s2 ZZ Qex T |0 |2 |1|2 - s2 ZZ Qex T (|0 |2 )|1|2 = 2s2 ZZ Qex T |0 |2 |1|2 + L.O.T. (3.36) 24 S. MITRA Computation of J32 : J32 = -2s2 2 ZZ Qex T |0 |2 |1|2 t = L.O.T. (3.37) Computation of J33 : J33 = -2s3 4 ZZ Qex T 3 |0 |4 |1|2 . (3.38) Combining the above computations (3.30)­(3.38), we obtain the following: ZZ Qex T P11P21 = 2s2 ZZ Qex T |0 · 1|2 + s2 ZZ Qex T |0 |2 |1|2 (3.39) + s3 3 ZZ Qex T |1|2 div(|0 |2 0 3 ) - 23 |0 |4 ! + L.O.T. Now, it is not hard to check that s3 3 ZZ Qex T |1|2 div(|0 |2 0 3 ) - 23 |0 |4 ! > cs3 4 ZZ Qex T 3 |0 |4 |1|2 - L.O.T. We thus immediately deduce from (3.4) and (3.29) that there exists C > 0 such that for all s and large enough, ZZ Qex T s2 |1|2 + ZZ Qex T s3 4 3 |1|2 6C ZZ Qex T e-2s |f3|2 + s3 4 ZZ T 3 |1|2 + s2 ZZ T |1|2 , (3.40) where T = (TL \ [-u1T, d + u1T] × (0, 1)) × (0, T). Now there are two steps to obtain (3.21) from (3.40): ­ Absorbing the observation in (3.40) involving 1 on T : This can be done classically by considering a slightly larger set of observation. For instance one can follow the arguments used in ([2], p. 565) or ([18], p. 461). ­ One should then come back to the original unknown from 1. This can be done as it is done classically by recalling that = es 1. This concludes the proof of Lemma 3.4. 3.3.2. Proof of Lemma 3.5 Proof. The proof of this lemma will follow the arguments used in ([2], Thm. 2.6) and ([18], p. 446) with some modifications. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 25 In order to solve (3.23) we will introduce a functional J whose Euler Lagrange equation provides a solution of (3.23). For smooth functions on Qex T with z = 0 on T1 T T0 T , let us define J() = 1 2 ZZ Qex T |(- t - )|2 e-2s + s3 4 2 ZZ T 3 ||2 e-2s - ZZ Qex T G. (3.41) We introduce the following space Xobs = { C(Qex T ) such that z = 0 on T1 T T0 T } k·kobs , (3.42) where k · kobs is the Hilbert norm defined by kk2 obs = ZZ Qex T |(- t - )|2 e-2s + s3 4 ZZ T 3 ||2 e-2s . (3.43) We endow the space Xobs with the Hilbert structure given by k · kobs. Of course, the fact that k · kobs is a norm follows from the Carleman estimate (3.21). Observe that, from the assumption (3.22) and the Carleman estimate (3.21), one has ZZ Qex T G 6 Ckkobs 1 s34 ZZ Qex T -3 |G|2 e2s 1/2 , (3.44) for some constant C > 0. Hence in view of (3.44), one observes that the functional J can be uniquely extended as a continuous functional on Xobs. We denote this extension by the notation J itself. The inequality (3.44) also infers the coercivity of J on Xobs. It is easy to check that J is strictly convex on Xobs. Hence J admits a unique minimizer min on Xobs. We further set Y = (- t - )mine-2s and H = -s3 4 3 mine-2s T . (3.45) From the Euler Lagrange equation of J at min, for all smooth functions on Qex T such that z = 0 on T 1 T T0 T , ZZ Qex T Y (- t - ) - ZZ Qex T G - ZZ T H = 0, (3.46) This equation easily implies that the solution Y of (3.23)(1,2,3) in the sense of transposition with source term G+H with H given by (3.45) coincides with the function Y given by (3.45) and satisfies the null controllability requirement Y (·, T) = 0 in TL × (0, 1). Note that, since G L2 (Qex T ) and H L2 (Qex T ), Y L2 (0, T; H2 (TL × (0, 1))) H1 (0, T; L2 (TL × (0, 1))) C0 ([0, T]; H1 (TL × (0, 1))). 26 S. MITRA Moreover since min is the minimizer of J on Xobs, using J(min) 6 J(0) = 0 and (3.44) one gets s3 4 ZZ Qex T |Y |2 e2s + ZZ T -3 |H|2 e2s 6 C ZZ Qex T -3 |G|2 e2s , (3.47) for large enough values of the parameters s and . One then follows the arguments used in proving ([2], Thm. 2.6), which mainly consists in a suitable energy estimate, i.e. a multiplication of (3.23) by -2 e2s Y , to show the following s2 ZZ Qex T -2 |Y |2 e2s 6 C ZZ Qex T -3 |G|2 e2s , (3.48) for large enough values of the parameters s and . In order to finish proving (3.24) one only needs to show s2 3 ZZ T1 T -1 |Y |2 e2s 6 C ZZ Qex T -3 |G|2 e2s , (3.49) for large enough values of the parameters s and . This can be achieved thanks to the estimates (3.47) and (3.48). In this direction we introduce a function = (x, z) = 0 z on TL × (0, 1). The function verifies · n = 1 on TL × {1}, 0 on TL × {0}, (3.50) where n denotes the unit outward normal to TL × {0, 1}. Now we perform the following calculations, using that z(-1 e2s ) = 0: s2 3 ZZ T1 T -1 |Y |2 e2s = s2 3 ZZ Qex T div(-1 |Y |2 e2s ) = s2 3 ZZ Qex T div(-1 e2s )|Y |2 + 2s2 3 ZZ Qex T -1 e2s Y · Y = s2 3 ZZ Qex T -1 e2s |Y |2 + 2s2 3 ZZ Qex T -1 e2s Y · Y 6 C s3 4 ZZ Qex T |Y |2 e2s + s2 ZZ Qex T -2 |Y |2 e2s . (3.51) Finally combining the estimates (3.47), (3.48) and (3.51) we prove (3.49). Hence we conclude the proof of the estimate (3.24). OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 27 3.3.3. Proof of Theorem 3.3 Proof. Since we only assume that f1 Qex T , f2 L2 (T1 T ), the solution of the problem (3.16) has to be considered in the sense of transposition. In particular, for Y L2 (0, T; H2 (TL × (0, 1))) H1 (0, T; L2 (TL × (0, 1))) C0 ([0, T]; H1 (TL × (0, 1))), satisfying zY = 0 on T1 T T0 T with Y (·, 0) = Y (·, T) = 0 in TL × (0, 1), we have ZZ Qex T f1Y = ZZ Qex T ( tY - Y )q - ZZ T1 T Y f2 (3.52) We thus choose a particular function G which satisfies (3.22), namely G = 3 qe-2s , (3.53) and Y the function given by Lemma 3.5, so that (3.52) yields: ZZ Qex T 3 |q|2 e-2s = ZZ Qex T f1Y + ZZ T1 T Y f2 - ZZ Qex T qH. (3.54) With the choice (3.53) of G, observe in particular that ZZ Qex T -3 |G|2 e2s = ZZ Qex T 3 |q|2 e-2s . Besides (3.54) furnishes, for all > 0, ZZ Qex T 3 |q|2 e-2s 6 C s3 4 ZZ Qex T |Y |2 e2s + 1 s34 ZZ Qex T |f1|2 e-2s + 1 ZZ T 3 |q|2 e-2s + ZZ T -3 |H|2 e2s + 1 s23 ZZ T1 T |f2|2 e-2s + s2 3 ZZ T1 T -1 |Y |2 e2s . (3.55) On the other hand with the particular choice (3.53) of G, the inequality (3.24) in particular takes the form s3 4 ZZ Qex T |Y |2 e2s + s2 3 ZZ T1 T -1 |Y |2 e2s + ZZ T -3 |H|2 e2s 6 C ZZ Qex T 3 |q|2 e-2s . (3.56) 28 S. MITRA Incorporating (3.56) in (3.55) and choosing small enough value for the parameter , we prove that s3 4 ZZ Qex T 3 |q|2 e-2s 6 C ZZ Qex T e-2s |f1|2 + s ZZ T1 T e-2s |f2|2 + s3 4 ZZ T e-2s 3 |q|2 . (3.57) Now one needs to estimate the integral of q on Qex T and the integral of q on T1 T to finish proving the inequality (3.18). In order to do that, we perform a weighted energy estimate on q by multiplying (3.16) by qe-2s . This and the estimates (3.8), (3.9) lead to ZZ Qex T |q|2 e-2s 6 C(1 + 1 )s2 ZZ Qex T 3 |q|2 e-2s + C ZZ Qex T |q|2 e-2s + C ZZ Qex T |q||f1|e-2s + C ZZ T1 T e-2s |f2||q| 6 C(1 + 1 )s2 ZZ Qex T 3 |q|2 e-2s + C ZZ Qex T |q|2 e-2s + C s2 ZZ Qex T |f1|2 e-2s + C ZZ T1 T e-2s |q|2 + C ZZ T1 T e-2s |f2|2 , (3.58) for some positive parameter . Now, similarly as in (3.51), we can obtain s ZZ T1 T e-2s |q|2 6 s ZZ T1 T e-2s 2 |q|2 (3.59) 6 C s2 2 (1 + 1 ) ZZ Qex T 3 |q|2 e-2s + ZZ Qex T |q|2 e-2s . Therefore, choosing > 0 small enough, we deduce from (3.58) and (3.59) that there exists a constant C > 0 such that for all s and large enough, ZZ Qex T |q|2 e-2s + s ZZ T1 T e-2s 2 |q|2 6 Cs2 2 ZZ Qex T 3 |q|2 e-2s + C s2 ZZ Qex T |f1|2 e-2s + C ZZ T1 T e-2s |f2|2 . With the estimate (3.57), this finishes the proof of Theorem 3.3. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 29 3.3.4. Proof of Corollary 3.6 Proof. Let us introduce the new unknown: q1 = 1 s q. From (3.16) we obtain the following system satisfied by q1: - tq1 - q1 = 1 s f1 + F5 in Qex T , zq1 = 1 s f2 on T1 T , zq1 = 0 on T0 T , q1(., T) = 1 s qT in TL × (0, 1), (3.60) where F5 = 1 s2 tq + 1 s2 q · + 1 s2 q + 2 s3 ||2 q - 1 s2 · q. (3.61) In view of the estimates (3.9), F5 satisfies ZZ Qex T e-2s |F5|2 6 C 2 s2 ZZ Qex T e-2s 1 |q|2 + 1 s2 ZZ Qex T e-2s 1 2 |q|2 . (3.62) Now we apply Theorem 3.3 for the system (3.60) to have the following ZZ Qex T e-2s (s2 |q1|2 + s3 4 3 |q1|2 ) + s2 3 ZZ T1 T e-2s 2 |q1|2 6 C 1 s22 ZZ Qex T e-2s 1 2 |f1|2 + ZZ Qex T e-2s |F5|2 + 1 s ZZ T1 T e-2s 1 2 |f2|2 +s3 4 ZZ T e-2s 3 |q1|2 6 C 1 s22 ZZ Qex T e-2s 1 2 |f1|2 + 2 s2 ZZ Qex T e-2s 1 |q|2 + 1 s2 ZZ Qex T e-2s 1 2 |q|2 + 1 s ZZ T1 T e-2s 1 |f2|2 + s2 ZZ T e-2s |q|2 , (3.63) 30 S. MITRA where the last step of (3.63) from the penultimate step follows by using (3.62) and the definition of q1. Now, using q = sq1, one further checks the following for sufficiently large vales of s and ZZ Qex T e-2s ( 1 s |q|2 + s2 |q|2 ) + ZZ T1 T e-2s |q|2 6 C ZZ Qex T e-2s (s2 |q1|2 + s3 4 3 |q1|2 ) + s2 3 ZZ T1 T e-2s 2 |q1|2 . (3.64) Combining (3.63) and (3.64), we get ZZ Qex T e-2s ( 1 s |q|2 + s2 |q|2 ) + ZZ T1 T e-2s |q|2 6 C 1 s22 ZZ Qex T e-2s 1 2 |f1|2 + 2 s2 ZZ Qex T e-2s 1 |q|2 + 1 s2 ZZ Qex T e-2s 1 2 |q|2 + 1 s ZZ T1 T e-2s 1 |f2|2 + s2 ZZ T e-2s |q|2 . Taking s and large enough, we conclude Corollary 3.6. 3.4. Observability of an adjoint transport equation In this section we derive an observability inequality for the adjoint transport equation ( -t - u1x + P 0 () = f4 in Qex T , (·, T) = T in TL × (0, 1). (3.65) Theorem 3.7. Let us recall the notations Qex T and T introduced in (3.13). There exists a positive constant C such that for all T L2 (TL × (0, 1)), f4 L2 (Qex T ) and for all values of the parameters s > 1 and > 1, the solution of (3.65) satisfies the following inequality k- 1 2 e-s k2 L2(Qex T ) 6 C(k- 1 2 f4e-s k2 L2(Qex T ) + k- 1 2 e-s k2 L2(T )). (3.66) There exists a positive constant C such that for all T H1 (TL × (0, 1)), f4 L2 (0, T; H1 (TL × (0, 1))), and for all values of the parameter s > 1 and > 1, the solution of (3.65) satisfies the following inequality k- 1 2 e-s k2 L2(Qex T ) 6 C(k- 1 2 f4e-s k2 L2(Qex T ) + k- 1 2 e-s k2 L2(T )). (3.67) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 31 Consequently there exists a positive constant C such that for all f4 L2 (0, T; H1 (TL × (0, 1))), and for all values of the parameter s > 1 and > 1, the solution of (3.65) satisfies the following inequality k- 1 2 te-s k2 L2(Qex T ) 6C(k- 1 2 f4e-s k2 L2(Qex T ) + k- 1 2 f4e-s k2 L2(Qex T ) + k- 1 2 e-s k2 L2(T ) + k- 1 2 e-s k2 L2(T )). (3.68) Proof. We first prove the inequality (3.66). The proof depends on a controllability estimate of the following problem te + u1xe + P0 () e = e f4 + ve in Qex T , e (·, 0) = 0 in TL × (0, 1), e (·, T) = 0 in TL × (0, 1). (3.69) and a duality argument. We suppose that the source term e f4 satisfies: 1 2 es e f4 L2(Qex T ) < . In order to obtain controllability estimates for the problem (3.69) we will follow the arguments used in ([14], Thm. 3.5) but here with different multipliers. The controlled trajectory e and the control function ve can be constructed by adapting ([14], eq. (3.26)) and ([14], eq. (3.28)) in a straight forward manner. To obtain (3.66) we recall at a glance the strategy to construct the controlled trajectory e . It is done by gluing the solutions to the forward problem te f + u1xe f + P0 () e f = e f4 in Qex T , e f (·, 0) = 0 in TL × (0, 1), (3.70) and the backward problem te b + u1xe b + P0 () e b = e f4 in Qex T , e (·, T) = 0 in TL × (0, 1), (3.71) using suitable cut off functions. One recalls the role of T0, T1 and from (3.5) and (3.6). Let 0 be a smooth cut off function taking value 1 on the set {x TL | x [-5 - u1T0, d + 5 + u1T0]} and value 0 on TL \ {x TL | x (-6 - u1T0, d + 6 + u1T0)}. Let us now introduce the function which solves: ( t + u1x = 0 in Qex T , (·, 0) = 0 in TL × (0, 1). (3.72) We set e = 2(x) (e f + (1 - )e b) + (1 - 2(x)) 1(t)e f in Qex T , (3.73) where, 1(t) is a smooth cut-off function taking value 1 on [0, T0] and vanishing for t > 2T0 and 2(x) is a smooth cut-off function taking value 1 on {x TL | x [-3, 3]} and vanishing on TL \ {x TL | x (-4, 4)}. Due 32 S. MITRA to the choice of suitable cut-off functions, one verifies that e given by (3.73) solves (3.69) with the following control function: ve = (2 - 1) e f4 + (1 - 2)1 e f4 + u1x2(e f + (1 - )e b) - 1u1x2e f + (1 - 2)t1e f . (3.74) Due to the choice of 2, the control function ve is supported on . Further we observe the following identities: 2(1 - ) = 0 in (TL × (0, 1)) × [0, T0] and 2 = 0 in (TL × (0, 1)) × [T - 2T1, T]. (3.75) Esimates on e : In view of the construction (3.73) of the controlled trajectory, one first proves estimates for e f and e b. In that direction we test (3.70)1 by e2s e f to furnish d dt 1 2 Z TL×(0,1) e2s |e f |2 6 1 2 Z TL×(0,1) |e f |2 -2 P0 () e2s + (t + u1x) (e2s ) + Z TL×(0,1) e2s | e f4|2 1 2 Z TL×(0,1) e2s |e f |2 1 2 . (3.76) Hence in view of the inequality (3.11), (3.76) and Gronwall's lemma we have k 1 2 es e f k2 L(0,T -2T1;L2(TL×(0,1))) 6 Ck 1 2 es e f4k2 L2(0,T -2T1;L2(TL×(0,1))). (3.77) Similarly one proves k 1 2 es e bk2 L(T0,T ;L2(TL×(0,1))) 6 Ck 1 2 es e f4k2 L2(T0,T ;L2(TL×(0,1))). (3.78) Using the identity (3.75), estimates (3.77) and (3.78) and adapting the expression (3.73) of the controlled trajectory, one at once renders k 1 2 es e k2 L(0,T ;L2(TL×(0,1))) 6 Ck 1 2 es e f4k2 L2(Qex T ). (3.79) Further one uses the expression (3.74) of the control function and estimates (3.77) and (3.78) to furnish k 1 2 es ve k2 L2(Qex T ) 6 Ck 1 2 es e f4k2 L2(Qex T ). (3.80) Now one recalls the following duality relation between (3.65) and (3.69) (which follows because of the initial and final time conditions (3.69)2 and (3.69)3): h e f4, iL2(Qex T ) + hve , iL2(Qex T ) = h(-t - u1x + P0 () ), e iL2(Qex T ) = hf4, e iL2(Qex T ). (3.81) Finally, the observability estimate (3.66) follows from (3.79), (3.80) and (3.81) by using the following arguments: k- 1 2 e-s kL2(Qex T ) = sup k 1 2 e f4eskL2(Qex T )61 |h e f4, iL2(Qex T )| OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 33 6 sup k 1 2 e f4eskL2(Qex T )61 (|he , f4iL2(Qex T )| + |hve , iL2(Qex T )|) 6 C(k- 1 2 f4e-s kL2(Qex T ) + k- 1 2 e-s kL2(T )). In the calculation above the second line is furnished from the first by using (3.81). This provides the inequality (3.66). The inequality (3.67) can be obtained by applying (3.66) to the system satisfied by . Once we have (3.67), the estimate (3.68) can be obtained directly by using the equation (3.65)1. The following result is a corollary to Theorem 3.7 and corresponds to the weighted estimates of and in L (L2 ) norms. The following result will be used in the next section, more precisely in the proof of Lemma 4.2, to obtain an estimate of (·, 2T0) in H1 (TL × (0, 1)) at a point 2T0 intermediate to 0 and T. Corollary 3.8. Let us recall the notations Qex T and T introduced in (3.13). There exists a positive constant C such that for all T L2 (TL × (0, 1)), f4 L2 (Qex T ) and for all values of the parameters s > 1 and > 1, the solution of (3.65) satisfies the following inequality k- 3 2 e-s k2 L(0,T ;L2(TL×(0,1))) 6 C(k- 1 2 f4e-s k2 L2(Qex T ) + sk- 1 2 e-s k2 L2(T )). (3.82) There exists a positive constant C such that for all T H1 (TL × (0, 1)), f4 L2 (0, T; H1 (TL × (0, 1))), and for all values of the parameter s > 1 and > 1, the solution of (3.65) satisfies the following inequality k- 3 2 e-s k2 L(0,T ;L2(TL×(0,1))) 6 C(k- 1 2 f4e-s k2 L2(Qex T ) + sk- 1 2 e-s k2 L2(T )). (3.83) Proof. We test (3.65)1 by -3 e-2s and use integration by parts to have: d dt 1 2 Z TL×(0,1) -3 e-2s ||2 6 1 2 Z TL×(0,1) ||2 2 P0 () -3 e-2s + (t + u1x) (-3 e-2s ) + Z TL×(0,1) -3 e-2s |f4|2 1 2 Z TL×(0,1) -3 e-2s ||2 1 2 . (3.84) Further using > 1 and the estimates (3.8) and (3.9) one has (t + u1x) (-3 e-2s ) 6 Cs-1 e-2s . Hence the R.H.S of (3.84) can be majorized by a constant multiple of s Z TL×(0,1) -1 e-2s ||2 + Z TL×(0,1) -1 e-2s |f4|2 . Now integrating both sides of (3.84) with respect to time in the interval (0, t), for any t (0, T), recalling that, -3 e-2s ||2 (·, 0) = 0 and making use of (3.66) we obtain (3.82). One proves (3.83) by applying (3.82) to the equation satisfied by . 34 S. MITRA 4. Observability and unique continuation of the system (1.22) The goal of this section is to prove Theorem 1.5 and Corollary 1.6. In order to do that, as for the proof of Theorem 1.3, the key step is first to prove an observability result for the system (2.3). In that direction we prove two lemmas, the first one is a Carleman estimate corresponding to the system (2.3) and the next uses this Carleman estimate to prove the desired observability inequality. We choose to state the Carleman estimate and the observability inequality in form of two different results since they might have independent interests. 4.1. Carleman estimate and observability inequality of the system (2.3) Lemma 4.1. There exists a constant C > 0 such that for all (, q, ) solving (2.3) with initial datum (T , qT , T , 1 T ) having the regularity (2.4) and satisfying the compatibility conditions (2.5), s7 8 ZZ T1 T 7 ||2 e-2s + s5 6 ZZ T1 T 5 |x|2 e-2s + s3 4 ZZ T1 T 3 (|xx|2 + |t|2 )e-2s + s2 ZZ T1 T (|tx|2 + |xxx|2 )e-2s + 1 s ZZ T1 T 1 (|tt|2 + |txx|2 + |xxxx|2 )e-2s + ZZ Qex T e-2s ( 1 s (|tq|2 + |q|2 ) + s2 (|q|2 + |tq|2 )) + 1 s ZZ Qex T 1 (||2 + ||2 + |t|2 )e-2s + 1 s22 sup (0,T ) Z TL×(0,1) 1 3 (||2 + ||2 )e-2s 6 Cs7 8 ZZ T 1 7 ||2 e-2s + Cs2 ZZ T e-2s (|q|2 + |xq|2 + |tq|2 ) (4.1) + C 1 s ZZ T 1 (||2 + ||2 )e-2s , where T = × (0, T), T b = b × (0, T) and the weight functions and were introduced in (3.7). Lemma 4.2. There exists a constant C > 0 such that for all (, q, ) solving (2.3) with initial datum (T , qT , T , 1 T ) having the regularity (2.4) and satisfying the compatibility conditions (2.5), k((·, 0), t(·, 0))kH3(TL×(0,1)))×H1(TL×(0,1)) + kq(·, 0)kH2(TL×(0,1)) + k(·, 0)kH1(TL×(0,1)) 6 CkkL2(T b ) + CkqkH1(T ) + CkkH1(T ), (4.2) where T = × (0, T), T b = b × (0, T). Lemmas 4.1 and 4.2 are proved respectively in Sections 4.1.1 and 4.1.2 below, and follows from a suitable use of the various Carleman estimates proved in the previous section. Based on Lemma 4.2, it will be rather easy to derive an estimate on v(·, 0) in H2 (TL × (0, 1)) and conclude the proof of Theorem 1.5, which will be done in Section 4.2. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 35 4.1.1. Proof of Lemma 4.1 Proof. Note that, from Lemma 2.2, for (T , qT , T , 1 T ) having the regularity (2.4) and satisfying the compati- bility conditions (2.5), we have that (t + u1x)q|TL×{1} belongs to L2 (T1 T ). We thus apply Theorem 3.2 with f = (t + u1x)q|TL×{1}: s7 8 ZZ T1 T 7 ||2 e-2s + s5 6 ZZ T1 T 5 |x|2 e-2s + s3 4 ZZ T1 T 3 (|xx|2 + |t|2 )e-2s + s2 ZZ T1 T (|tx|2 + |xxx|2 )e-2s + 1 s ZZ T1 T 1 (|tt|2 + |txx|2 + |xxxx|2 )e-2s 6 C ZZ T1 T |tq + u1xq|2 e-2s + Cs7 8 ZZ T b 7 ||2 e-2s . (4.3) One observes that (4.3) furnishes weighted estimates of t(t + u1x) and x(t + u1x). We also know from Lemma 2.2 that C0 ([0, T]; H1 (TL × (0, 1)) C1 ([0, T]; L2 (TL × (0, 1))), so we can apply Corollary 3.6 to q, tq and xq: ZZ Qex T e-2s ( 1 s |q|2 + s2 |q|2 ) (4.4) 6 C 1 s22 ZZ Qex T e-2s 1 2 ||2 + 1 s ZZ T1 T e-2s 1 |t + u1x|2 + s2 ZZ T e-2s |q|2 , ZZ Qex T e-2s ( 1 s |tq|2 + s2 |tq|2 ) + ZZ T1 T e-2s |tq|2 (4.5) 6 C 1 s22 ZZ Qex T e-2s 1 2 |t|2 + 1 s ZZ T1 T e-2s 1 |tt + u1xt|2 + s2 ZZ T e-2s |tq|2 , ZZ Qex T e-2s ( 1 s |xq|2 + s2 |xq|2 ) + ZZ T1 T e-2s |xq|2 (4.6) 6 C 1 s22 ZZ Qex T e-2s 1 2 |x|2 + 1 s ZZ T1 T e-2s 1 |tx + u1xx|2 + s2 ZZ T e-2s |xq|2 . 36 S. MITRA Now, we apply the observability estimates of Theorem 3.7 with f4 = q : 1 s ZZ Qex T 1 (||2 + ||2 + |t|2 )e-2s 6 C 1 s ZZ T 1 (||2 + ||2 )e-2s + C 1 s ZZ Qex T 1 (|q|2 + |q|2 )e-2s . (4.7) Further applying the estimates (3.82) and (3.83) of Corollary 3.8 with f4 = q : 1 s22 sup (0,T ) Z TL×(0,1) 1 3 (||2 + ||2 )e-2s 6 C 1 s ZZ T 1 (||2 + ||2 )e-2s + C 1 s22 ZZ Qex T 1 (|q|2 + |q|2 )e-2s . (4.8) Therefore, summing up (4.3)­(4.8), we obtain: s7 8 ZZ T1 T 7 ||2 e-2s + s5 6 ZZ T1 T 5 |x|2 e-2s + s3 4 ZZ T1 T 3 (|xx|2 + |t|2 )e-2s + s2 ZZ T1 T (|tx|2 + |xxx|2 )e-2s + 1 s ZZ T1 T 1 (|tt|2 + |txx|2 + |xxxx|2 )e-2s + ZZ Qex T e-2s ( 1 s (|q|2 + |tq|2 + |xq|2 ) + s2 (|q|2 + |tq|2 + |xq|2 )) + ZZ T1 T e-2s (|tq|2 + |xq|2 ) + 1 s ZZ Qex T 1 (||2 + ||2 + |t|2 )e-2s + 1 s22 sup (0,T ) Z TL×(0,1) 1 3 (||2 + ||2 )e-2s 6 C ZZ T1 T |tq + u1xq|2 e-2s + Cs7 8 ZZ T b 7 ||2 e-2s + C 1 s22 ZZ Qex T e-2s 1 2 (||2 + |x|2 + |t|2 ) + C 1 s ZZ T1 T e-2s 1 (|tt|2 + |xt|2 + |xx|2 + |t|2 + |x|2 ) + Cs2 ZZ T e-2s (|q|2 + |xq|2 + |tq|2 ) + C 1 s ZZ T 1 (||2 + ||2 )e-2s + C 1 s ZZ Qex T 1 (|q|2 + |q|2 )e-2s . Using that > > 1, taking s and large enough, we can absorb all the terms in the right hand side which are not localized in the observation set, so that we deduce (4.1). This finishes the proof of Lemma 4.1. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 37 Next using Lemma 4.1, we prove the Lemma 4.2. 4.1.2. Proof of Lemma 4.2 Proof. We now fix s and such that (4.1) holds. We then use the inequalities (3.12) to deduce from (4.1) that kkL2(2T0,T -2T1;H4(TL×{1}))H2(2T0,T -2T1;L2(TL×{1})) + kqkH1(2T0,T -2T1;H1(TL×(0,1))) + kkL(2T0,T -2T1;H1(TL×(0,1)))H1(2T0,T -2T1;L2(TL×(0,1))) 6 CkkL2(T b ) + CkqkL2(0,T ;H1())H1(0,T ;L2()) + CkkL2(0,T ;H1()). (4.9) The next step consists in proving that there exists a constant C such that k((·, 2T0), t(·, 2T0))kH3(TL×{1}))×H1(TL×{1}) + kq(·, 2T0)kH2(TL×(0,1)) + k(·, 2T0)kH1(TL×(0,1)) 6 CkkL2(T b ) + CkqkL2(0,T ;H1())H1(0,T ;L2()) + CkkL2(0,T ;H1()). (4.10) From (4.9), we have an estimate on in L2 (2T0, T - 2T1; H4 (TL × {1})) H1 (2T0, T - 2T1; H2 (TL × {1})) H2 (2T0, T - 2T1; L2 (TL × {1})) and thus by the interpolation result ([27], Thm. 3.1) on C0 ([2T0, T - 2T1]; H3 (TL × {1})) C1 ([2T0, T - 2T1]; H1 (TL × {1})), so that we obtain k((·, 2T0), t(·, 2T0))kH3(TL×{1}))×H1(TL×{1}) 6 CkkL2(T b ) + CkqkL2(0,T ;H1())H1(0,T ;L2()) + CkkL2(0,T ;H1()). (4.11) Then, from (4.9), we also have an estimate on q in C0 ([2T0, T - 2T1]; H1 (TL × (0, 1))), therefore kq(·, T - 2T1)kH1(TL×(0,1)) 6 CkkL2(T b ) + CkqkL2(0,T ;H1())H1(0,T ;L2()) + CkkL2(0,T ;H1()). (4.12) Now q satisfies the heat equation (2.3)(2,3,4,5) and has the regularity (2.6)2. We thus apply Lemma A.1 with f = -P 0 ()2 , g = -(t + u1x) on TL × {1}, 0 on TL × {0}, (4.13) in the time interval (2T0, T - 2T1) and further use (4.9) and (4.12) to deduce kq(·, 2T0)kH2(TL×(0,1)) 6 CkkL2(T b ) + CkqkL2(0,T ;H1())H1(0,T ;L2()) + CkkL2(0,T ;H1()). (4.14) Finally, in view of the assumptions (2.4) and (2.5) we know from (2.6) that C0 ([0, T]; H1 (TL × (0, 1))). Hence using (4.9), one at once deduce that k(·, 2T0)kH1(TL×(0,1)) 6 CkkL2(T b ) + CkqkL2(0,T ;H1())H1(0,T ;L2()) + CkkL2(0,T ;H1()). (4.15) The combination of (4.11)­(4.14)­(4.15) then concludes the proof of (4.10). We now use Lemma 2.2 to deduce (4.2). Since we are dealing with the solution (, q, ) of the system (2.3) in the strong regularity framework (2.6), compatibility conditions analogous to (2.5) is automatically satisfied at time 2T0. Hence using Lemma 2.2 to solve (2.3) starting from the time 2T0 and the estimate (4.10) we conclude the proof of (4.2). 38 S. MITRA 4.2. Proof of Theorem 1.5 Proof. For (T , vT , T , 1 T ) as in Theorem 1.5, we start by solving (2.3) with initial datum (T , qT , T , 1 T ), where qT = div vT + T . Using Lemma 4.2 and recalling that q = div v + , we deduce that k(·, 0)kH1(TL×(0,1)) + kdiv v(·, 0)kH1(TL×(0,1)) + k((·, 0), t(·, 0))kH3(TL×{1}))×H1(TL×{1}) 6 CkkL2(T b ) + CkqkL2(0,T ;H1())H1(0,T ;L2()) + CkkL2(0,T ;H1()) (4.16) 6 CkkL2(T b ) + CkvkL2(0,T ;H2())H1(0,T ;H1()) + CkkL2(0,T ;H1()). Thus, to obtain the result of Theorem 1.5, it only remains to estimate v(·, 0). As we already have an estimate on div v(·, 0), we first focus on getting an estimate on curl v. One now uses the system (2.8) to obtain the following set of equations solved by curlv: -(t(curlv) + u1x(curlv)) - µ(curlv) = 0 in (TL × (0, 1)) × (0, T), curlv = 0 on (TL × {0, 1}) × (0, T), curlv(·, T) = curlvT in TL × (0, 1). (4.17) Thus, curl v satisfies a parabolic heat type equation with homogeneous Dirichlet boundary condition. Classical observability estimates for the heat equation (see e.g. [23] or [18]) immediately yields kcurl v(·, 0)kH1(TL×(0,1)) 6 Ckcurl vkL2(T ) 6 CkvkL2(0,T ;H1()). (4.18) Now we recover v(·, 0) by solving the following elliptic problem at times t = 0: v(·, 0) = div (v(·, 0)) + z -x ! (curl v(·, 0)) in TL × (0, 1), v2(·, 0) = (·, 0) on TL × {1}, v2(·, 0) = 0 on TL × {0}, curl v(·, 0) = 0 on TL × {0, 1}. (4.19) One can use standard elliptic regularity to deduce that kv(·, 0)kH2(TL×(0,1)) 6 Ckcurl v(·, 0)kH1(TL×(0,1)) +Ckdiv v(·, 0)kH1(TL×(0,1)) +Ck(·, 0)kH3(TL×{1}). With the estimates (4.16)­(4.18) and this last estimate, we then deduce that kv(·, 0)kH1(TL×(0,1)) 6 CkkL2(T b ) + CkvkL2(0,T ;H2())H1(0,T ;H1()) + CkkL2(0,T ;H1()). (4.20) Together with (4.16), this concludes the proof of Theorem 1.5. 4.3. Proof of Corollary 1.6 Proof. The assumption that (, v, ) = 0 in (T )2 × T b at once gives (, q, ) = 0 in (T )2 × T b and hence from (4.1) we furnish: (, q, ) = 0 in ((TL × (0, 1)) × (0, T))2 × ((TL × {1}) × (0, T)). (4.21) The only task is to prove v = 0 on ((TL × (0, 1)) × (0, T)). From (4.21) it is easy to observe that div v = 0 in ((TL × (0, 1)) × (0, T)). Then one considers the system (4.17) solved by curl v use curl v = 0 in T and classical OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 39 Carleman estimate for heat type equation from [23] or [18] to infer curl v = 0 in ((TL × (0, 1)) × (0, T)). Since v C0 ([0, T]; H2 (TL × (0, 1))) (follows from (1.25)) it satisfies a system of the form (4.19) for each t (0, T) but with zero source terms since we already have: (div v, curl v, ) = 0 in ((TL × (0, 1)) × (0, T))2 × ((TL × {1}) × (0, T)). Considering this equations component wise and using that v = 0 in T one proves v = 0 in (TL × (0, 1)) × (0, T). This finishes the proof of Corollary 1.6. Appendix A. A.1 Formal derivation of the adjoint system (1.22) In this section we will explain the formal derivation of the adjoint system (1.22) from the primal system (1.21). Of course while deriving the adjoint system we neglect the role of the control functions ve , ve u and vb in the primal problem (1.21). In the following calculations we will use the short hand notations Qex T , T1 T and T0 T introduced in (3.13). We test (1.21)1, (1.21)2 and (1.21)8 respectively by the adjoint unknowns , v and , use the formal relation e u = div e u + z -x ! (curle u) in TL × (0, 1), integrate by parts and add the resulting expressions to have Z TL×(0,1) e (T)(T) - Z TL×(0,1) e 0(0) ! + ZZ Qex T (-t - u1x - P0 ()div v) e + ZZ T1 T e u2 + Z TL×(0,1) e u(T)v(T) - Z TL×(0,1) e u0v(0) ! + ZZ Qex T (-(tv + u1xv) -v - (µ + µ0 )div v - ) e u + ZZ T1 T T0 T (-(µ0 + 2µ)div e u + P0 ()e ) v2 + ZZ T1 T (µ0 + 2µ)div ve u2 + µ ZZ T1 T curl ve u2 + ZZ T1 T (tt + txx + xxxx) + Z TL×{1} t(T)(T) - Z TL×{1} 10 - Z TL×{1} (T)t(T) + Z TL×{1} 0t(0) ! = ZZ T1 T (-(µ0 + 2µ)div e u + P0 ()e ) . (A.1) 40 S. MITRA Further using the interface boundary condition (1.21)3 in the third and the seventh terms appearing in the left hand side of (A.1) and integrating by parts, we obtain: Z TL×(0,1) e (T)(T) - Z TL×(0,1) e 0(0) ! + ZZ Qex T (-t - u1x - P0 ()div v) e - ZZ T1 T ((t + u1x) ((µ0 + 2µ)div v + ) + (tt + txx + xxxx)) + Z TL×(0,1) e u(T)v(T) - Z TL×(0,1) e u0v(0) ! + ZZ Qex T (-(tv + u1xv) -v - (µ + µ0 )div v - ) e u + ZZ T1 T T0 T (-(µ0 + 2µ)div e u + P0 ()e ) v2 + Z TL×{1} (µ0 + 2µ)(T)div v(T) - Z TL×{1} (µ0 + 2µ)0div v(0) + µ ZZ T1 T curl ve u2 + Z TL×{1} (T)(T) - Z TL×{1} 0(0) + Z TL×{1} t(T)(T) - Z TL×{1} 10 - Z TL×{1} (T)t(T) + Z TL×{1} 0t(0) ! = ZZ T1 T (-(µ0 + 2µ)div e u + P0 ()e ) . (A.2) Next we set curl v = 0 on T1 T T0 T , v2 = on T1 T , v2 = 0 on T0 T to furnish the following from (A.2): Z TL×(0,1) e (T)(T) - Z TL×(0,1) e 0(0) ! + ZZ Qex T (-t - u1x - P0 ()div v) e - ZZ T1 T ((t + u1x) ((µ0 + 2µ)div v + ) + (tt + txx + xxxx)) + Z TL×(0,1) e u(T)v(T) - Z TL×(0,1) e u0v(0) ! + ZZ Qex T (-(tv + u1xv) -v - (µ + µ0 )div v - ) e u + Z TL×{1} (µ0 + 2µ)(T)div v(T) - Z TL×{1} (µ0 + 2µ)0div v(0) + Z TL×{1} (T)(T) - Z TL×{1} 0(0) + Z TL×{1} t(T)(T) - Z TL×{1} 10 - Z TL×{1} (T)t(T) + Z TL×{1} 0t(0) ! = 0. (A.3) Since the adjoint system defined in (0, T) must be independent of the initial conditions (e 0, e u0, 0, 1) and ((T), v(T), (T), t(T)) (for the primal and adjoint system respectively), we equate the coefficients of e , e u and appearing in the left hand side of (A.3) to formally obtain the system (1.22). OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 41 A.2 Proof of Lemma 2.2 This section is devoted for the proof of Lemma 2.2. Proof of Lemma 2.2. In this step we will prove the existence and regularity result for (, q, ) solving the system (2.3). In fact, we will prove that under the assumptions (2.4) and (2.5), the system (2.3) admits a unique solution in the following functional framework (2.6). We will first prove a local in time existence result for the problem (2.3). Then using the linearity of (2.3) we iterate the time steps in order to show (2.6). Since the problem (2.3) is posed backward in time by local in time existence, we first work in some time interval of the form (T - T0, T) for T0 sufficiently small. Let 0 < T0 < T. We consider the system (2.3). We are going to define a suitable map whose fixed point gives a solution to the system (2.3) in the time interval (T - T0, T). We define HT0 1 = (L2 (T - T0, T; H1 (TL × (0, 1))) H1 (T - T0, T; L2 (TL × (0, 1)))) × (H1 (T - T0, T; H3/2 (TL × {1})) H7/4 (T - T0, T; L2 (TL × {1})) L2 (T - T0, T; H5/2 (TL × {1})) H3/4 (T - T0, T; H1 (TL × {1}))), (A.4) and for (b , b ) HT0 1 satisfying the condition b (·, T) = T and ( b , t b )(·, T) = (T , 1 T ), (A.5) we solve the system -t - u1x + P0 () = P0 () q in (TL × (0, 1)) × (T - T0, T), -(tq + u1xq) - q - P0 () q = - P0 ()2 b in (TL × (0, 1)) × (T - T0, T), zq = -(t b + u1x b ) on (TL × {1}) × (T - T0, T), zq = 0 on (TL × {0}) × (T - T0, T), q(·, T) = qT in TL × (0, 1), (·, T) = T in TL × (0, 1), tt + txx + xxxx = (t + u1x)q on (TL × {1}) × (T - T0, T), (T) = T and t(T) = 1 T in TL × {1}, (A.6) This defines the following map: LT0 : (b , b ) HT0 1 7- (, ). (A.7) We will show that LT0 maps HT0 1 into itself and is a contraction there. Observe that a fixed point of the map infers a solution to the system (2.3) in the time interval (T - T0, T). In the sequel we will show that the map LT0 admits a fixed point in HT0 1 , for T0 sufficiently small. 42 S. MITRA A.3 LT0 maps HT0 1 to itself In that direction we first claim that there exists a positive constant C such that kqkL2(T -T0,T ;H3(TL×(0,1)))H3/2(T -T0,T ;L2(TL×(0,1))) 6 C(k(t b + u1x b )kL2(T -T0,T ;H3/2(TL×{1}))H3/4(T -T0,T ;L2(TL×{1})) + kb kL2(T -T0,T ;H1(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) + kqT kH2(TL×(0,1)))) (A.8) To begin with, we will just use the following regularities for the non-homogeneous source term, boundary data and the initial condition b L2 (T - T0, T; L2 (TL × (0, 1))), -(t b + u1x b ) L2 (T - T0, T; H1/2 (TL × {1})) H1/4 (T - T0, T; L2 (TL × {1})), qT H1 (TL × (0, 1)). (A.9) Hence using the regularities (A.9) one can apply ([28], Thm. 5.3, p. 32) to solve (A.6)2-(A.6)5 in the following functional framework q L2 (T - T0, T; H2 (TL × (0, 1))) H1 (T - T0, T; L2 (TL × (0, 1))). (A.10) Moreover there exists a positive constant C independent of T0 such that kqkL2(T -T0,T ;H2(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) 6 C(kb kL2(T -T0,T ;L2(TL×(0,1))) + k(div vT + T )kH1(TL×(0,1)) +k(t b + u1x b )kL2(T -T0,T ;H1/2(TL×{1}))H1/4(T -T0,T ;L2(TL×{1}))). (A.11) Let us explain how we obtain a constant C independent of T0 in the inequality (A.11). The technique is inspired from [36]. We extend b and (t b + u1x b ) in (0, T) by defining them zero in the time interval (0, T - T0). The extended functions are also denoted by the same notations b and (t b + u1x b ). It is easy to verify that b L2 (0, T; L2 (TL × (0, 1))) and (t b + u1x b ) L2 (0, T; H1/2 (TL × {1})) H1/4 (0, T; L2 (TL × {1})). One then has the following kqkL2(T -T0,T ;H2(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) 6 kqkL2(0,T ;H2(TL×(0,1)))H1(0,T ;L2(TL×(0,1))) 6 C(kb kL2(0,T ;L2(TL×(0,1))) + kqT kH1(TL×(0,1)) + k(t b + u1x b )kL2(0,T ;H1/2(TL×{1}))H1/4(0,T ;L2(TL×{1}))) = C(kb kL2(T -T0,T ;L2(TL×(0,1))) + kqT kH1(TL×(0,1)) + k(t b + u1x b )kL2(T -T0,T ;H1/2(TL×{1}))H1/4(T -T0,T ;L2(TL×{1}))), (A.12) where the constant C (might depend of T) is independent of T0. OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 43 Now to prove (A.8) we write the equations (A.6)2-(A.6)5 as follows: -tq - q = - P0 ()2 b + u1xq + P0 () q := Gb ,q in (TL × (0, 1)) × (T - T0, T), zq = -(t b + u1x b ) on (TL × {1}) × (T - T0, T), zq = 0 on (TL × {0}) × (T - T0, T), q(·, T) = qT = divvT + T in TL × (0, 1), (A.13) In view of (A.12) and the interpolation L2 (T - T0, T; H2 (TL × (0, 1))) H1 (T - T0, T; L2 (TL × (0, 1))) , H 1 2 (T - T0, T; H1 (TL × (0, 1))) one has the following regularity estimate of xq by interpolation kxqkL2(T -T0,T ;H1(TL×(0,1)))H1/2(T -T0,T ;L2(TL×(0,1))) 6 C(kb kL2(T -T0,T ;L2(TL×(0,1))) + kqT kH1(TL×(0,1)) + k(t b + u1x b )kL2(T -T0,T ;H1/2(TL×{1}))H1/4(T -T0,T ;L2(TL×{1}))), (A.14) for some positive constant C independent of T0. Indeed, this can be obtained by performing the interpolation process in the time interval (0, T) instead of (T - T0, T). Hence the assumption and for (b , b ) HT0 1 (HT0 1 is defined in (A.4)), the obtained regularity (A.12) and (A.14) implies that Gb ,q L2 (T - T0, T; H1 (TL × (0, 1))) H1/2 (T - T0, T; L2 (TL × (0, 1))) (A.15) and kGb ,qkL2(T -T0,T ;H1(TL×(0,1)))H1/2(T -T0,T ;L2(TL×(0,1))) 6 C(kb kL2(T -T0,T ;H1(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) + k(t b + u1x b )kL2(T -T0,T ;H1/2(TL×{1}))H1/4(T -T0,T ;L2(TL×{1})) + kqT kH1(TL×(0,1))), (A.16) for some positive constant C independent of T0. At this stage we will use the following regularities of the boundary and initial datum which follows from (A.4) and (2.4): -(t b + u1x b ) L2 (T - T0, T; H3/2 (TL × {1})) H3/4 (T - T0, T; L2 (TL × {1})), qT H2 (TL × (0, 1)). (A.17) Furthermore, in view of (2.5), (A.16) and (A.17), we can apply ([28], Thm. 5.3, p. 32) to solve (A.13) in the functional framework q L2 (T - T0, T; H3 (TL × (0, 1))) H3/2 (T - T0, T; L2 (TL × (0, 1))). (A.18) 44 S. MITRA There exists a positive constant C, such that the solution q of the heat equation (A.13) satisfies the following estimate kqkL2(T -T0,T ;H3(TL×(0,1)))H3/2(T -T0,T ;L2(TL×(0,1))) 6 C(kGb ,qkL2(T -T0,T ;H1(TL×(0,1)))H1/2(T -T0,T ;L2(TL×(0,1))) + k(t b + u1x b )kL2(T -T0,T ;H3/2(TL×{1}))H3/4(T -T0,T ;L2(TL×{1})) + kqT kH2(TL×(0,1))) 6 C(kb kL2(T -T0,T ;H1(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) + k(t b + u1x b )kL2(T -T0,T ;H3/2(TL×{1}))H3/4(T -T0,T ;L2(TL×{1})) + kqT kH2(TL×(0,1))). (A.19) In the final step of the estimate (A.19) we have used (A.16). Note that the constant C in (A.19) might depend on T0. The regularity (A.18) implies that q C0 ([T -T0, T]; H1 (TL ×(0, 1))). Hence using the regularity assumption T H1 (TL × (0, 1)), we obtain the following by solving (A.6)1 and (A.6)6, HT0 2 , where we have set HT0 2 = C0 ([T - T0, T]; H1 (TL × (0, 1))) C1 ([T - T0, T]; L2 (TL × (0, 1))). (A.20) Moreover, we have the estimate kkH T0 2 6 C(kqkL(T -T0,T ;H1(TL×(0,1))) + kT kH1(TL×(0,1))), (A.21) for some positive constant C independent of T0. For the proofs of (A.20) and (A.21) one can follow the arguments used in proving ([39], Lem. 2.4). Now one considers the equations (A.6)7­(A.6)8. Using standard trace theorem and (A.18) one in particular has (t + u1x)q |TL×{1} L2 (T - T0, T; L2 (TL × {1})). (A.22) Now the regularity (A.22) and the assumption (2.4) on (T , 1 T ) furnish the following regularities for : L2 (T - T0, T; H4 (TL × {1})) H1 (T - T0, T; H2 (TL × {1})) H2 (T - T0, T; L2 (TL × {1})). (A.23) The above regularity result for is a consequence of the fact that the corresponding damped beam operator is the generator of an analytic semigroup. This result can be found in [9] (see also [35]). Using standard interpolation results it is not hard to observe from (A.23) that HT0 3 , where we have set HT0 3 = H5/4 (T - T0, T; H3/2 (TL × {1})) H1 (T - T0, T; H2 ((TL × {1}))) H2 (T - T0, T; L2 (TL × {1})) H3/4 (T - T0, T; H5/2 ((TL × {1}))). (A.24) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 45 Furthermore, one has the following kkH T0 3 6 C(k(t + u1x)qkL2(T -T0,T ;L2(TL×{1})) + k(T , 1 T )kH3(TL×{1})×H1(TL×{1})), (A.25) for some positive constant C independent of T0. One can obtain a constant C independent of T0 in the inequality (A.25) by defining (t + u1x)q equal zero in the times interval (0, T - T0) and following the line of arguments already used in (A.12). Hence from (A.21) and (A.25) one obtains that (, ) HT0 2 × HT0 3 . (A.26) So far we have observed that for 0 < T0 < T, LT0 (defined in (A.7)) maps HT0 1 to HT0 2 × HT0 3 , which is obviously a subset of HT0 1 . A.4 Choice of T0 small enough such that the map LT0 admits a fixed point in the space HT0 1 Let us compare the space HT0 1 with HT0 2 × HT0 3 to observe that there exists a constant s > 0 such that k(·, ·)kH T0 1 6 CTs 0 k(·, ·)kH T0 2 ×H T0 3 , (A.27) for some positive constant C independent of T0. Now let (b 1, b 1) HT0 1 , (b 2, b 2) HT0 1 and both the pairs satisfy (A.5). Let the triplets (1, q1, 1) and (2, q2, 2) are solutions to (A.6) corresponding to (b 1, b 1) and (b 2, b 2) respectively. This implies (1, 1) = LT0 (b 1, b 1) and (2, 2) = LT0 (b 2, b 2). Considering the difference of the two linear systems solved by (1, q1, 1) and (2, q2, 2), one obtains that (d, qd, d) = (1 - 2, q1 - q2, 1 - 2) solves -td - u1xd + P0 () d = P0 () qd in (TL × (0, 1)) × (T - T0, T), -(tqd + u1xqd) - qd - P 0 () qd = -P 0 ()2 (b 1 - b 2) in (TL × (0, 1)) × (T - T0, T), zqd = -(t( b 1 - b 2) + u1x( b 1 - b 2)) on (TL × {1}) × (T - T0, T), zq = 0 on (TL × {0}) × (T - T0, T), qd(·, T) = 0 in TL × (0, 1), d(·, T) = 0 in TL × (0, 1), ttd + txxd + xxxxd = (t + u1x)qd on (TL × {1}) × (T - T0, T), d(T) = 0 and td(T) = 0 in TL × {1}, (A.28) The goal now is to show the following: kLT0 (b 1, b 1) - LT0 (b 2, b 2)kH T0 1 = k(1 - 2, 1 - 2)kH T0 1 = k(d, d)kH T0 1 6 CTs 0 k(b 1 - b 2, b - b 2)kH T0 1 , (A.29) for some positive constant C independent of T0. One recalls that during the analysis of the system (A.6), we obtained (A.8)­(A.25) with the constants in the estimates independent of T0, while the only exception was (A.19). Now we will explain how to obtain an 46 S. MITRA estimate for qd = q1 - q2, analogous to (A.19) but with a constant independent of T0. In that direction one first obtains an estimate similar to (A.12) with q replaced by qd, b replaced by (b 1 - b 2), qT by zero and b by ( b 1 - b 2). Then from (A.28) one computes the following system similar to (A.13): -tqd - qd = - P0 ()2 (b 1 - b 2) + u1xqd + P0 () qd = Gb 1-b 2,qd in (TL × (0, 1)) × (T - T0, T), zqd = -(t( b 1 - b 2) + u1x( b 1 - b 2)) on (TL × {1}) × (T - T0, T), zqd = 0 on (TL × {0}) × (T - T0, T), qd(·, T) = 0 in TL × (0, 1). (A.30) Similar to (A.15) and (A.17)1, one can verify that: Gb 1-b 2,qd L2 (T - T0, T; H1 (TL × (0, 1))) H1/2 (T - T0, T; L2 (TL × (0, 1))), zqd L2 (T - T0, T; H3/2 (TL × {0, 1})) H3/4 (T - T0, T; L2 (TL × {0, 1})). (A.31) One further observes that Gb 1-b 2,qd (·, T) = 0 in TL × (0, 1), zqd |TL×{0,1} (·, T) = 0. (A.32) We will now define extensions of Gb 1-b 2 and the normal trace zqd in (-T, T). We define: e Gb 1-b 2,qd (·, t) = Gb 1-b 2,qd (·, t) in (T - T0, T), Gb 1-b 2,qd (·, 2(T - T0) - t) in (T - 2T0, T - T0), 0 in (-T, T - T0) (A.33) In a similar way by reflection one defines g zqd, the extension of zqd in (-T, T). By virtue of (A.31) and (A.32) it is not hard to check that ( e Gb 1-b 2,qd L2 (-T, T; H1 (TL × (0, 1))) H1/2 (-T, T; L2 (TL × (0, 1))), g zqd L2 (-T, T; H3/2 (TL × {0, 1})) H3/4 (-T, T; L2 (TL × {0, 1})), (A.34) and most importantly k e Gb 1-b 2,qd kL2(-T,T ;H1(TL×(0,1)))H1/2(-T,T ;L2(TL×(0,1))) 6 2kGb 1-b 2,qd kL2(T -T0,T ;H1(TL×(0,1)))H1/2(T -T0,T ;L2(TL×(0,1))), k g zqdkL2(-T,T ;H3/2(TL×{0,1}))H3/4(-T,T ;L2(TL×{0,1})) 6 2kzqdkL2(T -T0,T ;H3/2(TL×{0,1}))H3/4(T -T0,T ;L2(TL×{0,1})), (A.35) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 47 with the constants in the inequalities (A.35) independent of T0. One can now solve (A.30) in the extended interval (-T, T) and furnish kqdkL2(T -T0,T ;H3(TL×(0,1)))H3/2(T -T0,T ;L2(TL×(0,1))) 6 kqdkL2(-T,T ;H3(TL×(0,1)))H3/2(-T,T ;L2(TL×(0,1))) 6 C k e Gb 1-b 2,qd kL2(-T,T ;H1(TL×(0,1)))H1/2(-T,T ;L2(TL×(0,1))) +k g zqdkL2(-T,T ;H3/2(TL×{0,1}))H3/4(-T,T ;L2(TL×{0,1})) 6 2C kGb 1-b 2,qd kL2(T -T0,T ;H1(TL×(0,1)))H1/2(T -T0,T ;L2(TL×(0,1))) +kzqdkL2(T -T0,T ;H3/2(TL×{0,1}))H3/4(T -T0,T ;L2(TL×{0,1})) 6 C kb 1 - b 2kL2(T -T0,T ;H1(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) +kt( b 1 - b 2) + u1x( b 1 - b 2)kL2(T -T0,T ;H3/2(TL×{1}))H3/4(T -T0,T ;L2(TL×{1})) , (A.36) where the fourth step of (A.36) from the third one follows by using (A.35), the final step from the penultimate one uses an estimate similar to (A.16) and (A.30)2. We specify that the constants in the inequalities of (A.36) only depends on the measure of (-T, T) i.e. only on T but not on T0. Using interpolation it is evident from (A.36) that qd C0 ([T - T0, T]; H1 (TL × (0, 1))). Further the interpolation can be first performed in the extended interval (-T, T) and then take its restriction on (T - T0, T) to furnish the following from (A.36): kqdkL(T -T0,T ;H1(TL×(0,1))) 6 C kb 1 - b 2kL2(T -T0,T ;H1(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) +kt( b 1 - b 2) + u1x( b 1 - b 2)kL2(T -T0,T ;H3/2(TL×{1}))H3/4(T -T0,T ;L2(TL×{1})) , (A.37) for some positive constant C independent of T0. In a similar spirit we obtain the following trace estimate from (A.36): k(t + u1x)qdkL2(T -T0,T ;L2(TL×{1})) 6 C kb 1 - b 2kL2(T -T0,T ;H1(TL×(0,1)))H1(T -T0,T ;L2(TL×(0,1))) +kt( b 1 - b 2) + u1x( b 1 - b 2)kL2(T -T0,T ;H3/2(TL×{1}))H3/4(T -T0,T ;L2(TL×{1})) , (A.38) for some positive constant C independent of T0. Similar to (A.21) and (A.25) one also obtains the following for d and d : ( kdkH T0 2 6 CkqdkL(T -T0,T ;H1(TL×(0,1))), kdkH T0 3 6 Ck(t + u1x)qdkL2(T -T0,T ;L2(TL×{1})), (A.39) for some positive constant C independent of T0. Using (A.37) and (A.38) into (A.39) one concludes that k(d, d)kH T0 2 ×H T0 3 6 Ck(b 1 - b 2, b 1 - b 2)kH T0 1 , (A.40) for some positive constant C independent of T0. Using (A.27) and (A.40) one concludes the proof of (A.29). In view of (A.29), for T - T0 close to T, i.e for T0 small enough, the map LT0 is a contraction from HT0 1 to itself. Hence applying the Banach fixed point theorem we obtain that for T0 small enough, the map LT0 admits a unique fixed point (b , b ) in HT0 1 . As (b , b ) = LT0 (b , b ), we also have that (b , b ) belongs to HT0 2 × HT0 3 and 48 S. MITRA the following regularities coming from (A.18) and (A.23): C0 ([T - T0, T]; H1 (TL × (0, 1))) C1 ([T - T0, T]; L2 (TL × (0, 1))), q L2 (T - T0, T; H3 (TL × (0, 1))) H3/2 (T - T0, T; L2 (TL × (0, 1))), L2 (T - T0, T; H4 (TL × {1})) H1 (T - T0, T; H2 (TL × {1})) H2 (T - T0, T; L2 (TL × {1})), (A.41) provided T0 is sufficiently small. Further (A.41) infers that C0 ([T - T0, T]; H1 (TL × (0, 1))]), q C0 ([T - T0, T]; H2 (TL × (0, 1))) C0 ([T - T0, T]; H3 (TL × {1})) C1 ([T - T0, T]; H1 (TL × (0, 1))). (A.42) The continuities (A.42) in time and the system (2.3) can be used to check the following compatibilities at time t = T - T0 : (i) zq(·, T - T0) = -(t + u1x)(·, T - T0) on TL × {1}, (ii) zq(·, T - T0) = 0 on TL × {0}. (A.43) Further one recalls that in proving (A.41) we did no assumption on the size of the initial datum. In view of (A.42), (A.43) and using that the linearity of the system (2.3) the solution (, q, ) can be extended to the time interval (0, T) by iteration in order to prove (2.6). This finishes the proof of (2.6) and thus of Lemma 2.2. A.5 A lemma on parabolic regularization In this section we prove a result on parabolic regularization for a heat type equation with non homoge- neous Neumann boundary condition. This result is in particular used in proving the inequality (4.14) from the information (4.12). Lemma A.1. We recall the notations introduced in (3.13). Let ( f L2 (0, T; H1 (TL × (0, 1))) H 1 2 (0, T; L2 (TL × (0, 1))), g L2 (0, T; H 3 2 (TL × {0, 1})) H 3 4 (0, T; L2 (TL × {0, 1})), (A.44) qT H1 (TL × (0, 1)) and the following compatibility is satisfied: zqT = g(·, T) in TL × {0, 1}. (A.45) Then for every 0 < 6 T, q(·, T - ) H2 (TL × (0, 1)), where q solves -(tq + u1xq) - q - P0 () q = f in Qex T , zq = g on T1 T T0 T , q(·, T) = qT in TL × (0, 1), (A.46) OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 49 with P, , u1 introduced respectively in (1.6) and (1.18). Further the following inequality holds: kq(·, T - )kH2(TL×(0,1)) 6 C kqT kH1(TL×(0,1)) + kfk L2(0,T ;H1(TL×(0,1)))H 1 2 (0,T ;L2(TL×(0,1))) +kgk L2(0,T ;H 3 2 (TL×{0,1}))H 3 4 (0,T ;L2(TL×{0,1})) , (A.47) for some positive constant C. Proof. We first apply ([28], Thm. 5.3, p. 32) (of course here backward in time) to get: kqkL2(0,T ;H2(TL×(0,1)))H1(0,T ;L2(TL×(0,1)))) 6 C kqT kH1(TL×(0,1)) + kfkL2(Qex T ) +kgk L2(0,T ;H 1 2 (TL×{0,1}))H 1 4 (0,T ;L2(TL×{0,1})) , (A.48) for some positive constant C. Now we introduce: q1(·, t) = (T - t)q(·, t) in Qex T . (A.49) One observes that q1 solves: -(tq1 + u1xq1) - q1 - P0 () q1 = (T - t)f + q in Qex T , zq1 = (T - t)g on T1 T T0 T , q1(·, T) = 0 in TL × (0, 1), (A.50) In view of (A.44) and (A.48): (T - t)f + q L2 (0, T; H1 (TL × (0, 1))) H 1 2 (0, T; L2 (TL × (0, 1))) and (T - t)g L2 (0, T; H 3 2 (TL × {0, 1})) H 3 4 (0, T; L2 (TL × {0, 1})). Of course the compatibility zq1(·, T) = 0 = (T - T)zg(·, T) is satisfied on TL × {0, 1}. Hence once again applying ([28], Thm. 5.3, p. 32), we furnish that q1 L2 (0, T; H3 (TL × (0, 1))) H 3 2 (0, T; (TL × (0, 1))). Consequently using ([28], Thm. 2.1, Sect. 2.2) in the interval (T - , T) one obtains that q1(·, T - ) H2 (TL × (0, 1)). Hence the definition (A.49) clearly implies that q(·, T - ) H2 (TL × (0, 1)), for every 0 < 6 T and the estimate (A.47) follows. Acknowledgements. The work was partially done when the author was a member of Institut de Mathematiques de Toulouse. The author wishes to thank the ANR project ANR-15-CE40-0010 IFSMACS as well as the Indo-French Centre for Applied Mathematics (IFCAM) for the funding provided during that period. The author is presently a member of 50 S. MITRA Institute of Mathematics, University of Wurzburg where the present article is finalized. The author deeply thank the anonymous referees for valuable comments which helped a lot in improving the earlier version of the article. References [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system. Electron. J. Differ. Equ. 2000 (2000) 1­15. [2] M. Badra, S. Ervedoza and S. 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Springer-Verlag, New York-Heidelberg (1972). OBSERVABILITY AND UNIQUE CONTINUATION OF THE ADJOINT 51 [29] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Vol. 10. Oxford Lecture Series in Mathematics and its Applications. Compressible Models, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1998). [30] S. Mitra, Analysis and Control of Some Fluid Models with Variable Density. Theses, Universite Toulouse III - Paul Sabatier. Available from https://tel.archives-ouvertes.fr/tel-01959694v1 (2018). [31] S. Mitra, Carleman estimate for an adjoint of a damped beam equation and an application to null controllability. J. Math. Anal. Appl. 484 (2020) 123718. [32] S. Mitra, Local existence of Strong solutions for a fluid-structure interaction model. J. Math. Fluid. Mech. 22 (2020) 60. [33] N. Molina, Local exact boundary controllability for the compressible Navier-Stokes equations. SIAM J. Cont. Optim. 57 (2019) 2152­2184. [34] K. 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Zajaczkowski, On nonstationary motion of a compressible barotropic viscous fluid with boundary slip condition. J. Appl. Anal. 4 (1998) 167­204. [41] X. Zhang, Exact controllability of semilinear plate equations. Asymptot. Anal. 27 (2001) 95­125. [42] X. Zhang and E. Zuazua, A sharp observability inequality for Kirchhoff plate systems with potentials. Comput. Appl. Math. 25 (2006) 353­373. [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system. Electron. J. Differ. Equ. 2000 (2000) 1–15. [2] M. Badra, S. Ervedoza and S. Guerrero, Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 33 (2016) 529–574. [3] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems. Systems & Control: Foundations & Applications, 2nd edn. Birkhäuser Boston, Inc., Boston, MA (2007). [4] M. Boulakia and S. Guerrero, Local null controllability of a fluid-solid interaction problem in dimension 3. J. Eur. Math. Soc. (JEMS) 15 (2013) 825–856. [5] M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM: COCV 14 (2008) 1–42. [6] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Vol. 183. Applied Mathematical Sciences. Springer, New York (2013). [7] M. Chapouly, On the global null controllability of a Navier-Stokes system with Navier slip boundary conditions. J. Differ. Equ. 247 (2009) 2094–2123. [8] F. W. Chaves-Silva, R. Lionel and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control. J. Math. Pures Appl. 101 (2014) 198–222. [9] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems. Pacific J. Math. 136 (1989) 15–55. [10] S. Chowdhury, D. Maity, M. Ramaswamy and J.-P. Raymond, Local stabilization of the compressible Navier-Stokes system, around null velocity, in one dimension. J. Differ. Equ. 259 (2015) 371–407. [11] S. Chowdhury, M. Ramaswamy and J.-P. Raymond, Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension. SIAM J. Control Optim. 50 (2012) 2959–2987. [12] J. M. Coron and A. V. Fursikov, Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429–448. [13] S. Ervedoza and M. Savel, Local boundary controllability to trajectories for the 1D compressible Navier Stokes equations. ESAIM: COCV 24 (2018) 211–235. [14] S. Ervedoza, O. Glass and S. Guerrero, Local exact controllability for the two- and three-dimensional compressible Navier-Stokes equations. Comm. Part. Differ. Equ. 41 (2016) 1660–1691. [15] S. Ervedoza, O. Glass, S. Guerrero and J.-P. Puel, Local exact controllability for the one-dimensional compressible Navier-Stokes equation. Arch. Ration. Mech. Anal. 206 (2012) 189–238. [16] C. Fabre and G. Lebeau, Prolongement unique des solutions de l’equation de Stokes. Comm. Part. Differ. Equ. 21 (1996) 573–596. [17] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3 (2001) 358–392. [18] E. Fernández-Cara, M. González-Burgos, S. Guerrero and J.-P. Puel. Null controllability of the heat equation with boundary Fourier conditions: the linear case. ESAIM: COCV 12 (2006) 442–465. [19] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. [20] F. Flori and P. Orenga, On a nonlinear fluid-structure interaction problem defined on a domain depending on time. Nonlinear Anal. 38 (1999) 549–569. [21] F. Flori and P. Orenga, Fluid-structure interaction: analysis of a 3-D compressible model. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 753–777. [22] X. Fu, X. Zhang and E. Zuazua, On the optimality of some observability inequalities for plate systems with potentials, in Phase Space Analysis of Partial Differential Equations, Vol. 69. Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, MA (2006) 117–132. [23] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Vol. 34. Lecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). [24] S. Guerrero, Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions. ESAIM: COCV 12 (2006) 484–544. [25] O. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. 87 (2007) 408–437. [26] O. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39–72. [27] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Translated from the French by P. Kenneth. In Vol. 181 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Heidelberg (1972). [28] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. II. Translated from the French by P. Kenneth, In Vol. 182 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Heidelberg (1972). [29] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Vol. 10. Oxford Lecture Series in Mathematics and its Applications. Compressible Models, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1998). [30] S. Mitra, Analysis and Control of Some Fluid Models with Variable Density. Thèses, Université Toulouse III - Paul Sabatier. 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COCV_2021__27_S1_A20_08f8e0a02-1f48-4170-93d7-8b73ed406beecocv19016010.1051/cocv/202006910.1051/cocv/2020069 Optimal control of the two-dimensional Vlasov-Maxwell system 0000-0003-3631-7491 Weber Jörg * Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany. *Corresponding author: [e-mail] SupplementS19 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF)Full (DJVU)

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

Relativistic Vlasov-Maxwell system optimal control with PDE constraints nonlinear partial differential equations calculus of variations 49J20 35Q61 35Q83 82D10 idline ESAIM: COCV 27 (2021) S19 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S19 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020069 www.esaim-cocv.org OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM Jorg Weber* Abstract. The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrody- namics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are real- izable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation. Mathematics Subject Classification. 49J20, 35Q61, 35Q83, 82D10. Received October 9, 2019. Accepted October 15, 2020. 1. Introduction 1.1. The system The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system. Collisions among the plasma particles can be neglected if the plasma is sufficiently rarefied or hot. The particles only interact through electromagnetic fields created collectively. We only consider plasmas consisting of just one particle species, for example, electrons. This work can immediately be adapted to the case of several particle species. For the sake of simplicity, we choose units such that physical constants like the speed of light, the charge and rest mass of an individual particle are normalized to unity. Also, for simplicity, we do not consider material parameters, for example for modeling superconductors in a fusion reactor, that is to say permittivity and permeability, which would appear in the Maxwell equations. Allowing the particles to move at relativistic speeds, the three-dimensional Vlasov-Maxwell system (on whole space) is given by tf + b p · xf + (E + b p × B) · pf = 0, (1.1a) tE - curlx B = -jf , (1.1b) Keywords and phrases: Relativistic Vlasov-Maxwell system, optimal control with PDE constraints, nonlinear partial differential equations, calculus of variations. Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 J. WEBER tB + curlx E = 0, (1.1c) divx E = , (1.1d) divx B = 0, (1.1e) f = 4 Z f dp, (1.1f) jf = 4 Z b pf dp. (1.1g) Here, the Vlasov equation is (1.1a) and the Maxwell equations of electrodynamics are (1.1b) to (1.1e). Vlasov and Maxwell equations are coupled via (1.1f) and (1.1g) rendering the whole system nonlinear due to the product term (E + b p × B) · pf. In particular, f = f(t, x, p) denotes the density of the particles on phase space, and E = E(t, x), B = B(t, x) are the electromagnetic fields, whereby t R, x, and p R3 stand for time, position in space, and momentum. The abbreviation b p = p 1+|p|2 denotes the velocity of a particle with momentum p. Furthermore, some moments of f appear as source terms in the Maxwell equations, that is to say jf and f which equal the current and charge density up to the constant 4. However, we have not readily explained the source term in (1.1d). If we would demand divx E = f this would lead to a seeming contradiction: Formally integrating this equation with respect to x (and assuming E 0 rapidly enough at ) leads to R f dx = 0 and hence f = 0 by ° f 0. This problem is caused by our simplifying restriction to one species of particles and is resolved by adding some terms to f , for example a neutralizing background density, so that we have a total charge density with vanishing space integral. However, usually this background density is neglected, see [6] for example. Considering the Cauchy problem for the above system, we moreover demand f(0, x, p) = ° f(x, p), E(0, x) = E(x), B(0, x) = B(x), where ° f 0, E, and B are some given initial data. Unfortunately, existence of global (i.e., global in time), classical (i.e., continuously differentiable) solutions for general (smooth) data is an open problem in the three-dimensional setting. It is only known that global weak solutions can be obtained. This was proved by R.J. Di Perna and P.L. Lions [1]. For a detailed insight concerning this matter we recommend the review article [14] by G. Rein. As for global existence of classical solutions, the strategy was to first consider lower dimensional settings. R. Glassey and J. Schaeffer proved global existence of classical solutions in the one and one-half [4], the two [6, 7], and the two and one-half dimensional setting [5]. Since it is convenient to have global existence of classical solutions on hand, we consider a two-dimensional version of the problem in this work. Notice that mutatis mutandis all results and techniques can be applied to the full three-dimensional setting once global existence of classical solutions has been proved. The restriction to `two-dimensionality' is to be understood in the following sense: All functions shall be independent of the third variables x3 and p3. This new model describes a plasma where the particles only move in the (x1, x2)-plane, but the plasma extends in the x3-direction infinitely. To ensure that these properties are preserved in time, we have to demand that the electric field lies in the plane and that the magnetic field is perpendicular to the plane so that E = (E1(t, x), E2(t, x), 0) and B = (0, 0, B(t, x)). Here and in the following, let x = (x1, x2) and p = (p1, p2) be two-dimensional variables. Note that hence the magnetic field is always divergence free with respect to x, so that (1.1e) is always satisfied and will no longer be mentioned. The two-dimensional Vlasov-Maxwell system reads tf + b p · xf + (E + (b p2, -b p1)B) · pf = 0, tE1 - x2 B = -jf,1, tE2 + x1 B = -jf,2, OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 3 tB + x1 E2 - x2 E1 = 0, divx E = , (f, E, B)|t=0 = ° f, E, B . The goal is to control the plasma properly. Thereto we add external currents U to the system, in applications generated by electric coils, that induce external electromagnetic fields affecting the plasma particles. These currents, like the electric field and the current density of the plasma particles, have to lie in the plane and have to be independent of the third space coordinate. Of course, there will be an external charge density ext corresponding to the external current. It is natural to assume local conservation of the external charge, i.e., text + divx U = 0. Hence, we can eliminate ext via ext = ext - Z t 0 divx U d. The initial value ext will be added to the background density, which is then neglected, as was already mentioned above. In the following, we consider the controlled relativistic Vlasov-Maxwell system tf + b p · xf + E - b p B · pf = 0, tE + x B = -jf - U, tB + curlx E = 0, divx E = f - Z t 0 divx U d, (f, E, B)|t=0 = ° f, E, B (CVM) on a finite time interval [0, T] with given T > 0; here we introduced the abbreviations a = (-a2, a1) for a R2 , x B = (-x2 B, x1 B), and the scalar curl operator curlx E = x1 E2 - x2 E1 in 2D. It is well-known that Lq -norms (with respect to (x, p), 1 q ) of f are preserved in time by f solving the Vlasov equation since the vector field b p, E - b p B is divergence free in (x, p). Therefore, especially, the L1 -norm (with respect to x) of the charge density f is constant in time. The outline of our work is the following: In the first part, we have to prove unique solvability of (CVM). Of course, some regularity assumptions on the external current and the initial data have to be made in order to prove existence of classical solutions. In the second part, we consider an optimal control problem. On the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, the external currents (the costs) shall be as small as possible. These two aims lead to minimizing some objective function. To analyze the optimal control problem, it is convenient to show differentiability of the control-to-state operator first. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation. The steps mentioned above were carried out by P. Knopf [10] and, only considering realizable control fields, by Knopf and the author [11] for the three-dimensional Vlasov-Poisson system with an external magnetic field. The consideration of the latter setting has the advantage of being able to work in three dimensions, but has the disadvantage of only imposing Poisson's equation, that is, Maxwell's equations with an internal magnetic field sufficiently small to be neglected, for the electromagnetic fields, which make things easier due to the elliptic nature of Poisson's equation in contrast to the hyperbolic nature of the (time evolutionary) Maxwell equations. 4 J. WEBER Also other approaches for controlling a Vlasov-Maxwell plasma have been considered in the literature, but they are different in nature compared to our approach. We refer to [3, 13] and the references therein. 1.2. Some notation and simple computations We denote by Br(x) the open ball with radius r > 0 and center x X where X is a normed space. Further- more, we abbreviate Br := Br(0). For a function g: [0, T] × Rj Rk we abbreviate g(t) := g(t, ·): Rj Rk for 0 t T. Also, we write supp g for the support of g, and suppx g (and likewise suppp g) for the support of a function g = g(t, x, p) with respect to x, that is, the closure of the set of all x such that there are t and p with g(t, x, p) 6= 0. Sometimes, denoting certain function spaces, we omit the set where these functions are defined. Which set is meant should be obvious, in fact the largest possible set like [0, T] × Rj (including time) or Rj (not including time). Moreover, Ck b denotes the space of k-times continuously differentiable functions (on a given set) such that all derivatives up to order k are bounded. The index c, as in Ck c , indicates that such functions are compactly supported. Furthermore, X , Y means that X is continuously embedded in Y . Finally, we denote by C > 0 some generic constant that may change from line to line (also inside a line) and may depend on some quantities; we will clarify at the beginning of each section on what quantities C may depend in this section or write the dependence explicitly as C(r), for example. 1.3. Maxwell equations We will have to consider first order and second order Maxwell equations. It is well-known that they are equivalent and that the divergence equations propagate in time if local conservation of charge holds, i.e., t + divx j = 0. (LC) In our two-dimensional setting with fields (E1, E2, 0) and (0, 0, B) we conclude: Lemma 1.1. Let E and B be of class C2 and E, B C2 , and , j C1 . If the conditions divx E = (0) (CC) and t + divx j = 0 (LC) are satisfied, then the systems of first order Maxwell equations tE + x B = -j, tB + curlx E = 0, (E, B)(0) = E, B , (1stME) OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 5 and second order Maxwell equations 2 t E - xE = -tj - x, E(0) = E, tE(0) = - x B - j(0), 2 t B - xB = curlx j, B(0) = B, tB(0) = - curlx E, (2ndME) are equivalent. Moreover, then also divx E = globally in time. We give a quite general condition that guarantees (LC). Lemma 1.2. Let g C0 , and f, d, and K of class C1 with divp K = 0 and f(t, x, ·) compactly supported for each t [0, T] and x R2 . Assume tf + b p · xf + K · pf = g and that R g dp = 0 holds. Then = f - R t 0 divx d d and j = jf + d satisfy (LC). Proof. First, t - R t 0 divx d d + divx d = 0 is obvious. Furthermore, integrating the Vlasov equation with respect to p instantly yields tf + divx jf = 0. Since (2ndME) consists of Cauchy problems for wave equations, we will need a solution formula for the 2D wave equation. In two dimensions, the (in C2 unique) solution of the Cauchy problem 2 t u - xu = f, u(0) = g, tu(0) = h, is given by the well known formula u(t, x) = 1 2 Z t 0 Z |x-y|<t- f(, y) q (t - ) 2 - |x - y| 2 dyd + 1 2 Z B1 g(x + ty) + tg(x + ty) · y + th(x + ty) q 1 - |y| 2 dy if the data are smooth. 1.4. Control space for classical solutions In the following let L > 0, U VL := d W2,1 0, T; C4 b R2 ; R2 | d(t, x) = 0 for |x| L , and let VL be equipped with the W2,1 0, T; C4 b R2 ; R2 -norm. 2. Existence results 2.1. Estimates on the fields 2.1.1. A generalized system The most important tool to get certain bounds is to have representations of the fields. One can use the solution formula for the wave equation and after some transformation of the integral expressions Gronwall-like 6 J. WEBER estimates on the density and the fields can be derived. These bounds, for instance, will imply that the sequences constructed in Section 2.3 converge in a certain sense. Having that in mind it is useful not to work with the system (CVM) but with a somewhat generalized one with second order Maxwell equations: tf + b p · xf + (p)K · pf = g, 2 t E - xE = -tjf - td - xf + x Z t 0 divx d d, 2 t B - xB = curlx jf + curlx d, (f, E, B)(0) = ° f, E, B , tE(0) = - x B - j° f - d(0), tB(0) = - curlx E, (GVM) with initial data ° f of class C1 c and E, B of class C2 b. We assume that we already have functions f, K of class C1 , E, B of class C2 , g of class C0 b, d of class C1 0, T; C2 b and of class C1 b satisfying (GVM). Furthermore, we assume that divp K = 0 and that there is a r > 0 such that f(t, x, p) = g(t, x, p) = 0 if |p| > r. 2.1.2. Estimates on the density Lemma 2.1. The density f and its (x, p)-derivatives are estimated by i) kf(t)k ° f + Z t 0 kg()kd if g C0 and ii) kx,pf(t)k x,p ° f + Z t 0 kx,pg()kd exp Z t 0 kx,p(K)()kd if g C1 . Proof. This is easily proved by considering the characteristics X = X(s, t, x, p), P = P(s, t, x, p) of the Vlasov equation in (GVM), which are defined via X = b P, P = (P)K(s, X, P) with initial condition (X, P)(t, t, x, p) = (x, p). Then f(t, x, p) = ° f((X, P)(0, t, x, p)) + Z t 0 g(s, (X, P)(s, t, x, p))ds and, if g C1 , x,pf(t, x, p) = x,p ° f ((X, P)(0, t, x, p)) + Z t 0 (x,pg)(s, (X, P)(s, t, z))ds - Z t 0 (x,pf)(s, (X, P)(s, t, z))(x,p(K))(s, (X, P)(s, t, z))ds; OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 7 see [12], Section 5. The asserted estimates are hence straightforwardly derived. The p-support condition on f is satisfied if supp BR for some R > 0: Obviously for |p| > max{R, r, r0} (where suppp ° f Br0 ) we have P(s, t, x, p) = 0, hence P(s, t, x, p) = p and therefore ° f((X, P)(0, t, x, p)) = g(s, (X, P)(s, t, x, p)) = 0. 2.1.3. Representation of the fields In the following the constant C may depend on T, r, and (i.e., its C1 b-norm), and we use the abbreviations = y - x t - , es = -2( + b p) 1 + b p · , bs = -2 · b p 1 + b p · , et = -2 1 - |b p| 2 ( + b p) (1 + b p · ) 2 , bt = -2 1 - |b p| 2 · b p (1 + b p · ) 2 , where t, [0, T], x, y, p R2 . We state some fundamental properties which will be used several times: Remark 2.2. i) For |p| r and || 1 we can estimate |p(bs)|, |p(es)|, |p(bs)|, |p(es)|, |bt|, |et|, (,p)(bt) , (,p)(et) by a constant C(r) > 0 only depending on r, since |1 + b p · | 1 - |b p||| 1 - r 1 + r2 > 0. ii) We compute Z |x-y|<t- dy q (t - ) 2 - |x - y| 2 = 2 Z t- 0 s (t - ) 2 - s2 - 1 2 ds = 2(t - ) and Z t 0 Z |x-y|<t- dyd (t - ) l+1 q 1 - || 2 = Z t 0 Z |x-y|<t- dyd (t - ) l q (t - ) 2 - |x - y| 2 = 2 Z t 0 (t - ) -l+1 d 2 2 - l T2-l = C(T, l) < for l < 2. Now we can derive integral expressions for the fields E and B proceeding similarly to [6]. Lemma 2.3. We have E = E0 + ES + ET + ED and B = B0 + BS + BT + BD where E0 , B0 are functionals of the initial data and d(0), and where ESj = Z t 0 Z |x-y|<t- Z (p(esj) + esj) · Kf + (esj)g q (t - ) 2 - |x - y| 2 dpdyd, BS = Z t 0 Z |x-y|<t- Z (p(bs) + bs) · Kf + (bs)g q (t - ) 2 - |x - y| 2 dpdyd, 8 J. WEBER ETj = Z t 0 Z |x-y|<t- Z etj (t - ) q (t - ) 2 - |x - y| 2 f dpdyd, BT = Z t 0 Z |x-y|<t- Z bt (t - ) q (t - ) 2 - |x - y| 2 f dpdyd, EDj = - 1 2 Z t 0 Z |x-y|<t- tdj - R 0 xj divx d ds q (t - ) 2 - |x - y| 2 dyd, BD = 1 2 Z t 0 Z |x-y|<t- curlx d q (t - ) 2 - |x - y| 2 dyd. Furthermore, the estimate kE(t)k + kB(t)k C ° f + E C1 b + B C1 b + kdkW1,1 (0,T ;C1 b) (2.1) + C Z t 0 ((1 + kK()k)kf()k + kg()k)d (2.2) holds. If additionally E, B C0 c, and d is compactly supported in x uniformly in t, so are also the fields. Proof. The representation formula are derived in much the same way as in [6], Theorem 1, the only difference is that here the source terms g and d appear. The support assertion is an immediate consequence of the representation formula. Physically, this is a result of the fact that electromagnetic fields can not propagate faster than the speed of light. Furthermore, the remaining estimate is a consequence of Remark 2.2. Remark 2.4. If f(t, x, ·) is compactly supported for every t, x, but not necessarily uniformly in t, x, nevertheless the fields are given by the formula above. For this, one does not need the uniformity. However, (2.1) can not be obtained in this situation. 2.1.4. First derivatives of the fields The next step is to differentiate these representation formulas and deriving certain estimates. The method is similar to the previous one. The constant C may now only depend on T, r, the initial data (i.e., their C2 b-norms), and kkC1 b . Lemma 2.5. If g C1 and d W2,1 0, T; C3 b , then the derivatives of the S-, T-, and D-terms are given by xi BS = Z t 0 Z |x-y|<t- Z (p(bs) + bs) · (fxi K + Kxi f) + bsxi g q (t - ) 2 - |x - y| 2 dpdyd, xi BT = Z t 0 Z |x-y|<t- Z bt (t - ) q (t - ) 2 - |x - y| 2 xi f dpdyd, xi BD = 1 2 Z t 0 Z |x-y|<t- xi curlx d q (t - ) 2 - |x - y| 2 dyd, OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 9 xi ES = Z t 0 Z |x-y|<t- Z (p(es) + es) · (fxi K + Kxi f) + esxi g q (t - ) 2 - |x - y| 2 dpdyd, xi ET = Z t 0 Z |x-y|<t- Z et (t - ) q (t - ) 2 - |x - y| 2 xi f dpdyd, xi ED = 1 2 Z t 0 Z |x-y|<t- txi d - R 0 xi x divx d ds q (t - ) 2 - |x - y| 2 dyd, tBS = Z t 0 Z |x-y|<t- Z (p(bs) + bs) · (ftK + Ktf) + bstg q (t - ) 2 - |x - y| 2 dpdyd + Z |x-y|<t Z (p(bs) + bs)|=0 · K(0)° f + bs|=0g(0) q t2 - |x - y| 2 dpdy, tBT = Z t 0 Z |x-y|<t- Z bt (t - ) q (t - ) 2 - |x - y| 2 tf dpdyd + Z |x-y|<t Z bt|=0 t q t2 - |x - y| 2 ° f dpdy, tBD = 1 2 Z t 0 Z |x-y|<t- t curlx d q (t - ) 2 - |x - y| 2 dyd + 1 2 Z |x-y|<t curlx d(0) q t2 - |x - y| 2 dy, tES = Z t 0 Z |x-y|<t- Z (p(es) + es) · (ftK + Ktf) + estg q (t - ) 2 - |x - y| 2 dpdyd + Z |x-y|<t Z (p(es) + es)|=0 · K(0)° f + es|=0g(0) q t2 - |x - y| 2 dpdy, tET = Z t 0 Z |x-y|<t- Z et (t - ) q (t - ) 2 - |x - y| 2 tf dpdyd + Z |x-y|<t Z et|=0 t q t2 - |x - y| 2 ° f dpdy, tED = - 1 2 Z t 0 Z |x-y|<t- 2 t d - x divx d q (t - ) 2 - |x - y| 2 dyd - 1 2 Z |x-y|<t tdj(0) q t2 - |x - y| 2 dy. Furthermore, the derivatives are estimated by kt,xE(t)k + kt,xB(t)k C(1 + kKk + kfk + kgk)(1 + kKk) 2 1 + ln+ |kx,pfk|[0,t] + Z t 0 kt,x,pK()kd + C Z t 0 kt,xg()kd + CkdkW2,1 (0,T ;C3 b) if kKk < . Here |kak|[0,t] := sup0t ka()k. Proof. Similarly as before, this is proved by following [6], now considering Theorem 3 therein. 10 J. WEBER 2.2. A priori bounds on the support with respect to p The most important property that is exploited later while showing global existence of a solution of (CVM), is to have a priori bounds on the p-support of f. This means: If we have a solution (f, E, B) of (CVM) on [0, T[ with f C1 and E, B of class C2 , we have to show that P(t) := inf{a > 0 | f(, x, p) = 0 for all |p| a, 0 t} + 3 is bounded, i.e., P(t) Q for 0 t < T where Q > 0 is some constant only dependent on T, the initial data (i.e., their C1 b-norms and P(0)), L, and kUkVL . In the following, the constant C may also only depend on these numbers. Note that, per definition, P is monotonically increasing and that |f| ° f . Moreover, P(t) < for each 0 t < T because we have an a priori estimate on the x-support of f via X 1, so that suppx f Bs, and on the compact set [0, t] × Bs the electromagnetic fields are bounded; hence the force field E - b p B is bounded there. Furthermore, (LC) holds by Lemma 1.2. Therefore, and with Remark 2.4 we have the representations of the fields as given in Lemma 2.3. Moreover, we can also demand that (f, E, B) solves tf + b p · xf + E - b p B · pf = 0, 2 t E - xE = -tjf - tU - xf + x Z t 0 divx U d, 2 t B - xB = curlx jf + curlx U, (f, E, B)(0) = ° f, E, B , tE(0) = - x B - j° f - U(0), tB(0) = - curlx E (CVM2nd) instead of (CVM) since both systems are equivalent by Lemma 1.1. We use the notation := y - x |y - x| , a b := a1b2 - a2b1, K := E - b p B and follow [7]. 2.2.1. Energy estimates The key in [7] is a sample of estimates that follow from the local energy conservation law t 1 2 |E| 2 + 1 2 B2 + 4 Z f q 1 + |p| 2 dp + divx -BE + 4 Z fp dp = 0. However, this equation is false in our situation due to the external currents U. But still we are able to prove an analogue of [7], Lemma 1: Lemma 2.6. Let 0 R T. The estimates i) sup xR2 Z |y-x|<R 1 2 |E| 2 + 1 2 B2 + 4 Z f q 1 + |p| 2 dp dy C, OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 11 ii) sup xR2 Z t 0 Z |y-x|=t-+R 1 2 (E · ) 2 + 1 2 (B + E) 2 + 4 Z f q 1 + |p| 2 (1 + b p · )dp dSyd C, iii) sup xR2 Z |y-x|<R 3 2 f dy C, iv) sup xR2 Z |y-x|<R Z f q 1 + |p| 2 dp 3 dy C hold for all t [0, T[. Proof. We split the electromagnetic fields into internal and external fields; precisely, they are defined by tEint + x Bint = -jf , tBint + curlx Eint = 0, (Eint, Bint)(0) = E, B and tEext + x Bext = -U, tBext + curlx Eext = 0, (Eext, Bext)(0) = 0. Indeed, the existence of (Eext, Bext) is guaranteed since the (time evolutionary) Maxwell equations form a linear, symmetric, hyperbolic system, see [9], Theorem I. Because of U VL we have Eext, Bext C0 0, T; H3 C1 0, T; H2 C1 ; furthermore k(Eext, Bext)(t)k Ck(Eext, Bext)(t)kH2 C Z T 0 kU()kH2 d CkUkVL = C by Sobolev's embedding theorem and the support condition on U. Because of the linearity of the Maxwell equations it holds that Eint := E - Eext and Bint := B - Bext solve their equations mentioned earlier and are of class C1 . Now let eint := 1 2 |Eint| 2 + 1 2 B2 int + 4 Z f q 1 + |p| 2 dp which is physically the energy density of the internal system and e := 1 2 |E| 2 + 1 2 B2 + 4 Z f q 1 + |p| 2 dp. 12 J. WEBER We have teint + divx -BintE int + 4 Z fp dp = Eint · tEint + BinttBint + 4 Z tf q 1 + |p| 2 dp + Eint,2x1 Bint + Bintx1 Eint,2 - Eint,1x2 Bint - Bintx2 Eint,1 + 4 Z xf · p dp = -Eint · jf - 4 Z K · pf q 1 + |p| 2 dp = -Eint · jf + 4E · Z fp q 1 + |p| 2 dp + 4B Z f divp p dp = Eext · jf where we made use of the respective Vlasov-Maxwell equations, p q 1 + |p| 2 = b p, and divp p = 0. We integrate this identity over a suitable set and arrive at Z t 0 Z |y-x|<t-+R Eext · jf dyd = Z t 0 Z |y-x|<t-+R eint + divy -BintE int + 4 Z fpdp dyd = - Z |y-x|<t+R eint(0, y)dy + Z |y-x|<R eint(t, y)dy + 1 2 Z t 0 Z |y-x|=t-+R eint + · -BintE int + 4 Z fpdp dSyd (2.3) after an integration by parts in (, y). The integrand of the last integral is non-negative because of 0 dint := 1 2 (Eint · ) 2 + 1 2 (Bint + Eint) 2 + 4 Z f q 1 + |p| 2 (1 + b p · )dp = 1 2 E2 int,12 1 + 1 2 E2 int,22 2 + 1 2 B2 int + Bint1Eint,2 - Bint2Eint,1 + 1 2 E2 int,22 1 + 1 2 E2 int,12 2 + 4 Z f q 1 + |p| 2 dp + · 4 Z fp dp = 1 2 E2 int,1 + 1 2 E2 int,2 + 1 2 B2 int + 4 Z f q 1 + |p| 2 dp + 1BintEint,2 - 2BintEint,1 + · 4 Z fp dp = eint + · -BintE int + 4 Z fp dp ; (2.4) note that 1 + b p · 1 - 1 · 1 = 0 and || = 1. The left hand side of (2.3) has to be investigated. The external fields are bounded by C, hence Z t 0 Z |y-x|<t-+R Eext · jf dyd C Z t 0 kjf ()kL1 d C Z t 0 kf ()kL1 d C (2.5) since the L1 -norm of f is constant in time. Now we can prove the assertions using (2.3), (2.4), and (2.5): OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 13 i) We have Z |y-x|<R eint dy Z |y-x|<t+R eint(0, y)dy + C C(R + t) 2 + C C since t, R T. Together with e 2eint + |Eext| 2 + |Bext| 2 2eint + C we conclude Z |y-x|<R e dy C + CR2 C. ii) Similarly, Z t 0 Z |y-x|=t-+R dint dSyd 2 Z |y-x|<t+R eint(0, y)dy + C C and d := 1 2 (E · ) 2 + 1 2 (B + E) 2 + 4 Z f q 1 + |p| 2 (1 + b p · )dp 2dint + 2|Eext| 2 + |Bext| 2 2dint + C yield Z t 0 Z |y-x|=t-+R d dSyd C + Ct(t + R) 2 C. iii) For r > 0 it holds that f = 4 Z f dp = 4 Z |p|<r f dp + 4 Z |p|r f dp Cr2 + 4r-1 Z |p|r f q 1 + |p| 2 dp C r2 + r-1 e . Now choose r := e 1 3 > 0 to derive f Ce 2 3 (if e = 0 then also f = 0) and hence Z |y-x|<R 3 2 f dy C Z |y-x|<R e dy C. iv) Similarly, Z f q 1 + |p| 2 dp C Z |p|<r 1 q 1 + |p| 2 dp + 1 1 + r2 Z |p|r f q 1 + |p| 2 dp C Z r 0 s 1 + s2 ds + 1 r2 e C r + r-2 e Ce 1 3 14 J. WEBER for again r := e 1 3 which yields Z |y-x|<R Z f q 1 + |p| 2 dp 3 dy C Z |y-x|<R e dy C. 2.2.2. Estimates on the fields The crucial problem is to estimate the fields properly. To this end, we use the representation formula stated in Lemma 2.3. Unfortunately, the estimates there can not be applied because, of course, we can not assume that P(t) is already bounded. Lemma 2.7. We have |ES1| + |ES2| + |BS| CP(t) ln P(t) + C Z t 0 (kE()k + kB()k)d, |ET1| + |ET2| + |BT| CP(t) ln P(t), |ED|, |BD| CkUkW1,1 (0,T ;C2 b) C, E0 , B0 C. Proof. The estimates on the S-and T-terms are derived in much the same way as in [7], Section 2. Note that the energy estimates of Lemma 2.6, that had to be modified in our situation, are enough to carry out the proofs therein. The estimate on the D-terms is derived straightforwardly, as well as the estimate on E0 , B0 , the latter parts only containing terms of the initial data and U(0). Now we can finally prove: Lemma 2.8. The a priori bound P(t) Q holds, where Q only depends on T, the C1 b-norms of the initial data, suppp ° f (which basically coincides with P(0)), L, and kUkVL . Proof. Collecting all bounds on the fields we arrive at kE(t)k + kB(t)k C + CP(t) ln P(t) + C Z t 0 (kE()k + kB()k)d. As in [7], this is enough to show that P(t) C + C Z t 0 P(s) ln P(s)ds, from which the assertion follows immediately. 2.3. Existence of classical solutions 2.3.1. The iteration scheme In the following we want to construct a solution of (CVM). We will only sketch the main ideas, since similar procedures have already been carried out in the literature, see for example [8], Section V. OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 15 We work with initial data ° f 0 of class C2 c, E, B of class C3 b, and control U VL that satisfy (CC), i.e., div E = ° f . We have to approximate these functions, so let ° fk ° f in C2 b, Ek E and Bk B in C3 b with ° fk C c , Ek, Bk C , and furthermore Uk U in VL with Uk C (note that C is dense in VL). The strategy to obtain a solution of (CVM) is the following: By iteration we construct densities fk and fields Ek, Bk in such a way that these functions will converge in a proper sense and that we may pass to the limit in (CVM). However, it is more convenient to work with a modified system. As the previous section suggests, it is crucial to control the p-support of f. For this reason we first consider a cut-off system on [0, T] where we modify the original Vlasov equation and use the second order Maxwell equations ((CC) and (LC) need not hold for the iterates): tf + b p · xf + (p) E - b p B · pf = 0, 2 t E - xE = -tjf - tU - xf + x Z t 0 divx U d, 2 t B - xB = curlx jf + curlx U, (f, E, B)(0) = ° f, E, B , tE(0) = - x B - j° f - U(0), tB(0) = - curlx E. (VM) Here, let the cut-off function be of class C c R2 with (p) = 1 for |p| 2Q. The property of the constant Q will imply that a solution of (VM) is also a solution of (CVM). We start the iteration with f0(t, x, p) := ° f0(x, p), E0(t, x) := E0(x), B0(t, x, p) := B0(x). The induction hypothesis is that fk, Ek, and Bk are of class C and that the fields are bounded. Given fk-1, Ek-1, and Bk-1, we firstly define fk as the solution of tfk + b p · xfk + (p) Ek-1 - b p Bk-1 · pfk = 0, fk(0) = ° fk, namely fk(t, x, p) = ° fk(Xk(0, t, x, p), Pk(0, t, x, p)) with the characteristics Xk = Xk(s, t, x, p), Pk = Pk(s, t, x, p) defined by Xk = b Pk, Xk(t, t, x, p) = x, Pk = (Pk) Ek-1 - b P k Bk-1 (s, Xk), Pk(t, t, x, p) = p, the dot referring to differentiation with respect to the first variable s. We conclude that Xk and Pk are of class C in all four variables by the induction hypothesis. This yields that even fk C . Since is compactly supported the p-support of fk is controlled by a constant C. Hence, fk and jfk are well-defined as C C1 b-functions. Secondly, we define Ek and Bk as the solution of 2 t Ek - xEk = -tjfk - tUk - xfk + x Z t 0 divx Uk d, 2 t Bk - xBk = curlx jfk + curlx Uk, (Ek, Bk)(0) = Ek, Bk , 16 J. WEBER tEk(0) = - x Bk - j° fk - Uk(0), tBk(0) = - curlx Ek. Indeed, we can solve these wave equations by applying the solution formula for the wave equation. Since the right-hand sides of the above equations are of class C and bounded, so are also Ek and Bk. Applying Lemmas 2.1, 2.3, and 2.5 then shows that the iterates are bounded in C1 b. As for the second derivatives, we differentiate (VM) and have, for example, txi fk + b p · xxi fk +Kk-1 · pxi fk = -xi Kk-1 · pfk, 2 t xi Ek - xxi Ek = -tjxi fk - txi Uk - xxi fk + x Z t 0 divx xi Uk d, 2 t xi Bk - xxi Bk = curlx jxi fk + curlx xi Uk, (xi fk, xi Ek, xi Bk)(0) = xi ° fk, xi Ek, xi Bk , txi Ek(0) = - x xi Bk - jxi ° fk - xi Uk(0), txi Bk(0) = - curlx xi Ek (2.6) and then apply the estimates of Lemmas 2.3 and 2.5. Note that for this we need four space derivatives in the definition of VL so that kxUkkW2,1 (0,T ;C3 b) is bounded. Likewise, one proceeds with the other second order derivatives. Altogether, the iterates are bounded in C2 b. After that, considering the difference of the iterates of the k-th step and the l-th step, Lemmas 2.1, 2.3, and 2.5 yield that the iteration sequences are even Cauchy sequences in C1 b, so that they converge to some (f, E, B) in the C1 b-norm. For later considerations it will be convenient that the density and the fields are even C2 b. Since all second derivatives are bounded in L [0, T] × Rj (j = 4 or 2 respectively) they converge, after extracting a suitable subsequence, in the weak-*-sense. Of course, these limits have to be the respective weak derivatives of f, E, and B. The remaining part is to show that the weak derivatives just obtained are in fact classical ones. For this sake, have a look at the representation formula for xi xj Bk; use system (2.6) and Lemma 2.5: xi xj Bk - xi B 0 k = Z t 0 Z |x-y|<t- Z bt (t - ) q (t - ) 2 - |x - y| 2 xi xj fk dpdyd + Z t 0 Z |x-y|<t- Z (p(bs) + bs) · xj fkxi Kk-1 q (t - ) 2 - |x - y| 2 dpdyd + Z t 0 Z |x-y|<t- Z (p(bs) + bs) · Kk-1xi xj fk q (t - ) 2 - |x - y| 2 dpdyd - Z t 0 Z |x-y|<t- Z (bs)xi xj Kk-1 · pfk q (t - ) 2 - |x - y| 2 dpdyd - Z t 0 Z |x-y|<t- Z (bs)xj Kk-1 · xi pfk q (t - ) 2 - |x - y| 2 dpdyd + 1 2 Z t 0 Z |x-y|<t- curlx xi Uk q (t - ) 2 - |x - y| 2 dyd. OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 17 Here, B 0 k is the `B0 ' of system (2.6) and converges to the respective expression without indices. We are allowed to pass to the limit in the integral expressions because all kernels are integrable, (fk, Ek, Bk) converge in C1 b, the second derivatives weak-* in L , and Uk in VL. Hence, we can omit the indices in the equation above or equivalently xi xj B - xi B 0 = Z t 0 Z |x-y|<t- Z bt (t - ) q (t - ) 2 - |x - y| 2 xi xj f dpdyd + Z t 0 Z |x-y|<t- Z (p(bs) + bs) · xi Kxj f q (t - ) 2 - |x - y| 2 dpdyd - Z t 0 Z |x-y|<t- Z (bs)xi xj K · pf q (t - ) 2 - |x - y| 2 dpdyd + 1 2 Z t 0 Z |x-y|<t- curlx xi U q (t - ) 2 - |x - y| 2 dyd and conclude that xi xj B is continuous which is an immediate consequence of U VL and the following lemma: Lemma 2.9. Denote M := {(s, z) [0, T] × Rn | 0 s T, |z| < s} and let h C0 ([0, T] × Rn+m ) with uni- form support in p Rm , i.e., suppp h Br for some r > 0, and let w C1 (M × Br) and {t, x1, . . . xn}. Furthermore, let one of the following options hold: i) h W1, ([0, T] × Rn+m ) and w L1 (M × Br), ii) h W1,1 (0, T; L (Rn+m )) if = t or h L 0, T; W1, (Rn+m ) if = xi respectively, and Z s-d<|z|<s Z Br |w(s, z, p)|dpdz 0 for d 0 uniformly in s [0, T]. Then H(t, x) := Z t 0 Z |x-y|<t- Z (h)(, y, p)w(t - , y - x, p)dpdyd = Z t 0 Z |z|<s Z (h)(t - s, x + z, p)w(s, z, p)dpdzds is continuous in (t, x) [0, T] × Rn . Proof. Let = xi and > 0 be given. For (t, x) [0, T] × Rn and d > 0 define Id(t, x) := Z t 0 Z s-d<|z|<s Z (xi h)(t - s, x + z, p)w(s, z, p)dpdzds and estimate in case i) |Id(t, x)| kxi hk Z T 0 Z s-d<|z|<s Z Br |w(s, z, p)|dpdzds 0 18 J. WEBER and in case ii) |Id(t, x)| Z T 0 kxi h(s)kds s 7 Z s-d<|z|<s Z Br |w(s, z, p)|dpdz 0 for d 0 uniformly in (t, x). Thus, we can choose d so that |Id(t, x)| < 4 for all (t, x). For now fixed d consider the remaining integral and integrate by parts Jd(t, x) := Z t 0 Z |z|<s-d Z (xi h)(t - s, x + z, p)w(s, z, p)dpdzds = Z t 0 Z |z|<s-d Z (zi h)(t - s, x + z, p)w(s, z, p)dpdzds = - Z t 0 Z |z|<s-d Z h(t - s, x + z, p)zi w(s, z, p)dpdzds + Z t 0 Z |z|=s-d Z h(t - s, x + z, p)w(s, z, p) 1 2 dpdSzds + Z |z|<t-d Z h(0, x + z, p)w(t, z, p)dpdz. This is allowed because the integration domain is away from the possibly singular set |z| = s. For that very reason Jd is obviously continuous by the standard theorem for parameter integrals, so if (t, x) is small enough (with t + t [0, T]) we have |Jd(t + t, x + x) - Jd(t, x)| < 2 . Finally with H = Id + Jd we conclude |H(t + t, x + x) - H(t, x)| |Id(t + t, x + x)| + |Id(t, x)| + |Jd(t + t, x + x) - Jd(t, x)| < . Analogously, one proves the assertion for = t. This lemma is applicable since f has uniform support in p, xf, pf, and xK are of class W1, , |bs|, |bt| C(r), and by Remark 2.2. Next, we have a representation formula for txj Bk according to Lemma 2.5. Analogously we conclude that txj B is continuous. For this, note that the terms without an R t 0 -integral are easy to handle since there only initial values appear. The procedure for E is nearly the same. The only critical point is to ensure that Z t 0 Z |x-y|<t- 2 t xj U q (t - ) 2 - |x - y| 2 dyd is continuous for U VL. To this end, we can apply Lemma 2.9 with h = txj U where = (p) C c R2 with R dp = 1. Note that txj U is continuous and of class W1,1 (0, T; L ) by U VL, and that Z s-d<|z|<s 1 q s2 - |z| 2 dz = 2 p 2sd - d21sd 2 T d, where 1sd denotes the indicator function of the set {s | s d}. So there only remain the 2 t -derivatives of E and B. By the known convergence, we can pass to the limit in (VM) so that the Vlasov equation holds everywhere OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 19 and the Maxwell equations almost everywhere. With this knowledge and the just proven fact that the second space derivatives of the fields are continuous, we conclude that also the 2 t -derivatives are continuous. Now the fact that all weak derivatives are continuous instantly implies that they are classical ones. Therefore, the fields are of class C2 . Thus, the characteristics X = b P, P = (P) E - b P B (s, X), (X, P)(t, t, x, p) = (x, p) are well defined and of class C2 in (t, x, p). Hence, f(t, x, p) = ° f((X, P)(0, t, x, p)) is also of class C2 . Therefore, we are able to pass to the limit in (VM), but actually (CVM) is to be solved: Obviously, (VM) coincides with (CVM2nd) as long as f vanishes for |p| Q. But this property is guaranteed by Lemma 2.8. Therefore, (f, E, B) is a solution of (CVM2nd) and hence of (CVM) by equivalence. We collect some properties of (f, E, B): Theorem 2.10. Let T, L > 0, ° f 0 of class C2 c, E, B of class C3 b, and U VL that satisfy div E = ° f . Then there is a solution (f, E, B) of (CVM) on [0, T] with: i) f, E, and B are of class C2 , ii) f vanishes for |p| Q or |x| R + T (where Q only depends on T, the initial data (their C1 b-norms and P(0)), and kUkVL , and where suppx ° f BR), iii) E, B vanish for |x| e R + L + R + T if their initial data are compactly supported, i.e., supp E, supp B Be R, iv) the C2 b-norms of the solution are estimated by a constant only depending on T, the initial data (their C2 b-norms and P(0)), L, and kUkVL . Proof. For ii) note that X 1, for iii) recall the representation formula of the fields, and iv) holds because it holds for all iterates, they converge in C1 b and their second derivatives weakly-* in L . 2.3.2. Uniqueness We prove uniqueness of the solution. Theorem 2.11. The obtained solution (f, E, B) of (CVM) is unique in C1 × C2 2 . Proof. The proof is standard. Consider the difference of two solutions and apply Lemmas 2.1 and 2.3 to show that the difference (measured in the C0 b-norm) vanishes after a Gronwall argument. Moreover, it is possible to show that the solution is unique in an even larger class. Here, the constructed solution satisfies the conditions if E and B are compactly supported. Theorem 2.12. A solution (f, E, B) of (CVM) with the properties i) f, E, and B are of class W1, H1 , ii) supp f [0, T] × B2 r for some r > 0, is unique (here, `solution' means that (CVM) holds pointwise almost everywhere). 20 J. WEBER Proof. Let e f, e E, e B (with the above properties) solve (CVM) too and define f := e f - f and so on. Then we have the system tf + b p · xf + e E - b p e B · pf = - E - b p B · pf, tE + x B = -jf , tB + curlx E = 0, f, E, B (0) = 0. Note that initial values make sense because of H1 H1 0, T; L2 , C0 0, T; L2 . We have 1 2 f(t) 2 L2 = Z t 0 Z Z ftf dpdxd = Z t 0 Z Z f -b p · xf - e E - b p e B · pf - E - b p B · pf dpdxd = Z t 0 Z Z - 1 2 divx b pf 2 - 1 2 divp e E - b p e B f 2 - f E - b p B · pf dpdxd = - Z t 0 Z Z f E - b p B · pf dpdxd kfkW1, Z t 0 f() L2 E() L2 + B() L2 d, which implies f(t) L2 kfkW1, Z t 0 E() L2 + B() L2 d via the quadratic version of Gronwall's inequality, cf. [2], Theorem 5. Similarly, 1 2 B(t) 2 L2 = Z t 0 Z BtB dxd = Z t 0 Z B -x1 E2 + x2 E1 dxd = Z t 0 Z E2x1 B - E1x2 B dxd = Z t 0 Z -E · tE - E · jf dxd. Note that in the integration by parts no surface terms appear because of E, B H1 . This computation leads to 1 2 E(t) 2 L2 + B(t) 2 L2 = Z t 0 Z -E · jf dxd Z t 0 E() L2 jf () L2 d C(r) Z t 0 E() L2 + B() L2 f() L2 d. Here, the last inequality holds because f vanishes as soon as |p| > r. Now again, the quadratic Gronwall lemma implies E(t) L2 + B(t) L2 C(r) Z t 0 f() L2 d C(r, T)kfkW1, Z t 0 E() L2 + B() L2 d. This yields E, B = 0 and hence also f = 0. OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 21 3. The control-to-state operator From now on the initial data always stay fixed with 0 ° f C2 c and E, B C3 c, and div E = ° f . As a result of the last section we may define the control-to-state operator via S : VL C2 b [0, T] × R4 × C2 b [0, T] × R2 ; R2 × C2 b [0, T] × R2 , U 7 (f, E, B). The goal is to show that S is differentiable with respect to suitable norms. 3.1. Lipschitz continuity First we show that S is Lipschitz continuous; to be more precise, locally Lipschitz continuous. Let U, U VL and denote (f, E, B) = S(U), f, E, B = S(U + U), and e f, e E, e B = S(U + U) - S(U). We arrive at the system t e f + b p · x e f + E - b p B · p e f = - e E - b p e B · pf, t e E + x e B = -je f - U, t e B + curlx e E = 0, e f, e E, e B (0) = 0, which is equivalent to the system with second order Maxwell equations because of Lemmas 1.1 and 1.2. Note that the x- and p-support of the density and the C1 b-norm of the solution is controlled by a constant dependent on T, the initial data, L, and the VL-norm of the control, see Theorem 2.10. Therefore, we can perform the same estimates also on the `bar'-solution with a constant dependent on T, the initial data, L, and kUkVL because, for instance, for kUkVL 1 we have kU + UkVL kUkVL + 1. Hence, we will only show the locally Lipschitz continuity of S. Indeed, using again the estimates of Lemmas 2.1, 2.3, and 2.5, we see that e f, e E, e B C1 b CkUkVL . Thus, we have proved: Lemma 3.1. The control-to-state map S : VL C1 b [0, T] × R4 × C1 b [0, T] × R2 3 is locally Lipschitz continuous. 3.2. Solvability of a linearized system To show even differentiability of S we will have to analyze a linearized system of the form tf + b p · xf + G · pf = E - b p B · g + a, tE + x B = -jf - h, tB + curlx E = 0, (f, E, B)(0) = 0 (LVM) with already given functions a L1 0, T; L2 , G C2 b with divp G = 0, g C1 b with g = pe g for some e g C2 b and g(t, x, p) = 0 for |x| r or |p| r for some r > 0, and h VL. We call (f, E, B) a solution of (LVM) if f, 22 J. WEBER E, and B are of class C0 H1 , the equalities hold pointwise almost everywhere, and f vanishes for |p| R for some R > 0. A crucial estimate is the following: Lemma 3.2. Let (f, E, B) be a solution of (LVM). Then kf(t)kL2 + kE(t)kL2 + kB(t)kL2 C(R, kgk, T) Z t 0 (ka()kL2 + kh()kL2 )d. Proof. The proof is very similar to that of Theorem 2.12 and is omitted. We approximate G, e g, and h with smooth functions Gk, e gk, and hk which are converging to G, e g, and h in C2 b and VL respectively, and define gk := pe gk. To show solvability of (LVM) for a = 0 we proceed similarly as before. Define f0 = E0,1 = E0,2 = B0 = 0 and solve in the k-th step tfk + b p · xfk + Gk · pfk = Ek-1 - b p Bk-1 · gk, fk(0) = 0 by defining fk(t, x, p) = Z t 0 Ek-1 - b p Bk-1 · gk (Xk(0, t, x, p), Pk(0, t, x, p))d with the characteristics Xk = b Pk, Xk(t, t, x, p) = x, Pk = Gk(s, Xk, Pk), Pk(t, t, x, p) = p, and then solving 2 t Ek - xEk = -tjfk - thk - xfk + x Z t 0 divx hk d, 2 t Bk - xBk = curlx jfk + curlx hk, (Ek, Bk)(0) = 0, tEk(0) = -Uk(0), tBk(0) = 0. All iterates are again of class C . Furthermore, the characteristics are independent of the solution sequence (fk, Ek, Bk). Thus, we instantly have Pk C, so |Pk - p| CT. Having a look at the formula for fk we conclude that fk vanishes as soon as |p| 2r + CT =: Q (3.1) since then the integrand vanishes as a result of |Pk(s, t, x, p)| |p| - |Pk - p| 2r + CT - CT = 2r. OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 23 The same can be done for the x-coordinate starting with Xk 1; hence fk(t, x, p) = 0 for |x| 2r + T. The assertions of Section 2.1 are directly applicable. We do not have to insert some because of the already known bound on the p-support of fk. Therefore, (LC) holds for the iterated system, and we can thus switch between first order and second order Maxwell equations; note that Ek-1 - b p Bk-1 · gk = divp Ek-1 - b p Bk-1 e gk . We proceed like in Section 2.3: The iterates are bounded in C1 b and are Cauchy with respect to the C0 b-norm. However, after that there appears a difference: Unfortunately, we can not show the Cauchy property with respect to the C1 b-norm. For this we would first have to bound second derivatives of fk which would require control of second derivatives of gk. This, on the other hand, would require a smoother g. But for the later application we will not have more regularity of g than C1 b. Thus, we have to proceed differently: Since fk, Ek, and Bk are bounded in the C1 b-norm, their first derivatives converge, after extracting a suitable subsequence, to the respective derivatives of f, E, and B in L in the weak-*-sense. Because of Z T 0 Z Z (Gk · pfk - G · pf)dpdxd Z T 0 Z Z |Gk - G||pfk|||dpdxd + Z T 0 Z Z G(pfk - pf) dpdxd CkGk - GkkkL1 + Z T 0 Z Z G(pfk - pf) dpdxd 0 for k for any test function , (f, E, B) satisfies (LVM) pointwise almost everywhere; the other terms are obviously easier to handle. Altogether we have found a solution of (LVM) of class C0 W1, . Furthermore, it is also of class H1 because all sequence elements have compact support with respect to x, p or x respectively uniformly in t and k; for the fields recall the representation formula. For uniqueness, let (f1, E1, B1) be a solution of (LVM) too and define f2 := f - f1 and so on which yields tf2 + b p · xf2 + G · pf2 = E2 - b p B2 · g, tE2 + x B2 = -jf2 , tB2 + curlx E2 = 0, (f2, E2, B2)(0) = 0. Applying Lemma 3.2 this instantly implies that f2, E2, and B2 vanish. 3.3. Differentiability We want to study the differentiability of S : VL C0 0, T; L2 R4 × C0 0, T; L2 R2 3 . Let U VL and let U VL be some perturbation. In the following denote (f, E, B) = S(U) and f, E, B = S(U + U). The candidate for the linearization is S0 (U)U = (f, E, B) where the right hand side satisfies tf + b p · xf + E - b p B · pf = - E - b p B · pf, tE + x B = -jf - U, tB + curlx E = 0, (f, E, B)(0) = 0. 24 J. WEBER Indeed, this system can be solved because of G := E - b p B C2 b (note that divp G = 0), g := -pf C1 b, and h := U VL. First we note that S0 (U) is linear and that by Lemma 3.2 k(f, E, B)kC0(0,T ;L2) C Z T 0 kU(t)kL2 dt CkUkVL (3.2) which says that S0 (U) is bounded. The last inequality holds because of supp U(t) BL. The next step is to show that S(U + U) - S(U) - S0 (U)U is `small'. Defining e f := f - f - f and so on and subtracting the respective equations yield t e f + b p · x e f + E - b p B · p e f = - e E - b p e B · pf - E - E - b p B - B · p f - f , t e E + x e B = -je f , t e B + curlx e E = 0, e f, e E, e B (0) = 0. Applying Lemma 3.2 we conclude e f, e E, e B C0(0,T ;L2) C Z T 0 ka(t)kL2 dt where a := - E - E - b p B - B · p f - f . Here we have to exploit the Lipschitz property of S. Lemma 3.1 yields ka(t)kL2 C E - E + B - B f - f C1 b CkUk 2 VL . Note that for the first inequality the fact was used that f and f have compact support in x and p uniformly in t and independent of kUkVL for, for instance, kUkVL 1 (recall Theorem 2.10 and the reasoning in Sect. 3.1). Finally we arrive at e f, e E, e B C0(0,T ;L2) CkUk 2 VL (3.3) which proves part of i) of the following theorem: Theorem 3.3. The following maps are continuously Frechet-differentiable with locally Lipschitz derivative: i) S : VL W := C0 0, T; L2 R4 × C0 0, T; L2 R2 3 , ii) := S1 : VL C0 0, T; L2 R2 , U 7 f , iii) := S1 : VL C0 0, T; L1 R2 , U 7 f . Proof. For part ii) define 0 (U)U := f . (3.4) OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 25 Now it is crucial to bound the p-support of f, f, and f by a constant C > 0 only depending on T, the initial data, L, and kUkVL . We first consider f. The control of the p-support in (3.1) holds for all iterates and hence for f. The constant there only depends on T, kGk = E - b p B , the p-support of pf, and L. Because of Theorem 2.10 the absolute values of the fields E and B and the p-support of f are controlled by some constant only depending on T, the initial data, L, and kUkVL . Hence, we have together with (3.2) kf (t)kL2 = Z Z f dp 2 dx !1 2 C Z Z |f| 2 dpdx 1 2 CkUkVL which implies that 0 (U) is bounded. Furthermore, the p-supports of f and f only depend on T, the initial data, L, and kUkVL (for again kUkVL 1 for example). Hence, the same assertion holds for e f = f - f - f and therefore with (3.3) e f (t) L2 = Z Z e f dp 2 dx !1 2 C Z Z e f 2 dpdx 1 2 CkUk 2 VL . Together with the equality (U + U) - (U) - 0 (U)U = f - f - f = e f this instantly yields that 0 (U) is indeed the Frechet-derivative of in U. Part iii) is an instant consequence of ii) and the support assertions discussed above. The derivative of is given by (3.4) as before. To show continuity of S0 , let V VL with kV kVL 1. We have to investigate f, E, B := f1 , E1 , B1 - f0 , E0 , B0 := S0 (U + U)V - S0 (U)V. Applying the previously given formula for S0 we arrive at t f + b p · x f + E - b p B · p f = - E - b p B · pf - E0 - b p B0 · p f - f - E - E - b p B - B · pf0 , tE + x B = -j f , tB + curlx E = 0, f, E, B (0) = 0. We know that the p-support of f0 and the absolute values of E0 and B0 are controlled by a constant only depending on T, the initial data, L, kUkVL , and kV kVL (the latter can be neglected, of course). The dependence on some terms in f, E, and B can be eliminated like in the beginning of this proof. Hence, proceeding as before and using Lemma 3.2 and the locally Lipschitz continuity of S, we conclude f, E, B W CkUkVL where C only depends on T, the initial data, L, and kUkVL . This leads to kS0 (U + U) - S0 (U)kL(VL,W ) CkUkVL which says that S0 is even locally Lipschitz continuous. 26 J. WEBER Using the assertions for the p-support of f0 and f1 (controlled by a constant only depending on T, the initial data, L, and kUkVL if kUkVL 1) we conclude f C0(0,T ;L2) , f C0(0,T ;L1) C f C0(0,T ;L2) CkUkVL as before. This implies that 0 and 0 are locally Lipschitz continuous. 4. Optimal control problem Now we consider some optimal control problems. We want to minimize some objective function that depends on the external control U and the state (f, E, B). The control and the state are coupled via (CVM) so that (CVM) appears as a constraint. We first give thought to a problem with general controls and a general objective function. Then we proceed with optimizing problems where the objective function is explicitly given and where the control set is restricted to such controls that are realizable in applications concerning the control of a plasma. 4.1. General problem 4.1.1. Control space Until now we have worked with the control space VL = U W2,1 0, T; C4 b R2 ; R2 | U(t, x) = 0 for |x| L . To apply standard optimization techniques it is necessary that the control space is reflexive. Hence, we choose UL := U H2 0, T; W5, R2 ; R2 | U(t, x) = 0 for |x| L , where > 2 is fixed, equipped with the H2 0, T; W5, -norm. By Sobolev's embedding theorems, UL is continuously embedded in VL. In accordance with Theorems 2.10 and 3.3, we have already proved that there is a continuously differentiable control-to-state operator S : VL C2 b [0, T] × R4 × C2 b [0, T] × R2 ; R2 × C2 b [0, T] × R2 , k·kC0(0,T ;L2) , U 7 (f, E, B), such that (CVM) holds for (f, E, B) and control U. Furthermore, the map U 7 f is continuously differentiable with respect to the C0 0, T; L2 - and C0 0, T; L1 -norm in the image space. Moreover, the C2 b-norm and the x- and p-support of (f, E, B) are controlled by a constant only depending on T, L, the initial data, and kUkVL . By UL , VL, these assertions also hold with UL instead of VL. We are aware that the regularity assumed for U UL is quite high. However, to derive the assertions mentioned above, U VL was really necessary. 4.1.2. Existence of minimizers We consider the general problem min (f,E,B)(C2 H1 ) 3 ,UUL (f, E, B, U) s.t. (f, E, B) = S(U). (GP) OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 27 We have to specify some assumptions on : Condition 4.1. i) : C2 H1 3 × UL R {} and 6 , ii) is coercive in U UL, i.e., in general: Let X, Y be normed spaces; : X × Y R is said to be coercive in y Y iff for all sequences (yk) Y with kykkY , k , then also (xk, yk) , k , for any sequence (xk) X, iii) is weakly lower semicontinuous, i.e.: if (fk, Ek, Bk) * (f, E, B) in H1 and Uk * U in UL, then (f, E, B, U) lim infk (fk, Ek, Bk, Uk). These assumptions allow us to prove existence of a (not necessarily unique) minimizer. We will first prove a lemma that will be useful later: Lemma 4.2. Let (Uk) VL be bounded and (fk, Ek, Bk) = S(Uk). Then, after extracting a suitable subsequence, it holds that: i) The sequences (fk), (Ek), and (Bk) converge weakly in H1 , weakly-* in W1, , and strongly in L2 to some f, E, and B. ii) There is r > 0 such that f, E, B, and, for all k N, fk, Ek, and Bk vanish if |x| r or |p| r. iii) If additionally Uk U in the sense of distributions for some U VL for k , then (f, E, B) = S(U) and f, E, and B are of class C2 b. Proof. By Theorem 2.10, on the one hand, (fk, Ek, Bk) is bounded in the C1 b-norm. On the other hand, fk vanishes as soon as |p| is large enough uniformly in k. Moreover, fk, Ek, and Bk vanish as soon as |x| is large enough. Hence, (fk, Ek, Bk) is also bounded in H1 and in H1 0, T; L2 . Together with the boundedness in C1 b, (fk, Ek, Bk) converge, after extracting a suitable subsequence, to some (f, E, B), namely weakly in H1 , and weakly-* in W1, . This proves ii) and part of i). For the remaining part of i) (strong convergence in L2 ) we have to exploit some compactness. This compactness is guaranteed by the theorem of Rellich-Kondrachov. By the reasoning above, (fk, Ek, Bk) are bounded in H1 and in fact, only a bounded subset of the x- and p-space matters. Hence, (a subsequence of) (fk, Ek, Bk) converges strongly in L2 to the limit (f, E, B). For iii), we have to pass to the limit in (CVM). First, the initial conditions are preserved in the limit since H1 , H1 0, T; L2 , C0 0, T; L2 . Furthermore, the Vlasov and Maxwell equations hold pointwise almost everywhere for the limit functions: The only difficult part is the nonlinear term in the Vlasov equation. To handle this, we have to make use of the strong convergence in L2 obtained above. We find for each C c ]0, T[ × R4 that Z T 0 Z Z Ek - b p Bk · pfk - E - b p B · pf dpdxdt Z T 0 Z Z E - b p B · (pfk - pf) dpdxdt + kpfkk Z T 0 Z Z (|Ek - E| + |Bk - B|)||dpdxdt. Both terms converge to 0 for k since fk * f in H1 , Ek E, Bk B in L2 , and fk is bounded in C1 b. Therefore, altogether, (CVM) holds pointwise almost everywhere. Now we can apply Theorem 2.12 to conclude (f, E, B) equals S(U) and is hence of class C2 b. Theorem 4.3. Let satisfy Condition 4.1. Then there is a minimizer of (GP). Proof. We consider a minimizing sequence (fk, Ek, Bk, Uk) with (fk, Ek, Bk) = S(Uk) and lim k (fk, Ek, Bk, Uk) = m := inf UUL,(f,E,B)=S(U) (f, E, B, U) R {-}. 28 J. WEBER By coercivity in U, cf. Condition 4.1 ii), (Uk) is bounded in UL and therefore in VL. Hence, we may extract a weakly convergent subsequence (also denoted by Uk) since H2 0, T; W5, is reflexive. The weak limit U is the candidate for being an optimal control. Of course, by weak convergence, U vanishes for |x| L; hence U UL. Because of UL , L1 we also get Uk * U in L1 and hence Uk U in the sense of distributions. Lemma 4.2 yields (fk, Ek, Bk) * (f, E, B) in H1 (after extracting a suitable subsequence) and (f, E, B) = S(U). Together with the weak lower semicontinuity of , see Condition 4.1 iii), we instantly get (f, E, B, U) = m which proves optimality. In order to be able of examining some problem that is somehow application-oriented, we first have to think about possible problems concerning the conditions on the objective function . Especially the coercivity in U will make some trouble since the UL-norm is pretty strong. One can try to guarantee these conditions in various ways, for example if (f, E, B, U) = (f, E, B) + kUk 2 UL ; the objective function contains some cost term of the control in the full UL-norm. But typically in applications, such a strong cost term makes no sense. Furthermore, first order optimality conditions would contain a differential equation of very high order, which is hard to solve. On the other hand, we can not simply use a less regular control space. Firstly, we need UL , VL to ensure that the control-to-state operator is differentiable; this will be useful later. Secondly, UL needs to be reflexive to extract (in some sense) converging subsequences from a minimizing sequence. Here we should remark that we also could demand W2,p -regularity in time for p > 1 instead of H2 -regularity which would allow more controls if 1 < p < 2. However, working in a H2 -setting (at least in time) is more convenient. 4.2. An optimization problem with realizable external currents 4.2.1. Motivation As the previous considerations suggest, it would be nice if we somehow eliminated the variability of the control with respect to the space coordinate. This can be achieved by only considering controls of the form U(t, x) = N X j=1 uj(t)zj(x) where the functions 0 6 zj C4 b R2 ; R2 with zj vanishing for |x| rj > 0 are fixed, and we only vary the functions uj H2 ([0, T]). From a physical point of view, this model describes an ensemble of N coils with `size' rj, that stay fixed in time. Obviously, U is an element of VL if we set L = max{rj | j = 1, . . . N}. Each coil generates a current zj at full capacity that is tangential to the plane and that extends infinitely in the third space dimension. We control the system by turning these coils on whereby the capacity uj is suitably adjusted as a function of time. Hence, we will have to consider an additional constraint |uj| 1. Physically, the consideration only of controls of the above form is no substantial restriction at all because only such control fields are realizable in applications. A similar approach was done by P. Knopf and the author [11]. 4.2.2. Formulation The problem to be considered is the following: min (f,E,B)(C2 H1 ) 3 , uH2 ([0,T ])N 1 2 kf - dk 2 L2([0,T ]×R2) + 2 N X j=1 cj kujk 2 L2([0,T ]) + 1ktujk 2 L2([0,T ]) + 2 2 t uj 2 L2([0,T ]) s.t. (f, E, B) = S N X j=1 ujzj , |uj| 1 (P) OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 29 where cj := kzjk 2 L2(R2;R2). We give some comments on the objective function: ­ The charge density shall be as close as possible to some given desired density d L2 [0, T] × R2 . One could consider the L2 -norm of some f - fd instead but the space coordinates of the particles are of actual interest rather than their momenta. ­ Furthermore, the cost term containing the control shall be as small as possible. We have to use the full H2 -norm (an equivalent norm, to be more precise) of the uj in the regularization term so that the objective function is coercive in u H2 ([0, T]) N . However, the L2 -norms of the uj itself are more interesting than the ones of their derivatives. Hence, it is suitable to choose 0 < 1, 2 1. ­ The parameter > 0 indicates which of the two aims mentioned above shall rather be achieved. 4.2.3. Existence of minimizers Section 4.1.2 is useful for showing existence of minimizers of (P). Theorem 4.4. There is a minimizer of (P). Proof. The objective function, abbreviated by = (f, E, B, u) = 1(f)+2(u) (let 1 be the term with f -d and 2 the remaining sum), is coercive in u H2 ([0, T]) N because of (f, E, B, u) 2 min{1, 1, 2} min{cj | j = 1, . . . , N}kuk 2 (H2)N , where kuk 2 (H2)N = PN j=1 kujk 2 H2([0,T ]). Hence, considering a minimizing sequence fk, Ek, Bk, uk (we use upper indices for uk to avoid confusion with the components) with (fk, Ek, Bk) = S PN j=1 uk j zj and uk j 1, we conclude that uk is bounded in H2 N ; hence uk * u in H2 N for some u H2 N for k , possibly after extracting a suitable subsequence. The constraint |uj| 1 is obviously preserved by weak convergence. Furthermore, the sequence (Uk) := PN j=1 uk j zj is bounded in VL because of H2 ([0, T]) , W2,1 ([0, T]). Clearly, Uk U := PN j=1 ujzj in the sense of distributions by uk j * uj in H2 . Therefore, Lemma 4.2 is applicable and delivers some f, E, and B so that (CVM) is preserved in the limit. The remaining part is to show that U is indeed an optimal control. Firstly, uk * u in H2 N instantly implies 2(u) lim infk 2 uk . Secondly, by Lemma 4.2, all fk and f have compact support with respect to p uniformly in k, and fk f in L2 . These properties yield fk f in L2 by Holder's inequality and therefore 1(f) = limk 1(fk). This finally proves the desired optimality. 4.2.4. Differentiability of the objective function Next we study the differentiability of the objective function. Theorem 4.5. i) The solution map : H2 ([0, T]) N C0 0, T; L2 R4 × C0 0, T; L2 R2 3 , u 7 (f, E, B) = S N X j=1 ujzj 30 J. WEBER is continuously Frechet-differentiable and 0 (u)u = (f, E, B) satisfies tf + b p · xf + E - b p B · pf = - E - b p B · pf, tE + x B = -jf - U, tB + curlx E = 0, (f, E, B)(0) = 0 where U = PN j=1 ujzj. ii) The maps : H2 ([0, T]) N C0 0, T; L2 R2 , u 7 f and : H2 ([0, T]) N C0 0, T; L1 R2 , u 7 f are continuously Frechet-differentiable and 0 (u)u = f with f from above. iii) The objective function : H2 ([0, T]) N R, u 7 1 2 kf - dk 2 L2 + 2 N X j=1 cj kujk 2 L2 + 1ktujk 2 L2 + 2 2 t uj 2 L2 is continuously Frechet-differentiable and 0 (u)u = hf - d, f iL2 + N X j=1 cj huj, ujiL2 + 1htuj, tujiL2 + 2 2 t uj, 2 t uj L2 with f from above. Proof. Clearly, u 7 PN j=1 ujzj is differentiable by linearity and boundedness. Hence, all assertions follow immediately by Theorem 3.3 and the chain rule. 4.2.5. Optimality conditions Now we want to deduce first order optimality conditions for a (local) minimizer of (P). First we write (P) in the equivalent form min uH2([0,T ])N 1 2 k(u) - dk 2 L2([0,T ]×R2) + 2 N X j=1 cj kujk 2 L2([0,T ]) + 1ktujk 2 L2([0,T ]) + 2 2 t uj 2 L2([0,T ]) s.t. - uj + 1 0, uj + 1 0. (P') Here, the objective function = (u) = ((u), u) is a function of only the control. OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 31 The constraints will lead to corresponding Lagrange multipliers. In general, to prove their existence, some condition on the constraints is necessary. On this account we verify the constraint qualification of Zowe and Kurcyusz, see [16], which is based on a fundamental work of Robinson [15]. We rewrite the constraints: g(u) K, where g(u) = (-u + 1, u + 1) K, K denoting the cone of component-wise positive functions in C0 ([0, T]) 2N . The constraint qualification we have to verify is g0 (u) H2 ([0, T]) N - {k - g(u) | k K, 0} = C0 ([0, T]) 2N . In other words, for given (w+ , w- ) C0 ([0, T]) 2N we have to find u H2 ([0, T]) N , R0, and k = (+ , - ) C0 ([0, T]) 2N with + j , - j 0, satisfying (-u, u) - + , - + (-u + 1, u + 1) = w+ , w- . (4.1) We abbreviate + := max i=1,...,N w+ i , - := max i=1,...,N w- i . Now let := + + - 2 + 1, + j := + - uj + 1 - w+ j , - j := - + uj + 1 - w- j , uj := - 1 2 + (uj + 1) + - (uj - 1) . Obviously, 0 and uj is of class H2 . Furthermore, + j , - j C0 ([0, T]) and are 0 by choice of + , - , and feasibility of u. Thereby (4.1) can easily be verified. Thus, we deduce the following KKT-conditions for a minimizer of (P'). We denote by M([0, T]) = C0 ([0, T]) the set of regular Borel measures on [0, T]. Theorem 4.6. Let u be a local minimizer of (P'). Then there are Lagrange multipliers + j (corresponding to the constraint uj 1), - j M([0, T]) (corresponding to uj -1), j = 1, . . . , N, satisfying: i) (Primal feasibility): |uj| 1. ii) (Dual feasibility): + j , - j 0, i.e., + j v, - j v 0 for all v C0 ([0, T]) with v 0. iii) (Complementary slackness): + j (uj - 1) = 0, - j (uj + 1) = 0. iv) (Stationarity): For all u H2 ([0, T]) N it holds that D f - d, f E L2 + N X j=1 cj huj, ujiL2 + 1htuj, tujiL2 + 2 2 t uj, 2 t uj L2 = N X j=1 - j - + j uj where f is obtained by solving tf + b p · xf + E - b p B · pf = - E - b p B · pf, tE + x B = -jf - U, tB + curlx E = 0, (f, E, B)(0) = 0 with U = PN j=1 ujzj and f, E, B = (u). 32 J. WEBER 4.2.6. Adjoint equation Considering the optimality conditions above, we note that we have to compute 0 and thus the whole derivative 0 at an optimal point u. However, there is a more efficient way, the adjoint approach, that is to say firstly solve the adjoint equation yF((u), u) q = -y((u), u) for the adjoint state q and secondly compute 0 (u) = uF((u), u) q + u((u), u). (4.2) Here, y = (f, E, B) denotes the state and F(y, u) = 0 the PDE system. In order to apply these considerations to our problem we have to define F suitably. Here, `suitably' means that the differentiability of F and the differentiability of the control-to-state operator have to fit together. In other words, F(y, u) should be differentiable with respect to the C0 0, T; L2 -norm in the state variable y = (f, E, B). In the following let MR := (f, E, B) C2 c [0, T] × R4 × C2 c [0, T] × R2 ; R2 × C2 c [0, T] × R2 | f(t, x, p) = 0 for all |p| R for some R > 0, and let MR be equipped with the C0 0, T; L2 -norm. Here, the index `c' means `compactly supported with respect to x and p' (or x respectively). Furthermore, let Z := H1 [0, T] × R4 × H1 [0, T] × R2 3 × L2 R4 × L2 R2 3 . Now define FR : MR × H2 ([0, T]) N Z via FR((f, E, B), u)(g, h1, h2, h3, a1, a2, a3, a4) = - Z T 0 Z Z tg + b p · xg + E - b p B · pg f dpdxdt + hg(T), f(T)iL2 - hg(0), f(0)iL2 , Z T 0 Z (-E1th1 + Bx2 h1 + jf,1h1 + U1h1)dxdt + hh1(T), E1(T)iL2 - hh1(0), E1(0)iL2 , Z T 0 Z (-E2th2 - Bx1 h2 + jf,2h2 + U2h2)dxdt + hh2(T), E2(T)iL2 - hh2(0), E2(0)iL2 , Z T 0 Z (-Bth3 - E2x1 h3 + E1x2 h3)dxdt + hh3(T), B(T)iL2 - hh3(0), B(0)iL2 , Z Z f(0) - ° f a1 dpdx, Z E1(0) - ° E1 a2 dx, Z E2(0) - E a3 dx, Z B(0) - B a4 dx ! where U = PN j=1 ujzj. After several integrations by parts, it is obvious that (f, E, B) solves (CVM) with control U iff FR((f, E, B), u) = 0 for any R > 0 with suppp f BR. Since no derivatives of the state y = (f, E, B) appear OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 33 above and the state is of class C0 b, yFR exists and is given by yFR((f, E, B), u)(f, E, B)(g, h1, h2, h3, a1, a2, a3, a4) = - Z T 0 Z Z tg + b p · xg + E - b p B · pg f + E - b p B f · pg dpdxdt + hg(T), f(T)iL2 - hg(0), f(0)iL2 , Z T 0 Z (-E1th1 + Bx2 h1 + jf,1h1)dxdt + hh1(T), E1(T)iL2 - hh1(0), E1(0)iL2 , Z T 0 Z (-E2th2 - Bx1 h2 + jf,2h2)dxdt + hh2(T), E2(T)iL2 - hh2(0), E2(0)iL2 , Z T 0 Z (-Bth3 - E2x1 h3 + E1x2 h3)dxdt + hh3(T), B(T)iL2 - hh3(0), B(0)iL2 , Z Z f(0)a1 dpdx, Z E1(0)a2 dx, Z E2(0)a3 dx, Z B(0)a4 dx ! for (f, E, B) MR. Note that it is important that f vanishes for |p| R so that for i = 1, 2 the linear map (f, E, B) 7 h 7 Z T 0 Z jf,ih dxdt ! H1 [0, T] × R2 is bounded due to Z T 0 Z jf,ih dxdt C(T, R)kfkC0(0,T ;L2)khkH1 and hence differentiable. On the other hand, we have y((f, E, B), u)(f, E, B) = hf - d, f iL2 . Here again, the support condition given in the definition of MR is important to estimate Z T 0 Z (f - d)f dxdt C(T, R)kf - dkL2 kfkC0(0,T ;L2) and Z T 0 Z 2 f dxdt C(T, R)kfk 2 C0(0,T ;L2). Now we search for an adjoint state q = (g, h1, h2, h3, a1, a2, a3, a4) Z = H1 [0, T] × R4 × H1 [0, T] × R2 3 × L2 R4 × L2 R2 3 34 J. WEBER satisfying the adjoint system. In other words, after integrating by parts once, - Z T 0 Z Z tg + b p · xg + E - b p B · pg - 4(b p1h1 + b p2h2) f dpdxdt + Z T 0 Z -th1 + x2 h3 + Z gp1 f dp E1 dxdt + Z T 0 Z -th2 - x1 h3 + Z gp2 f dp E2 dxdt + Z T 0 Z -th3 + x2 h1 - x1 h2 - Z gb p · pf dp B dxdt + hg(T), f(T)iL2 - hg(0) - a1, f(0)iL2 + hh1(T), E1(T)iL2 - hh1(0) - a2, E1(0)iL2 + hh2(T), E2(T)iL2 - hh2(0) - a3, E2(0)iL2 + hh3(T), B(T)iL2 - hh3(0) - a4, B(0)iL2 = - Z T 0 Z Z 4(f - d)f dpdxdt (4.3) for all (f, E, B) MR. Therefore, the adjoint state solves the adjoint system tg + b p · xg + E - b p B · pg = 4(b p1h1 + b p2h2) + 4(f - d), th1 - x2 h3 = Z gp1 f dp0 , th2 + x1 h3 = Z gp2 f dp0 , th3 - x2 h1 + x1 h2 = - Z gb p0 · pf dp0 , (g, h1, h2, h3)(T) = 0 (Ad) for |p| < R. Since R > 0 (with suppp f BR) is arbitrary, it is natural to demand (Ad) holds globally on [0, T] × R4 . Conversely, if (Ad) holds for all p, then (4.3) holds for all (f, E, B) MR for any R > 0 if we simply set a1 = g(0), (a2, a3, a4) = (h1, h2, h3)(0). The latter equations are unsubstantial and can be ignored. In accordance with (4.2), we compute the derivative of via 0 (u)u = Z T 0 Z (U1h1 + U2h2)dxdt + N X j=1 cj huj, ujiL2 + 1htuj, tujiL2 + 2 2 t uj, 2 t uj L2 where U = PN j=1 ujzj. System (Ad) has to be investigated. It is a final value problem which can easily be turned into an initial value problem via e g(t, x, p) = g(T - t, -x, -p) and e h(t, x) = h(T - t, -x), so that the left-hand sides of the differential equations in (Ad) do not change. In other words, the hyperbolic system (Ad) is time reversible. To show unique solvability of (Ad), one can proceed similar to the dealing with (LVM). Yet there are some differences, which we will briefly sketch. Firstly, the source terms in the Maxwell equations are not the current densities induced by g but some other moments of g. Additionally, even in the fourth equation of (Ad) a source term appears. Hence, we have to prove analogues of Lemmas 2.3 and 2.5 with more general source terms. Secondly, the right-hand side of the Vlasov equation (and hence a solution g) does not have compact support with respect to p. But this will not cause any problems since in a representation formula for h there will appear a factor pf (or first derivatives of pf). Because of the known fact that f is compactly supported with respect to p uniformly in t, x, we do not have to demand that g has this property. In Section 2.1 we had to assume this property for the density since the integral defining the current density induced by this density contains the factor b p which is obviously not compactly supported in p. OPTIMAL CONTROL OF THE TWO-DIMENSIONAL VLASOV-MAXWELL SYSTEM 35 References [1] R.J. Di Perna and P.L. Lions, Global weak solutions of Vlasov-Maxwell systems. Commun. Pure Appl. Math. 42 (1989) 729­757. [2] S.S. Dragomir, Some Gronwall Type Inequalities and Applications. Nova Science Publishers (2003). [3] O. Glass and D. Han-Kwan, On the controllability of the relativistic Vlasov-Maxwell system. J. de Math. Pures Appl. 103 (2015) 695­740. [4] R.T. Glassey and J. Schaeffer, On the `one and one-half dimensional' relativistic Vlasov­Maxwell system. Math. Methods Appl. Sci. 13 (1990) 169­179. [5] R.T. Glassey and J. Schaeffer, The "two and one­half dimensional" relativistic Vlasov Maxwell system. Commun. Math. Phys. 185 (1997) 257­284. [6] R.T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions: part I. Arch. Ration. Mech. Anal. 141 (1998) 331­354. [7] R.T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions: part II. Arch. Ration. Mech. Anal. 141 (1998) 355­374. [8] R.T. Glassey and W.A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities. Arch. Ration. Mech. Anal. 92 (1986) 59­90. [9] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58 (1975) 181­205. [10] P. Knopf, Optimal control of a Vlasov­Poisson plasma by an external magnetic field. Cal. Variat. Part. Differ. Equ. 57 (2018) 134. [11] P. Knopf and J. Weber, Optimal control of a Vlasov­Poisson plasma by fixed magnetic field coils. Appl. Math. Optim. 81 (2020) 961­988. [12] M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system. Electron. J. Differ. Equ. 2005 (2005) 1­17. [13] T.T. Nguyen, T.V. Nguyen and W.A. Strauss, Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinet. Relat. Models 8 (2015) 53. [14] G. Rein, Global weak solutions to the relativistic Vlasov-Maxwell system revisited. Commun. Math. Sci. 2 (2004) 145­158. [15] S. Robinson, Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13 (1976) 497­513. [16] J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5 (1979) 49­62. [1] R. J. Di Perna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems. Commun. Pure Appl. Math. 42 (1989) 729–757. [2] S. S. Dragomir, Some Gronwall Type Inequalities and Applications. Nova Science Publishers (2003). [3] O. Glass and D. Han-Kwan, On the controllability of the relativistic Vlasov-Maxwell system. J. de Math. Pures Appl. 103 (2015) 695–740. [4] R. T. Glassey and J. Schaeffer, On the 'one and one-half dimensional' relativistic Vlasov–Maxwell system. Math. Methods Appl. Sci. 13 (1990) 169–179. [5] R. T. Glassey and J. Schaeffer, The “two and one–half dimensional” relativistic Vlasov Maxwell system. Commun. Math. Phys. 185 (1997) 257–284. [6] R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions: part I. Arch. Ration. Mech. Anal. 141 (1998) 331–354. [7] R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions: part II. Arch. Ration. Mech. Anal. 141 (1998) 355–374. [8] R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities. Arch. Ration. Mech. Anal. 92 (1986) 59–90. [9] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58 (1975) 181–205. [10] P. Knopf, Optimal control of a Vlasov–Poisson plasma by an external magnetic field. Cal. Variat. Part. Differ. Equ. 57 (2018) 134. [11] P. Knopf and J. Weber, Optimal control of a Vlasov–Poisson plasma by fixed magnetic field coils. Appl. Math. Optim. 81 (2020) 961–988. [12] M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system. Electron. J. Differ. Equ. 2005 (2005) 1–17. [13] T. T. Nguyen, T. V. Nguyen and W. A. Strauss, Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinet. Relat. Models 8 (2015) 53. [14] G. Rein, Global weak solutions to the relativistic Vlasov-Maxwell system revisited. Commun. Math. Sci. 2 (2004) 145–158. [15] S. Robinson, Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13 (1976) 497–513. [16] J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5 (1979) 49–62. COCV_2021__27_S1_A21_09b5d384c-5b21-472b-b908-27c97844d1d9 cocv190218 10.1051/cocv/202006810.1051/cocv/2020068 On minimizers of an anisotropic liquid drop model Misiats Oleksandr 0000-0003-4584-156X Topaloglu Ihsan * Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA. *Corresponding author: [e-mail] 01 03 2021 01 03 2021 2021 cocv/2021/01 Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science OuverteSupplementS20 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF) Full (DJVU)

We consider a variant of Gamow’s liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in C¹-norm and quantify the rate of convergence. We also obtain a quantitative expansion of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.

Liquid drop model anisotropic Wulff shape quasi-minimizers of anisotropic perimeter 35Q40 35Q70 49Q20 49S05 82D10 idline ESAIM: COCV 27 (2021) S20 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S20 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020068 www.esaim-cocv.org ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL Oleksandr Misiats and Ihsan Topaloglu* Abstract. We consider a variant of Gamow's liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in C1 -norm and quantify the rate of convergence. We also obtain a quantitative expansion of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter. Mathematics Subject Classification. 35Q40, 35Q70, 49Q20, 49S05, 82D10. Received December 23, 2019. Accepted October 7, 2020. 1. Introduction In this paper we consider an anisotropic nonlocal isoperimetric problem given by inf n E(F) |F| = 1 o , (1.1) over sets of finite perimeter F Rn where E(F) := Z F f(F ) dHn-1 + Z F Z F 1 |x - y| dxdy, with 0 < < n and | · | denotes the Lebesgue measure. The first term in E is the anisotropic surface energy Pf (F) := Z F f(F ) dHn-1 , defined via a one-homogeneous and convex surface tension f : Rn [0, ) that is positive on Rn \ {0}. Here Hn-1 is the (n - 1)-dimensional Hausdorff measure, and F denotes the reduced boundary of F, which consists Keywords and phrases: Liquid drop model, anisotropic, Wulff shape, quasi-minimizers of anisotropic perimeter. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 O. MISIATS AND I. TOPALOGLU of points x F where the limit F (x) = lim0 -F (B(x)) |F |(B(x)) exists and has length one. This limit is called the measure-theoretic outer unit normal of F. The second term in the energy E is given by the Riesz interactions V(F) := Z F Z F 1 |x - y| dxdy, for 0 < < n. The minimization problem (1.1) is equivalent (via the rescaling = m(n+1-)/n ) to the anisotropic liquid drop model inf n E(E) := Pf (E) + V(E) |E| = m o , (1.2) introduced by Choksi, Neumayer and the second author in [9] as an extension of the classical liquid drop model. Gamow's liquid drop model, initially developed to predict the mass defect curve and the shape of atomic nuclei, dates back to 1930 [18]; however, it recently has generated considerable interest in the calculus of variations community (see e.g. [1, 5, 7, 17, 20­23] as well as [8] for a review). The version of this model in the language of the calculus of variations includes two competing forces: an attractive isotropic surface energy associated with the depletion of nucleon density near the nucleus boundary, and a repulsive Coulomb energy due to the interactions of positively charged protons. These two forces are in direct competition. The surface energy prefers uniform, symmetric and connected domains whereas the repulsive term is minimized by a sequence of sets diverging infinitely apart. The parameter of the problem ( in (1.1) or m in (1.2)) sets a length scale between these competing forces. As such, the liquid drop model is a paradigm for shape optimization via competitions of short- and long-range interactions and it appears in many different systems at all length scales. In the anisotropic extension of the liquid drop model the global minimizer of the surface energy Pf (E) over sets |E| = m is (a dilation or translation of) the Wulff shape Kf associated with f (cf. [6, 15, 16]), where Kf := \ Sn-1 x Rn x · < f() . (1.3) Properties of Kf depend on the regularity of the surface tension f. In the literature two important classes of surface tensions are considered: ­ We say that f is a smooth elliptic surface tension if f C (Rn \ {0}) and there exist constants 0 < 6 < such that for every Sn-1 , ||2 6 2 f()[, ] 6 ||2 , for all Rn with · = 0. For such surface tensions, the corresponding Wulff shape has C boundary and is uniformly convex. ­ We say that f is a crystalline surface tension if for some N finite and xi Rn , f() = max 16i6N xi · . For crystalline surface tensions, the corresponding Wulff shape K is a convex polyhedron. In the anisotropic liquid drop model (1.2) the competition which leads to an energy-driven pattern formation is not only between the attractive and repulsive forces, as they scale differently in terms of the mass m, but there is also a competition between the anisotropy in the surface energy and the isotropy in the Coulomb-like energy. As shown in ([9], Thm. 3.1), the problem (1.2) admits a minimizer when m is sufficiently small and fails ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 3 to have minimizers for large values of m. However, ([9], Thm. 1.1) shows that when f is smooth the Wulff shape Kf is not a critical point of the energy E for any m > 0. On the other hand, for particular crystalline surface tensions the authors prove that the corresponding Wulff shape is the unique (modulo translations) minimizer for sufficiently small m. This demonstrates a fundamentally interesting situation: the regularity and ellipticity of the surface tension f determines whether the isoperimetric set Kf is also a minimizer of the perturbed problem (1.1). As stated in [9], while the regularity and ellipticity of the surface tension affect typically quantitative aspects of anisotropic isoperimetric problems, here, due to the incompatibility of the Wulff shape with the Riesz energies, qualitative aspects of the problem are effected, too. Motivated by the results in [9], we study qualitative properties of the minimizers of (1.1) for smooth anisotropies in the asymptotic regime 0, and obtain ­ the convergence of the minimizers to the Wulff shape in strong norms, providing the rate of convergence, and ­ an expansion of the energy around the energy of a Wulff shape in terms of . In particular, our first main result shows that the minimizers of E are close to the Wulff shape in C1 -norm in the small regime. Further we obtain quantitative estimates on how much a minimizer F of E differs from the Wulff shape when is sufficiently small. Theorem 1.1. Let f be a smooth elliptic surface tension and F be a minimizer of the problem (1.1). Let K denote the Wulff shape corresponding to f rescaled so that |K| = 1. Then we have the following two statements. (i) For > 0 sufficiently small there exists C1 (K) such that F = x + (x)K(x) x K , and |F4K| . kkC1(K) . |F4K|1/(n+1) . (ii) For > 0 sufficiently small, we have that |F4K| ' . Combining parts (i) and (ii) of the theorem, we conclude that . kkC1(K) . 1/(n+1) . Using the elliptic regularity theory, via Schauder estimates on the Euler­Lagrange equation, implies that 0 in some C2, -norm as 0; however, finding the rate of convergence explicitly in terms of seems to be a challenging task. As for quantifying the convergence rate in stronger norms, adapting arguments from Figalli and Maggi's work on the shapes of liquid drops (cf. [13]), it is possible to obtain quantitative convexity estimates on minimizers F ultimately yielding C2 -control on the function via an upper bound that depends on (see Rem. 2.6). Although the result above only establishes an explicit C1 -control of the function , our proof relies only on a simple geometric argument we present in the next section. Next we show that the energy difference between a minimizer and the Wulff shape scales as 2 . 4 O. MISIATS AND I. TOPALOGLU Theorem 1.2. Suppose f is a smooth elliptic surface tension that is not a constant multiple of the Euclidean distance. Let F be a minimizer of the energy E. Then for sufficiently small, E(K) - E(F) ' 2 , (1.4) where K is the Wulff shape corresponding to f rescaled so that |K| = 1 and translated to have the same barycenter as F. Combined with the estimate on the symmetric difference, this expansion also yields a geometric stability estimate of the form E(K) - E(F) > C|F4K|2 , for the minimizer F in the small regime. We prove Theorems 1.1 and 1.2 in Section 2. In two dimensions, when the Wulff shape is given by a particular perturbation of a set that is symmetric with respect to the coordinate axes and lines y = ±x, it is possible to determine the constant in the lower bound E(Kf ) - E(F) > C 2 , explicitly. This result is independent of the regularity of the surface tension f and applies to both smooth and crystalline cases. Furthermore, when the surface tension is given by f() = 1 2 a0| · e1| + a-1 0 | · e2| for some a0 > 1, we show that the minimizer of E is a rectangle with dimensions determined explicitly in terms of a0 and , and we obtain an expansion of the energy E of a minimizer in terms of and a0 only. We prove these results in Section 3. Finally, we would like to note that a similar incompatibility occurs also in a nonlocal isoperimetric problem considered by Cicalese and Spadaro [10] where the authors study the isotropic version of the energy E (i.e., with f given by the Euclidean distance) on a bounded domain . Here the incompatibility is due to the boundary effects. As a result of the boundary effects the isoperimetric region (in this case a ball) is not a critical point of the nonlocal term, and the authors study the asymptotic properties of the minimizers in the small limit. Notation Throughout the paper we use the notation f . g to denote that f 6 Cg for some constant C > 0 independent of f. We also write f ' g to denote that c g 6 f 6 C g for constants c, C > 0 independent of f. The constants C we use might change from line to line unless defined explicitly. Also, when necessary, we emphasize the dependence of the constants to the parameters. In order to simplify notation we will denote the Wulff shape by K, suppressing the dependence on the surface tension f. 2. Proofs of Theorem 1.1 and Theorem 1.2 The proof of the first part of Theorem 1.1 relies on a result which is independent of the optimality of the set F, and is rather a general property of two sets where the boundary of one of the sets is expressed as a graph over the boundary of the other set. We state this geometric result as a separate lemma since it might be of interest to readers beyond its connection to the anisotropic liquid drop model. Lemma 2.1. Suppose E and F are bounded subsets of Rn with C1 boundaries such that F = x + (x)E(x) x E , for some function C1 (E). (i) Then 1 Hn-1(E) |E4F| 6 kkC1(E). ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 5 (ii) If, in addition, E C2 and F is convex, then there exits a constant C > 0, depending only on the maximal principle curvature and the perimeter of E (see (2.3) for the explicit dependence) such that kkC1(E) 6 C|E4F| 1 n+1 . (2.1) Remark 2.2. In fact, the simple estimate below shows that the symmetric difference can be controlled by the C0 -norm of . Namely, |E4F| 6 Hn-1 (E) kkC0(E). Proof. The proof of (i) is straightforward: |E4F| = Z E |(x)| dHn-1 6 kkC1(E)Hn-1 (E). To show (ii), let z = (z 1 , ..., z n) = arg maxE |(x)|. Suppose z E F. Further, assume that in some neighborhood Uz Rn there exist f, g : Rn-1 R such that F = x = (x1, ..., xn) Uz xn = f(x1, ..., xn-1) , and E = x = (x1, ..., xn) Uz xn = g(x1, ..., xn-1) . Since the mean curvature of E is bounded, we can choose the neighborhood Uz such that |Uz | > cE > 0 for some constant cE depending only on , the maximal principle curvature of E. Finally, without loss of generality, we may choose an appropriate rotation and translation to have z = 0, g(0) = f(0) = 0 and f(0) = 0. (2.2) In this case max E |(x)| = |g(0)|. Denote x := (x1, ..., xn-1) and U0 = x Rn-1 : (x, 0) U0 . If we expand g in the neighborhood U0 of 0, we get g(x) = g(0) · x + 1 2 x · 2 g(0) · xT + o(|x|2 ). Furthermore x · 2 g(0) · xT > -c2 |x|2 , where c > 0 depends only on . Hence in U0 we have g(x) > g(0) · x - c2 |x|2 = |g(0)|2 4c2 - cx - g(0) 2c 2 . On the other hand, due to the convexity of F, we have f(x) 6 0 in U0. Therefore, |E4F| > Z U0 |g(x) - f(x)| dHn-1 x > Z U0{g(x)>0} g(x) dHn-1 x > Z U0 |cx- g(0) 2c | 2 6 |g(0)|2 4c2 |g(0)|2 4c2 - cx - g(0) 2c 2 ! dHn-1 x 6 O. MISIATS AND I. TOPALOGLU = 1 cn-1 Z U0 |y|26 |g(0)|2 4c2 |g(0)|2 4c2 - |y| 2 dHn-1 y = cE n-1 cn-1 |g(0)| 2c n+1 - (n - 1) cEn-1 cn-1 Z |g(0)| 2c 0 rn dr = cE n-1|g(0)|n+1 (n + 1)2nc2n . If z E \ F, on the other hand, we may again choose an appropriate rotation and translation so that z = 0; however, now the functions f and g in (2.2) satisfy g(0) = cg, f(0) = 0 and f(0) = 0. This, in turn, implies that g(x) > cg + |g(0)|2 4c2 - cx - g(0) 2c 2 , and estimating as above we get |E4F| > |cg|cE + cE n-1|g(0)|n+1 (n + 1)2nc2n > cE n-1|g(0)|n+1 (n + 1)2nc2n . Thus max E |(x)| 6 2n c2n (n + 1) cE n-1 1 n+1 |E4F| 1 n+1 . Under the assumption that (0) = 0, we have max E |(x)| 6 max E |(x)|Hn-1 (E), hence altogether kkC1(E) 6 C |E4F| 1 n+1 , where C := 1 + Hn-1 (E) 2n c2n (n + 1) cE n-1 1 n+1 , (2.3) is independent of F. Remark 2.3. The constants c and cE differ from the actual maximal principle curvature of E by a factor, proportional to (1 + |g(0)|2 )3/2 . Remark 2.4. The inequality (2.1) holds true for any C2 smooth set F, not necessarily convex. However, in this case, the constant C in (2.3) will depend on the principle curvatures of both F and E. In our case, we will apply the lemma in the situation, where the curvature of E is known while the curvature of F is not known a priori, hence the convexity of F is crucial. ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 7 For a smooth elliptic surface tension f the first variation of Pf (E) with respect to a variation generated by X C1 c (Rn , Rn ) is given by Pf (E)[X] = Z E div E f E X · E dHn-1 . Here div E denotes the tangential divergence along E. The function Hf E : E R defined by Hf E = div E f E is called the anisotropic mean curvature of the reduced boundary of E. The first variation of V(E) with respect to a variation generated by X C1 c (Rn , Rn ), on the other hand, is given by V(E)[X] = Z E vE(x)X · E dHn-1 , with vE(x) = R E |x - y|- dy. We say that a set E is a critical point of (1.1) if (Pf (E) + V(E))[X] = 0 for all variations with R E X · E dHn-1 = 0, i.e., variations that preserve volume to first order. Hence, a volume-constrained critical point E of (1.1) satisfies the Euler-Lagrange equation Hf E(x) + vE(x) = for all x E, (2.4) where the constant is the Lagrange multiplier associated with the volume constraint |E| = 1. In order to obtain the rate of the L1 -convergence of the minimizing sets in terms , we will utilize the following lemma, which provides a lower bound on the energy deficit and is also a fundamental part of the proof of Theorem 1.2. Lemma 2.5. Suppose f is a smooth elliptic surface tension that is not a constant multiple of the Euclidean distance. Let F be a minimizer of the energy E. Then for sufficiently small, E(K) - E(F) > C2 , (2.5) where K is the Wulff shape corresponding to f rescaled so that |K| = 1 and translated to have the same barycenter as F and the constant C depends only on f, , and d. Proof. First note that since f is not a multiple of the Euclidean distance the corresponding Wulff shape K is not a ball but since K minimizes the perimeter functional its anisotropic mean curvature Hf K is constant on K. On the other hand, characterization results ([9], Thm. 1.3) and ([19], Thm. 4.2.) state that the only sets E for which the Riesz potential vE is constant on E are given by balls. Hence, K does not satisfy (2.4), and therefore it is not a critical point of the energy E for any . This implies that there exists a function : K R such that the first variation of V in the normal direction K is negative for small perturbations by the function . That is, µ2(K) := V(K) = [K] = d d V(K,) =0 < 0, where K, := {x + (x)K(x) x K}. Now, let µ1(K) := Z K D2 f(, ) - 2 tr(D2 fA2 K) dHn-1 , 8 O. MISIATS AND I. TOPALOGLU where AK denotes the second fundamental form of K. Then µ1(K) = 2 Pf (K)[K], the second variation of Pf at K (cf. ([11], Thm. 4.1)). Since f is uniformly elliptic by assumption, using ([24], Lem. 4.1) and arguing as in the proof of ([24], Prop. 1.9) (where we also use that bar K = bar F with bar denoting the barycenter of a set), we have that µ1(K) > C Z K ||2 dHn-1 > 0. Using the minimality of F and expanding the energies in terms of we obtain E(K) - E(F) > E(K) - E(K,) = Pf (K) - Pf (K,) + V(K) - V(K,) = -µ2(K) - µ1(K) 2 2 - o(). Optimizing in we let = - µ2(K)/µ1(K) . Then E(K) - E(F) > C2 for some constant C > 0; hence, we obtain the lower bound as claimed. Another important ingredient in the proof of the theorem is the regularity of quasiminimizers of the surface energy Pf . We say that F is a q-volume-constrained quasiminimizer of Pf if Pf (F) 6 Pf (E) + q|F4E| for all E with |E| = |F|. We are now ready to prove the theorem. Proof of Theorem 1.1. We start by noting that the nonlocal functional V is Lipschitz continuous with respect to the symmetric difference. To see this let (0, n) and let vF : Rn R denote the Riesz potential of F given by vF (x) = R F |x - y|- dy. Hence, V(F) = R F vF (x) dx. Let r = -1/n n , where n denotes the volume of the unit ball in Rn . Then kvF kL(Rn) 6 kvBr(0)kL(Rn) = vBr(0)(0) = n 1-(n-)/n n n - . In fact, by ([25], Lem. 3) and ([5], Prop. 2.1), vF is Holder continuous with kvF kCk, (Rn) 6 C(n, |F|, k, ), for k = bn - c and (0, 1) with k + < n - . A direct calculation shows that V(E) - V(F) = Z Rn Z Rn E(x)E(y) - F (x)F (y) |x - y| dxdy = Z Rn vE(y)(E(y) - F (y)) dy + Z Rn vF (x)(E(x) - F (y)) dx 6 2n /n n n - |E4F|. ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 9 Hence, for any E, F Rn with |E| 6 |F| the functional V is Lipschitz continuous with respect to the symmetric difference with Lipschitz constant given by cn, := 2n /n n n - . Now, for any minimizer F with |F| = 1 of the energy E, we have that Pf (F) 6 Pf (E) + V(E) - V(F) 6 Pf (E) + cn,|E4F|, for any competitor E with |E| = 1. Thus F is a cn,-volume-constrained quasiminimizer of the surface energy Pf . Classical arguments and regularity results for quasiminimizers in the literature (see e.g. [2­4, 12]) imply that for sufficiently small F is a C2, -hypersurface for all (0, 0) with 0 := min{1, n - }. In fact, F can locally be written as a small C2, -graph over the boundary of the Wulff shape K of mass 1, and F is uniformly convex (see also [9], Thm. 2.2 for a precise statement of this regularity result). Therefore, there exists C1 (K) such that F = x + (x)K(x) x K . Since both F and K are uniformly convex, we can apply Lemma 2.1 to conclude that |F4K| . kkC1(K) . |F4K|1/(n+1) . This establishes part (i) of the theorem. In order to prove the second part, first we note that by the quantitative Wulff inequality ([14], Thm. 1.1), Pf (F) - Pf (K) > C |F4K|2 , for some constant C > 0 depending only on n and K. Then, by minimality of F and by Lipschitzianity of V, we get |F4K|2 6 C Pf (F) - Pf (K) 6 C V(K) - V(F) 6 C |F4K|. Hence, |F4K| 6 C, and we obtain the upper bound in part (ii). In order to prove the lower bound, first suppose that |Pf (K) - Pf (F)| 2 for any > 0. Note that V(K) - V(F) > C for some C > 0 since otherwise -1 k V(K) - V(F) 0 along a subsequence k, which would imply that E(K) - E(F) = o(2 ) and this would contradict the estimate (2.5). Therefore, there exists a constant C > 0 such that C2 6 Pf (K) - Pf (F) + V(K) - V(F) . Since K minimizes the surface energy Pf , again using the Lipschitzianity of V, we can estimate the right-hand side by V(K) - V(F) 6 C |F4K|. Combining these two estimates yields |F4K| > C. 10 O. MISIATS AND I. TOPALOGLU If Pf (F) - Pf (K) > C2 , on the other hand, then using minimality of F as above yields C2 6 Pf (F) - Pf (K) 6 C V(K) - V(F) 6 C |F4K|. Hence, we again obtain the lower bound |F4K| > C, and combined with the upper bound this concludes the proof of the theorem. Remark 2.6 (Quantification of convexity). For f C (Rn \ {0}) and (0, n - 1) it is possible to adapt the arguments in ([13], Thm. 2 and Rem. 2) to include the nonlocal Riesz kernel V as the perturbation of Pf , and obtain quantitative estimates in terms of regarding the convexity of F. Namely, one can prove that max F |2 f(F )F - IdTxF | 6 C 2n (n+2)(n+1-) , where F denotes the second fundamental form of F. This, ultimately, provides a quantitative estimate on kkC2(F ) in terms of . We finish this section with an expansion of the energy of a minimizer of E around the energy of the Wulff shape corresponding to smooth elliptic anisotropies that are not given by the Euclidean distance. The key idea here is that for such surface tensions the Wulff shape is not a critical point (in the sense of first variations by smooth perturbations) of the nonlocal part V. Therefore, the energy expansion does not vanish at the first order, and contribution at order is present. Proof of Theorem 1.2. We will prove this theorem in two parts. The upper bound follows by the Lipschitzianity of the nonlocal term V and the result of Theorem 1.1(ii). Namely, for sufficiently small we have E(K) - E(F) 6 V(K) - V(F) 6 C|F4K| 6 C2 . The lower bound, on the other hand, follows directly from Lemma 2.5. 3. Explicit constructions in two dimensions For certain surface tensions in two dimensions a constant in the lower bound of the expansion (1.4) can be computed explicitly by a particular choice of small perturbations for which the first variation of the nonlocal part is negative. In order to determine these constants quantitatively, we consider one dimensional transformations of the Wulff shape. We will denote by Ea the one-dimensional stretching of any set E R2 with barycenter zero by a factor a > 0, i.e., Ea := nx a , ay R2 (x, y) E o . (3.1) Our first result gives an explicit lower bound of the energy expansion when the Wulff shape is such a trans- formation of a diagonally symmetric set. Examples of such symmetric sets include sets with smooth boundaries as well as regular polygons such as octagons, and they can be written as Wulff shapes of functions which possess dihedral symmetry. That is, if D4 denotes the set of eight matrices in the dihedral group, then we will consider functions f : R2 R satisfying f(Ax) = f(x) for all A D4 and x R2 . (3.2) We also note that the proposition below does not make any assumptions on the regularity of the surface tension f, and applies to both the smooth and crystalline cases. ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 11 Proposition 3.1. Let f : R2 R be a surface tension (either smooth elliptic or crystalline) satisying (3.2). Let fa(x1, x2) = f(ax1, x2/a) for any a > 0, and let Ka0 be the Wulff shape corresponding to fa0 for some a0 > 0. Then for any minimizer F of the energy E defined via the surface tension fa0 , and for sufficiently small, we have E(Ka0 ) - E(F) > C 2 where the constant C is determined explicitly in terms of the second variation of Pfa0 and the first variation of V around Ka0 . Proof. Let K be the Wulff shape corresponding to the function f. Since f satisfies (3.2), K is symmetric with respect to the rotations and reflections in D4. The symmetry of K implies E(Ka0 ) = E(K1/a0 ); hence, without loss of generality, we can take a0 > 1. For any fa let Kfa be the corresponding Wulff shape. We claim that the set Kfa equals Ka where Ka is obtained from K via the transformation (3.1). Since the sets Kfa and Ka are convex it suffices to show that the boundary is mapped to the boundary. To see this, for any [-, ] \ {±/2}, let = arctan(a2 tan ), and = arccot(a-2 cot ) if = ±/2. Then we have cos = a-1 cos p a-2 cos2 + a2 sin2 and sin = a sin p a-2 cos2 + a2 sin2 . This yields, Ka = \ [-,] n (x, y) R2 : a-1 x cos + ay sin = f(cos , sin ) o = \ [-,] ( (x, y) R2 : a-1 x cos p a-2 cos2 + a2 sin2 + ay sin p a-2 cos2 + a2 sin2 = f cos p a-2 cos2 + a2 sin2 , sin p a-2 cos2 + a2 sin2 !) = \ [-,] n (x, y) R2 : x cos + y sin = f(a cos , a-1 sin ) o = \ [-,] n (x, y) R2 : x cos + y sin = fa(cos , sin ) o = Kfa . For any a0, let Ka0 be the Wulff shape determined by the surface tension fa0 . Then for any a close to a0 the perimeter can be expanded as Pf (Ka) = Pf (Ka0 ) + µ1(Ka0 ) 2 (a - a0)2 + O (a - a0)3 , (3.3) where µ1(Ka0 ) := d2 /da2 Pf (Ka) a=a0 . 12 O. MISIATS AND I. TOPALOGLU Note that d/daPf (Ka) a=a0 = 0 since Ka0 is the corresponding Wulff shape; hence, it is a critical point. Moreover, as both Ka0 and Ka are convex and perturbations of K, in two dimensions they intersect at at most four points. Hence, using at most four functions, it is possible to express Ka locally as a graph over Ka0 . Therefore, arguing as in the proof of Theorem 1.2 we get that µ1(Ka0 ) > 0. On the other hand, expanding the nonlocal term, we get V(Ka) = V(Ka0 ) + µ2(Ka0 )(a - a0) + O (a - a0)2 , (3.4) where µ2(Ka0 ) := d da V(Ka) a=a0 . In order to explicitly evaluate the first variation of the nonlocal energy with respect to these special perturbations, we introduce the change of variables xi = xi/a and yi = yi/a for i = 1, 2. This yields V(Ka) = Z K Z K a2 (x1 - x2)2 + a-2 (y1 - y2)2 - 2 dx1dx2dy1dy2. Hence, µ2(Ka0 ) = - a0 Z K Z K a2 0(x1 - x2)2 - a-2 0 (y1 - y2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2. Changing the variables once again, we get µ2(Ka0 ) = - a0 Z Ka0 Z Ka0 (x1 - x2)2 - (y1 - y2)2 (x1 - x2)2 + (y1 - y2)2 1+ 2 dx1dx2dy1dy2. This show that µ2(Ka0 ) < 0 due to the fact that the deformation of K into Ka0 stretches the domain in the x-direction, hence increasing the first term in the integral, and at the same time shrinks it in the y-direction, thus decreasing the second term in the integral. Referring back to the expansions (3.3) and (3.4), there exists two positive constants C1 and C2 such that E(Ka) 6 E(Ka0 ) + 1 2 µ1(Ka0 ) (a - a0)2 + µ2(Ka0 ) (a - a0) + C1(a - a0)3 + C2(a - a0)2 . Optimizing in (a - a0) we let a - a0 = - µ2(Ka0 ) µ1(Ka0 ) , and note that the coefficient is positive since µ2(Ka0 ) < 0. Then using the fact that F is a minimizer, we get E(F) 6 E(Ka) 6 E(Ka0 ) - µ2 2(Ka0 ) 4µ1(Ka0 ) 2 + C2 µ2 2(Ka0 ) 4µ2 1(Ka0 ) - C1 µ3 2(Ka0 ) 8µ3 1(Ka0 ) 3 6 E(Ka0 ) - µ2 2(Ka0 ) 8µ1(Ka0 ) 2 , ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 13 for > 0 sufficiently small. Hence, E(Ka0 ) - E(F) > µ2 2(Ka0 ) 8µ1(Ka0 ) 2 , with the constant depending only on the set K (that is, on the surface tension f) and a0. Remark 3.2 (More general sets). The proposition above can be stated for more general Wulff shapes which are not necessarily perturbations via (3.1) of a set symmetric with respect to the dihedral group D4. In fact, a sufficient condition on a set S for the above proof to work is that Z S Z S (x1 - x2)2 - (y1 - y2)2 (x1 - x2)2 + (y1 - y2)2 1+ 2 dx1dx2dy1dy2 6= 0. Sets considered in the above proposition are S = Ka0 where K can be a disk, square, regular octagon, etc. As mentioned before, the map K Ka0 for a0 6= 1 is stretching (shrinking) the set K in x direction while shrinking (stretching) the set in y direction, and is one of the examples of the perturbation for which d da0 V(Ka0 ) a0=1 6= 0. We conjecture that one can perform a similar shrinking/stretching deformation K Ka0 along some direction such that d da0 V(Ka0 ) a0=1 6= 0. for any K different from a ball. Remark 3.3 (The constants µ1(Ka0 ) and µ2(Ka0 )). While approximate values of µ1(Ka0 ) and µ2(Ka0 ) can be found numerically, finding their exact values analytically is a challenging task. Although the perturbation of the Wulff shape is given by a simple transformation, determining the exact value of µ1 would require an explicit formula for the surface tension f corresponding to K in order to write f(Ka ) in terms of f(K). For the constant µ2, on the other hand, we can derive estimates in different a0 regimes, using the properties of the set K. We list these estimates here. (i) For a0 1, we have µ2(Ka0 ) = - a1+ 0 Z K Z K dx1dx2dy1dy2 (x1 - x2) + o a -(1+) 0 . Hence, lima0 µ2(Ka0 ) = 0. (ii) Note that µ2(K) = - Z K Z K (x1 - x2)2 - (y1 - y2)2 (x1 - x2)2 + (y1 - y2)2 1+ 2 dx1dx2dy1dy2. Since both the denominator of the above integral and the set K is symmetric with respect to swapping the variables xi and yi, we get that µ2(K) = 0. Hence, K is a critical point of V with respect to this special class of perturbations. 14 O. MISIATS AND I. TOPALOGLU Figure 1. For any Wulff shape K which is convex and has 8-fold symmetry, we can find squares Smin and Smax as depicted above. (iii) For a0 close to 1, µ2(Ka0 ) = d da µ2(Ka) a=1 (a0 - 1) + O(a0 - 1)2 where d da µ2(Ka) a=1 = -2 Z K Z K - (x1 - x2)4 + (y1 - y2)4 + (4 + )(x1 - x2)2 (y1 - y2)2 (x1 - x2)2 + (y1 - y2)2 2+/2 dx1dx2dy1dy2 (iv) We may estimate µ2(Ka0 ) and d da µ(Ka) a=1 independently of K. Suppose K is an arbitrary convex set which is symmetric with respect to the lines y = ±x, such that K passes through (2p, 0) for some p > 0. Then Smin K Smax where Smax = [-2p, 2p] × [-2p, 2p], and Smin = [-p, p] × [-p, p] (see Fig. 1). Moreover, Z Smax Z Smax a2 0(x1 - x2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2 > Z K Z K a2 0(x1 - x2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2 > Z Smin Z Smin a2 0(x1 - x2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2 = 2-4 Z Smax Z Smax a2 0(x1 - x2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2. ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 15 Analogously, Z Smax Z Smax a-2 0 (y1 - y2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2 > Z K Z K a-2 0 (y1 - y2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2 > 2-4 Z Smax Z Smax a-2 0 (y1 - y2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2. Therefore, - a0 Z Smax Z Smax 2-4 a2 0(x1 - x2)2 - a-2 0 (y1 - y2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2 > µ2(Ka0 ) > - a0 Z Smax Z Smax a2 0(x1 - x2)2 - 2-4 a-2 0 (y1 - y2)2 a2 0(x1 - x2)2 + a-2 0 (y1 - y2)2 1+ 2 dx1dx2dy1dy2, Similar upper and lower bounds can be found for d da µ2(Ka) a=1 as well. When the Wulff shape is given by a rectangle in two dimensions, due to a rigidity theorem by Figalli and Maggi, we obtain a quantitative description of the minimizers as well as an asymptotic expansion of its energy in terms of and the Wulff shape. Proposition 3.4. Let S = [-1/2, 1/2] × [-1/2, 1/2] be the square of area 1. For a0 > 1, let f() = 1 2 a0| · e1| + a-1 0 | · e2| be the surface tension whose corresponding Wulff shape is Sa0 obtained via the transformation (3.1). Then there exists > 0 such that for < any minimizer of E is a rectangle Sa where a = a0 - µ2(a0)a2 0 2 , (3.5) and E(Sa) = E(Sa0 ) - µ2(a0)a0 2 2 2 + µ2(a0)a0 2 3 + µ2 2(a0)µ3(a0)a4 0 8 ! 3 + o(3 ), (3.6) with µ2(a0) = d da V(Sa) a=a0 and µ3(a0) = d2 da2 V(Sa) a=a0 . Proof. Let F be a minimizer of E. As shown in the proof of Theorem 1.1 above, for sufficiently small, F is a cn,-volume-constrained quasiminimizer of the surface energy Pf . Then, by the two dimensional rigidity theorem ([13], Thm. 7) of Figalli and Maggi, which states that if f is a crystalline surface tension then any q-volume-constrained quasiminimizer with sufficiently small q is a convex polygon with sides aligned with those of the Wulff shape, we get that F is a rectangle with side parallel to Sa0 . Thus, there exists > 0, such that for < we have F = Sa for some a > 1. In order to find the optimal scaling a, we expand the perimeter and the nonlocal term around a0 and get E(Sa) = E(Sa0 ) + 1 a2 0 (a - a0)2 + µ2(a0)(a - a0) + 2 µ3(a0)(a - a0)2 - 1 a3 0 (a - a0)2 + · · · . 16 O. MISIATS AND I. TOPALOGLU Optimizing at the second-order (i.e., the second and third terms in the expansion above) yields, as before, a - a0 = - µ2(a0)a2 0 2 , and we obtain (3.5). Plugging this back into E(Sa) we get (3.6), i.e., an exact expansion of the energy of a minimizer in . Remark 3.5. While we cannot determine the constant explicitly, the expansion (3.6) yields an explicit upper bound on . Namely E(Sa) < E(Sa0 ) implies that < 2(µ2(a0)a0)2 (µ2(a0)a0)3 + µ2 2(a0)µ3(a0)a4 0 . Acknowledgements. The authors would like to thank Gian Paolo Leonardi for bringing the question of energy expansion around the energy of the Wulff shape to our attention and to Marco Bonacini, Riccardo Cristoferi and Robin Neumayer for their valuable suggestions and comments. Finally, the authors are grateful to the referees for their careful reading of the manuscript and for their detailed suggestions. References [1] S. Alama, L. Bronsard, R. Choksi and I. Topaloglu, Droplet breakup in the liquid drop model with background potential. Commun. Contemp. Math. 21 (2019) 1850022. [2] F.J. Almgren Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem. Ann. Math. 84 (1966) 277­292. [3] F.J. Almgren Jr., R. Schoen and L. 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Var. doi:10.1515/acv- 2019-0088 (2020). [10] M. Cicalese and E. Spadaro, Droplet minimizers of an isoperimetric problem with long-range interactions. Comm. Pure Appl. Math. 66 (2013) 1298­1333. [11] U. Clarenz and H. von der Mosel, On surfaces of prescribed F-mean curvature. Pacific J. Math. 213 (2004) 15­36. [12] F. Duzaar and K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546 (2002) 73­138. [13] A. Figalli and F. Maggi, On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201 (2011) 143­207. [14] A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182 (2010) 167­211. [15] I. Fonseca. The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432 (1991) 125­145. [16] I. Fonseca and S. Muller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 125­136. [17] R.L. Frank and E.H. Lieb, A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J. Math. Anal. 47 (2015) 4436­4450. [18] G. Gamow, Mass defect curve and nuclear constitution. Proc. R. Soc. Lond. A 126 (1930) 632­644. [19] J. Gomez-Serrano, J. Park, J. Shi and Y. Yao, Symmetry in stationary and uniformly-rotating solutions of active scalar equations. Preprint arXiv:1908.01722 (2019). [20] V. Julin, Isoperimetric problem with a Coulomb repulsive term. Indiana Univ. Math. J. 63 (2014) 77­89. [21] H. Knupfer and C.B. Muratov, On an isoperimetric problem with a competing nonlocal term II: The general case. Comm. Pure Appl. Math. 67 (2014) 1974­1994. [22] H. Knupfer, C.B. Muratov and M. Novaga, Low density phases in a uniformly charged liquid. Comm. Math. Phys. 345 (2016) 141­183. ON MINIMIZERS OF AN ANISOTROPIC LIQUID DROP MODEL 17 [23] J. Lu and F. Otto, Nonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsacker model. Comm. Pure Appl. Math. 67 (2014) 1605­1617. [24] R. Neumayer, A strong form of the quantitative Wulff inequality. SIAM J. Math. Anal. 48 (2016) 1727­1772. [25] W. Reichel, Characterization of balls by Riesz-potentials. Ann. Mat. Pura Appl. 188 (2009) 235­245. [1] S. Alama, L. Bronsard, R. Choksi and I. Topaloglu, Droplet breakup in the liquid drop model with background potential. Commun. Contemp. Math. 21 (2019) 1850022. [2] F. J. Almgren Jr. Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. Math. 84 (1966) 277–292. [3] F. J. Almgren Jr. R. Schoen and L. Simon, Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. Acta Math. 139 (1977) 217–265. [4] E. Bombieri, Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal. 78 (1982) 99–130. [5] M. Bonacini and R. Cristoferi, Local and global minimality results for a nonlocal isoperimetric problem on ℝ$$. SIAM J. Math. Anal. 46 (2014) 2310–2349. [6] J. E. Brothers and F. Morgan, The isoperimetric theorem for general integrands. Michigan Math. J. 41 (1994) 419–431. [7] R. Choksi and M. A. Peletier, Small volume fraction limit of the diblock copolymer problem: I. Sharp-interface functional. SIAM J. Math. Anal. 42 (2010) 1334–1370. [8] R. Choksi, C. B. Muratov and I. Topaloglu, An old problem resurfaces nonlocally: Gamow’s liquid drops inspire today’s research and applications. Notices Amer. Math. Soc. 64 (2017) 1275–1283. [9] R. Choksi, R. Neumayer and I. Topaloglu, Anisotropic liquid drop models. To appear in: Adv. Calc. Var. doi: (2020). [10] M. Cicalese and E. Spadaro, Droplet minimizers of an isoperimetric problem with long-range interactions. Comm. Pure Appl. Math. 66 (2013) 1298–1333. [11] U. Clarenz and H. Von Der Mosel, On surfaces of prescribed F-mean curvature. Pacific J. Math. 213 (2004) 15–36. [12] F. Duzaar and K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546 (2002) 73–138. [13] A. Figalli and F. Maggi, On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201 (2011) 143–207. [14] A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182 (2010) 167–211. [15] I. Fonseca. The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432 (1991) 125–145. [16] I. Fonseca and S. Müller, 2016 A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 125–136. [17] R. L. Frank and E. H. Lieb, A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J. Math. Anal. 47 (2015) 4436–4450. [18] G. Gamow, Mass defect curve and nuclear constitution. Proc. R. Soc. Lond. A 126 (1930) 632–644. [19] J. Gómez-Serrano, J. Park, J. Shi and Y. Yao, Symmetry in stationary and uniformly-rotating solutions of active scalar equations. Preprint (2019).. [20] V. Julin, Isoperimetric problem with a Coulomb repulsive term. Indiana Univ. Math. J. 63 (2014) 77–89. [21] H. Knüpfer and C. B. Muratov, On an isoperimetric problem with a competing nonlocal term II: The general case. Comm. Pure Appl. Math. 67 (2014) 1974–1994. [22] H. Knüpfer, C. B. Muratov and M. Novaga, Low density phases in a uniformly charged liquid. Comm. Math. Phys. 345 (2016) 141–183. [23] J. Lu and F. Otto, Nonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsäcker model. Comm. Pure Appl. Math. 67 (2014) 1605–1617. [24] R. Neumayer, A strong form of the quantitative Wulff inequality. SIAM J. Math. Anal. 48 (2016) 1727–1772. [25] W. Reichel, Characterization of balls by Riesz-potentials. Ann. Mat. Pura Appl. 188 (2009) 235–245. COCV_2021__27_S1_A22_055c6460d-17e1-4548-8552-52739e24046c cocv190201 10.1051/cocv/202006610.1051/cocv/2020066 Cost for a controlled linear KdV equation Krieger Joachim 0000-0002-8404-5016 Xiang Shengquan * Bâtiment des Mathématiques, EPFL, Station 8, 1015 Lausanne, Switzerland. *Corresponding author: [e-mail]; [e-mail]. 01 03 2021 01 03 2021 2021 cocv/2021/01 Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science OuverteSupplementS21 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF) Full (DJVU)

The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [L. Rosier, ESAIM: COCV 2 (1997) 33-55.]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.

Korteweg-de Vries controllability cost observability 35Q53 34H05 idline ESAIM: COCV 27 (2021) S21 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S21 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020066 www.esaim-cocv.org COST FOR A CONTROLLED LINEAR KDV EQUATION Joachim Krieger and Shengquan Xiang* Abstract. The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [L. Rosier, ESAIM: COCV 2 (1997) 33-55.]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant. Mathematics Subject Classification. 35Q53, 34H05. Received November 22, 2019. Accepted October 3, 2020. 1. Introduction The goal of this paper is to give a quantitative cost estimate of the controlled system ut + ux + uxxx = 0, u(t, 0) = u(t, L) = 0, ux(t, L) = a(t). Theorem 1.1. Let L > 0. There exist effectively computable T0 = T0(L) > 0 and c = c(L) > 0 such that for any T T0 the solution u of ut = -ux - uxxx, u(t, 0) = u(t, L) = ux(t, L) = 0, u(0, x) = u0(x), satisfies Z T 0 |ux(t, 0)|2 dt cku0k2 L2(0,L), u0 L2 , if L / N; (1.1) Z T 0 |ux(t, 0)|2 dt cku0k2 L2(0,L), u0 H L2 , if L N. (1.2) Here H is the controllable subspace, and N is the so called critical length set that is given by N = ( 2 r k2 + kl + l2 3 ; k, l N ) . Keywords and phrases: Korteweg-de Vries, controllability, cost, observability. Batiment des Mathematiques, EPFL, Station 8, 1015 Lausanne, Switzerland. * Corresponding author: [e-mail]; [e-mail]. Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 J. KRIEGER AND S. XIANG Actually following Lions' H.U.M. [22] (see also [11]) the optimal estimate of the observability inequality (1.1) (or (1.2)) implies the exact controllability of the KdV equation with some optimal control a(t) L2 (0, T). Rosier [25] proved that such linear controlled system is exactly controllable if and only if L / N, therefore found the existence of such c that satisfying the observability inequality (1.1). Though not controllable in critical cases (i.e. L N), we can decompose L2 by H M, where the subspaces H and M are controllable and uncontrollable parts respectively, and Rosier's method also provides the existence of c in (1.2). From then on, based on this observability result, rich control theories have been developed on KdV model concerning linear and nonlinear controllability, stability and stabilization, while the value of the observability constant played a significant role in these studies. For example, in [24] this value is directly used to get expo- nential (energy) stability on L2 for non-critical cases and exponential stability on H for critical cases; later on it is proved successively in [5, 7, 12] that the nonlinear controlled KdV system, i.e. ut + ux + uxxx + uux = 0, is locally controllable with some a(t) = ux(t, L) despite M, where the cost can be estimated by the related observability in H; though the finite dimensional central manifold M makes the linear system not asymptoti- cally stable, it is shown in [10] that the nonlinear term as well as the exponential decay on H (which is exactly the observability inequality) lead to polynomial stability of the system; more recently, in [16] exponential sta- bilization is achieved by quadratic structure on M and of course the exponential decay on H (more precisely, the observability (1.2)). In [25] Rosier used a method due to Bardos-Lebeau-Rauch [2], and as we mentioned above this way only provided the existence of such constant, while the value of it remained open. Thus it is important and interesting to give an explicit observability estimate. Typical and classical ways of solving cost problems are moment methods [27], Lebeau-Robbiano strategy type methods [21], and Carleman estimates [19]. The first two consist in investigating the eigenfunctions and decomposing the states by them, see for example [9, 23]. However, in our case the related eigenfunctions do not form a Riesz basis, due to the fact that the operator is neither self-adjoint nor skew-adjoint. In fact they are not even complete in L2 (0, L), see [29], which prevents us from directly applying those methods. Due to the existence of the critical length set, it does not seem natural to consider Carleman estimates. In this paper, we introduce a constructive approach that quantifies the observability constant. We concentrate on the proof of (1.1) for non-critical cases, mainly presented in Section 3. Then we comment in Section 4 that almost the same proof leads to inequality (1.2) for critical cases. More precisely, inequality (1.1) can be achieved in two steps. Let us denote by S(t) the corresponding semi-group of the operator Au := -ux - uxxx, u(t, 0) = u(t, L) = ux(t, L) = 0. Proposition 1.2. Let K1 1, L / N. There exists = (L, K1) > 0 effectively computable such that the set B(K1), B = B(K1) := u H3 (0, L; C); kukL2 = 1, kukH3 K1, u(0) = u(L) = ux(L) = 0, |ux(0)| < , inf C ku - ux - uxxxkL2 < is empty. Proposition 1.3. There exist K1(L) and T0(L) such that for any > 0 there is = (L, ) > 0 effectively computable with the property that, if there are u L2 (0, L)\{0}, K1 K1(L), and T T0(L) satisfying Z T 0 S(t)u x (t, 0) 2 dt < kuk2 L2(0,L), (1.3) then B(K1) is not empty. In conjunction with the preceding propositions, this then implies that we can set c = L, (L, K1(L) in (1.1) for Theorem 1.1. COST FOR A CONTROLLED LINEAR KDV EQUATION 3 Remark 1.4. We only prove Propositions 1.2 and 1.3 for L 4, though the same way of the proof also applies to the other cases. In fact, when L is below 3 (which is small than the first critical length 2), an alternative simple proof in [6] gives an explicit observability constant, which, for the completeness of the paper, is also presented, see Appendix A. 2. Some properties of S(t) From now on we always assume that L 4. The goal of this section is to develop several properties concerning the smoothing effect of S(t). All the results stated here will be demonstrated, and all the constants will be explicitly characterized, in Appendix B. Due to some compatibility issues, we define the following Sobolev spaces Hk (0) satisfying natural compatibility conditions on the boundary, H0 (0)(0, L) := L2 (0, L); H1 (0)(0, L) := {f H1 , f(0) = f(L) = 0}; H2 (0)(0, L) := {f H2 , f(0) = f(L) = f0 (L) = 0}; H3 (0)(0, L) := {f H3 , f(0) = f(L) = f0 (L) = 0}; H4 (0)(0, L) := {f H4 H3 (0), (Af)(0) = (Af)(L) = 0}; H5 (0)(0, L) := {f H5 H3 (0), (Af)(0) = (Af)(L) = (Af)x(L) = 0}; H6 (0)(0, L) := {f H6 H3 (0), (Af)(0) = (Af)(L) = (Af)x(L) = 0}, with the same norm as Hk : kfk2 Hk(0,L) := Z L 0 |f(k) (x)|2 + |f(x)|2 dx. Lemma 2.1. There is a constant En m which only depends on n < m such that Z L 0 |f(n) (x)|2 dx En m m-n Z L 0 |f(m) (t)|2 dt + -n Z L 0 |f(t)|2 dt ! , (0, 1]. Now we are ready to prove the following properties concerning regularities of the flow S(t). Suppose that f0 L2 , f(t, x) = S(t)f0, then simple integration by parts yields Z T 0 Z L 0 f2 x(t, x) dx dt T + L 3 Z L 0 f2 0 (x) dx, (2.1) Z L 0 f2 (t, x) dx = Z L 0 f2 0 (x) dx - Z t 0 f2 x(s, 0) ds Z L 0 f2 0 (x) dx. (2.2) The preceding two inequalities tell us that starting from some L2 data the solution will stay in the same space, moreover, on almost every (time) t [0, T] the solution becomes H1 (0, L) thus gains regularity. Actually, similar regularity results hold for arbitrary order: Lemma 2.2. Let k {0, 1, 2, 3, 4, 5, 6}. If the initial data f0 belongs to Hk (0), then the flow S(t)f0 stays in C([0, T]; Hk (0)(0, L)) L2 (0, T; Hk+1 (0) (0, L)). Moreover, there exist constants Fk 0 (L) and Fk 1 (L) independent of 4 J. KRIEGER AND S. XIANG the choice of f Hk (0) and T (0, L] such that kS(t)f0kC([0,T ];Hk (0) (0,L)) Fk 0 kfkHk (0) (0,L), (2.3) kS(t)f0kL2(0,T ;Hk+1 (0) (0,L)) Fk 1 kfkHk (0) (0,L). (2.4) Remark 2.3. The same type of smoothing results also hold for the nonlinear KdV flow (e.g. [3]). But this phenomenon only appears for initial boundary value problems, it does not exist for KdV flow on whole space. An immediate consequence is the following smoothing effect result. Lemma 2.4. Let k {1, 2, 3, 4, 5, 6}. There exists a constant Fk s = Fk s (L) only depending on L such that kS(t)f0kHk (0) (0,L) Fk s tk/2 kf0kL2(0,L), t (0, T], T L, (2.5) kS(t)f0kHk (0) (0,L) Fk s Lk/2 kf0kL2(0,L), t [L, +). (2.6) Remark 2.5. The rate t-k/2 in Lemma 2.4 is optimal by assuming Lemma 2.2. Moreover, both of them can be generalized to k N, while more (but similar) efforts are required to get explicit values. For any given K > 0, let A = AK be A := {u H3 (0, L), u(0) = u(L) = 0, kukH3(0,L) K}. Then we have the following simple Lemma 2.6. There exist B = B(L, K) and a set {f1, f2, . . . , fB} H3 (0, L), such that for each f A, there is fj with kf - fjkL2(0,L) < 2 2 . An immediate consequence is the Corollary 2.7. Assume that {g1, g2, . . . , gP } A is orthonormal. Then P B(L, K). 3. Proofs of Proposition 1.2 and Proposition 1.3 This section is devoted to the proof of the two main propositions of this paper. In the following we will work with a parameter K much bigger than 8, which will eventually determine K1 = K1(L, K) = B 1 2 (L, K)K. For ease of notations, from now on, let k · k refers to the L2 -norm, and h·i refers to the L2 -inner product; {y1,...,yn} refers to {span{y1,...,yn}} ; f = O(a) if |f| |a|, g = OH(a) if kgkH |a|, etc. First, we prove Proposition 1.2, following the procedure in Rosier's proof. Observe that this proposition is ineffectively true, since if it's false, we can find a sequence n 0 as well as functions un with associated n COST FOR A CONTROLLED LINEAR KDV EQUATION 5 as in the definition of B with kunkH3 K1, and so in particular a subsequence will have n as well as un, un,x converge point wise, and also in H3 weak sense, to some u which is as in Rosier's result, and hence results in a contradiction. Rendering this effective will require `perturbing Rosier's proof'. Proof of Proposition 1.2. Assume that L / N and let u B as in the statement of that proposition, thus there exist and f(x) such that u(x) + u0 (x) + u000 (x) = f(x), x (0, L), (3.1) kukL2 = 1, kukH3 K1, u(0) = u(L) = ux(L) = 0, |u0 (0)| < , kfkL2 < . (3.2) At first we can get some information about from the above equation on u. In fact, we get from the preceding equation that ||kukL2 - 1 + q E1 3 kukH3 ||kukL2 - kAukL2 ku + u0 + u000 kL2 < , thus is bounded by || < + 1 + q E1 3 K1 < 1 + 1 + q E1 3 K1 =: K2. Moreover, direct integration by parts from equation (3.1) yields hu, ui = h-u0 - u000 + f, ui, which, combined with (3.2), leads to Re(hu0 , ui) = 0, Re(hu000 , ui) = 1 2 |u0 |2 (0). Therefore, is close to the imaginary axis, |Re()| 2. Then we derive further information on u from classical complex analysis. By extending u and f trivially past the endpoints of the interval [0, L], we obtain a function u H 3 2 - (R), which satisfies the relation u + u0 + u000 = f + u00 (0)0 + u0 (0)0 0 - u00 (L)L, x R. Then, the extended function u, via the Fourier(­Laplace) transformation, further satisfies b u() · + (i) + (i)3 = - e-iL + i + b f(), || < , | b f()| < L1/2 eL|im| , C, where := u00 (0), := u00 (L), := u0 (0). It is followed from Paley-Wiener theorem that b u() and b f() are holomorphic functions when extended on complex valued , as u(x) and f(x) are compactly supported. 6 J. KRIEGER AND S. XIANG We conclude that away from the zeroes of the polynomial + (i) + (i)3 , we have the representation b u() = i - e-iL + i + b f() p - + 3 , p = i. Then observe that (, ) 6= (0, 0) since otherwise we cannot possibly have the normalisation condition kukL2 = 1 (or kb ukL2 = 2), provided is small enough. In fact, if the function i i + b f() p - + 3 L2 (R), then the polynomial p - + 3 has to divide the numerator, in the sense that the quotient is an entire function as well. But since |p| = || < K2, the roots of this polynomial lie in a disc of radius R = R(K1) := 1+3K2/2 1/3 > 1 in the complex plane centered at the origin: ||3 = | - p| < K2 + || < K2 + 1 3 ||3 + 2 3 . Choose D2R(K1) and 3R = D3R(K1), we get from Cauchy's integral formula that |b u()| = i i + b f() (p - + 3) = 1 2i Z 3R i i + b f() (p - + 3)( - ) d . On the one hand, since | b f()| e3LR Z L 0 |f(x)| dx < e3LR L1/2 , D3R, |(p - + 3 )( - )| (26K2 + 52 3 )R = 52 3 R4 , D3R, D2R, we have |b u()| 1 2 Z 3R i (p - + 3)( - ) d + 1 2 Z 3R b f() (p - + 3)( - ) d , 27 52R2 + 9L1/2 e3LR 52R3 . (3.3) On the other hand, for D2R c we have |p - - 3 | 2 3 ||3 - 1 3 || 16 3 R3 - 2 3 R > 14 3 R3 , || |p - - 3| || 2 3 ||3-1 3 || < 12 7||2 , COST FOR A CONTROLLED LINEAR KDV EQUATION 7 thus |b u()| 12 7||2 + 3 14 b f() , D2R c , which, together with (3.3), yields Z R |b u()|2 d = Z [-2R,2R] |b u()|2 d + Z [-2R,2R]c |b u()|2 d, 4R 27 52R2 + 9L1/2 e3LR 52R3 2 2 + 24 49 2 + 9 98 2 , 4R 27 52R2 + 9L1/2 e3LR 52R3 2 + 57 98 ! 2 . This contradicts our assumption that kukL2 = 1, provided small enough: 4R 27 52R2 + 9L1/2 e3LR 52R3 2 + 57 98 ! 2 < 2. Furthermore, by a simple variation of the preceding argument, we infer the existence of (L, K1) > 0 such that || + || (L, K1) is forced by the normalisation condition on u. Indeed, based on the above estimates with (, ) = (0, 0), when (, ) 6= (0, 0), for D2R we have |b u()| = i - e-iL + i + b f() (p - + 3) = 1 2 Z 3R - e-iL + i + b f() (p - + 3)( - ) d , 27 52R2 + 9L1/2 e3LR 52R3 + 9 52R3 || + e3LR || , and for (D2R)c R we have |b u()| 12 7||2 + 3 14 b f() + 12(|| + ||) 7||3 , which imply that Z R |b u()|2 d = Z [-2R,2R] |b u()|2 d + Z [-2R,2R]c |b u()|2 d, 8R 27 52R2 + 9L1/2 e3LR 52R3 2 2 + 8R 9 52R3 || + e3LR || 2 , + 36 49 2 + 27 196 2 + 27(|| + ||)2 245 , 8 J. KRIEGER AND S. XIANG 8R 27 52R2 + 9L1/2 e3LR 52R3 2 + 171 196 ! 2 , + 81e6LR 338R5 + 27 245 (|| + ||)2 . Thus 81e6LR 169R5 + 54 245 (|| + ||)2 1, = 81e6LR 169R5 + 54 245 -1/2 , provided that 8R 27 52R2 + 9L1/2 e3LR 52R3 2 + 171 196 ! 2 2 - 1. Moreover, by shrinking 0(L, K1) if necessary this can be easily improved to ||/2 || 2|| and min{||, ||} 1 3 (L, K1). Since else we can arrange the numerator not to have any zeroes at all on or near the real axis, while as shown in the beginning that = iR + O(2), whence there exists at least one root of the denominator that is near the real axis for small . More precisely, we only need to show the former relation as it leads to the latter one, if which is not true, then either || < ||/2 with || 2/3, or || > 2|| with || 2/3. For the first case, the zeroes of the numerator that lie in DR satisfy e-iL = + i + b f(), (3.4) thus ||eLIm() |||e-iL | || 2 + R + L1/2 eLR , therefore, Im() 1 L log( 3 4 ), COST FOR A CONTROLLED LINEAR KDV EQUATION 9 provided that satisfies R + L1/2 eLR 6 . While for the latter case, - = -e-iL + i + b f(), || || 2 eLIm() + R + L1/2 eLR , therefore Im() 1 L log( 3 2 ), provided that satisfies R + L1/2 eLR 6 . Hence, the zero in DR, which exists as b u is holomorphic, should verify L|Im()| log(4 3 ). On the other hand, we turn to the zeroes of the denominator, which all lie in DR, p - + 3 = 0, where p R + iO(2), DR. Suppose that p is given by a + ib with some |a| < K2 and |b| < 2, we can find some 0 DR R as solution of a-0 +3 0 = 0. Therefore, there exists a solution = 0 +r of a+ib- +3 = 0 with |r| < 3, thus |Im()| < 3, which is in contradiction with L|Im()| log(4 3 ) if verifies 3L log(4 3 ). Consider then the numerator - e-iL + i + b f(), we can then assume that all the roots of - e-iL + i + b f() in DR are simple, and have to be of distance OK1,L() from the roots of - e-iL , which, thanks to the fact that ||/2 || 2||, are of the form µ0 + 2n L , with |Re(µ0)| L , |Im(µ0)| log 2 L , and n Z. (3.5) In fact, as - e-iL + i + b f() 0 = iLe-iL + i - i d (xf)(), if there exists some double solution in DR, then L 3 e-LR |iLe-iL | 1 + L3 3 1/2 eLR ! , 10 J. KRIEGER AND S. XIANG contradiction when 1 + L3 3 1/2 eLR ! < L 3 e-LR . Furthermore, if µ is such a zero of - e-iL that is in DR, we pick a circle r(µ) in the complex plane centred at µ of radius r (0, min{ 8L , R}) = (0, 8L ). We prove that under certain conditions, which will be chosen later on, there is only one solution of - e-iL + i + b f() that lies in the domain - L + Re(µ), L + Re(µ) × R, actually this solution is inside r(µ). At first for any r(µ) we have - e-iL 2 = ( - e-iL ) - ( - e-iLµ ) 2 = e-iLµ e-iLr - 1 2 , = ||2 (cos(Lra)eLrb - 1)2 + (sin(Lra)eLrb )2 , where r = - µ = r(a + ib) with a2 + b2 = 1. If |a| 1 8 , then (sin(Lra)eLrb )2 e-Lr Lr 16 2 Lr 48 2 . If |a| 1 8 , then |b| 7 8 . If further b < 0, then (cos(Lra)eLrb - 1)2 1 - e- 7Lr 8 2 Lr 48 2 . Else b > 0, thus (cos(Lra)eLrb - 1)2 1 2 Lr 2 . Therefore, - e-iL ||Lr 48 Lr 144 , r(µ), which yields - e-iL + i + b f() Lr 288 , r(µ), if R + L1/2 eLR Lr 288 . Moreover, under the above condition, there is no solution in - L + Re(µ), L + Re(µ) × R \ Dr(µ). Indeed, for any DR - L + Re(µ), L + Re(µ) × R, we estimate - e-iL by two situations: if |Im( - µ)| r/2, then - e-iL = || e-iL(-µ) - 1 3 |eLIm(-µ) - 1| Lr 48 , if |Re( - µ)| r/2 and |Im( - µ)| r/2, then - e-iL = || e-iL(-µ) - 1 ||eLIm(-µ) | sin(Re( - µ)L) Lr 24 . Next we prove that, shrinking the upper bound on if necessary, there is exactly one solution inside r(µ). As demonstrated before, there is no solution on r(µ), therefore the number of solutions (counting multiplicity) COST FOR A CONTROLLED LINEAR KDV EQUATION 11 inside which is given by 1 2i Z r(µ) - e-iL + i + b f() 0 - e-iL + i + b f() d = 1 2i Z r(µ) iLe-iL + i - i d (xf)() - e-iL + i + b f() d. As µ is the only solution of - e-iL = 0 inside r(µ), we also have 1 = 1 2i Z r(µ) - e-iL 0 - e-iL = 1 2i Z r(µ) iLe-iL - e-iL d. It suffices to find a sufficient condition such that Nr : = 1 2i Z r(µ) iLe-iL + i - i d (xf)() - e-iL + i + b f() d - 1 2i Z r(µ) iLe-iL - e-iL d , = 1 2i Z r(µ) - (iLe-iL )(i + b f()) ( - e-iL + i + b f())( - e-iL) + i - i d (xf)() - e-iL + i + b f() d , is strictly smaller than 1, since Nr takes value from integer. In fact, under the above conditions, thanks to the above known estimates, we have Nr 1 2 Z r(µ) (iLe-iL )(i + b f()) ( - e-iL + i + b f())( - e-iL) d + 1 2 Z r(µ) i - i d (xf)() - e-iL + i + b f() d, r (LeLR )(R + L1/2 eLR ) Lr 48 Lr 288 + 1 + L3 3 1/2 eLR Lr 288 , r · 288 · 48eLR (R + L1/2 eLR ) L + · 288 1 + L3 3 1/2 eLR L , 288 96eLR (R + L1/2 eLR ) Lr + 1 + L3 3 1/2 eLR L . Thus it suffices to let satisfy 288 96eLR (R + L1/2 eLR ) Lr + 1 + L3 3 1/2 eLR L < 1. We conclude that all the zeroes of the numerator - e-iL + i + b f() in DR are of the form µ0 + k 2 L + O(r), k Z, where µ0 satisfies |µ0| 2 L . Because all the zeroes of the denominator, which are in DR, should also be solutions of the numerator, amongst those solutions have to be the roots of the polynomial p - + 3 . Picking 0 suitably, 12 J. KRIEGER AND S. XIANG we may assume that the roots of this polynomial in DR are then of the form 0, 1 = 0 + k 2 L + 2O(r), 2 = 0 + (k + l) 2 L + 2O(r), where k, l are positive integers. Observe that necessarily we have |i| R, |(k + l) 2 L | 2R + 1/4. Then one infers the system 0 + 1 + 2 = 0, 01 + 02 + 12 = -1, 012 = -p, and the first two of these equations yield 30 + 2 L (2k + l) + 4O(r) = 0, 3 = ( 2 L )2 (k2 + kl + l2 ) + 2 L (28k + 10l)O(r) + 36O(r2 ). Thus L2 - 2 r k2 + l2 + kl 3 !2 56L2 (R + 1/8)r + 36L2 r2 56L2 (R + 1)r. In particular, if 56L2 (R + 1)r < mink,lN L2 - 2 q k2+l2+kl 3 2 , then we have B = . Let us remark here that the existence of such r satisfying the preceding condition is guaranteed by the selection of L. In conclusion, we can set = (L, K1) that satisfies, K2 = 1 + 1 + q E1 3 K1, R = 1 + 3K2/2 1/3 , = 81e6LR 169R5 + 54 245 -1/2 , r < 8L , 56L2 (R + 1)r < min k,lN L2 - 2 r k2 + l2 + kl 3 !2 , 8R 27 52R2 + 9L1/2 e3LR 52R3 2 + 171 196 ! 2 2 - 1, R + L1/2 eLR 6 , 1 + L3 3 1/2 eLR ! < L 3 e-LR , COST FOR A CONTROLLED LINEAR KDV EQUATION 13 R + L1/2 eLR Lr 288 , 288 96eLR (R + L1/2 eLR ) Lr + 1 + L3 3 1/2 eLR L < 1. Now we turn to the second proposition and begin with outlining the idea of the proof. Assume that there is a u, kukL2 = 1 as in Proposition 1.3 satisfying flux inequality (1.3). Heuristically, we shall now construct a finite dimensional vector space V H3 (0, L) of functions satisfying the desired boundary vanishing conditions and such that kV Af - Afk < kfk. (3.6) Moreover, this vector space admits an orthonormal basis in A, such that CV A|CV has a normalised (complex) eigenfunction with kukH3 B 1 2 (L, K)K =: K1, and which then implies Proposition 1.3. More precisely, thanks to Corollary 2.7, suppose that {g1, g2, ..., gp} with p B(L, K) is an orthonormal basis of V, (3.7) gj(0) = gj(L) = (gj)x(L) = 0, |(gj)x(0)| < p B(L, K) , (3.8) kgjk = 1, kgjkH3 K, (3.9) Agj V, j {1, 2, ..., p - 1}, (3.10) kAgp - V Agpk < , (3.11) then the vector space V and the complex vector space CV satisfies kV Af - Afk < kfk, f V, kCV Af - Afk < kfk, f CV, CV A : CV - CV. As CV is of finite dimension, the map CV A admits at least one eigenvalue: g := p X j=1 ajgj CV, CV Ag = g, p X j=1 |aj|2 = 1, which further satisfies, kg - Agk = kCV Ag - Agk < kgk = , kgkH3 p X j=1 |aj|kgjkH3 B 1 2 (L, K)K = K1, g(0) = g(L) = gx(L) = 0, |gx(0)| p X j=1 |aj||(gj)x(0)| < , therefore g B. 14 J. KRIEGER AND S. XIANG Keeping in mind the above essential observation, in the following complete proof we will only need to construct orthonormal functions {gj} verifying conditions (3.7)­(3.11). Proof of Proposition 1.3. Step 0: In the first part, we present some basic properties of the flow, while some of them are based on the "smallness" of the flux. The remaining parts of the proof are basically repeating these key observations. Observation (i). Set K0 = K0(L) by 2F3 s /K0 2/3 = 1, define K1(L) := K1(L, K0), and pick t1 = t1(K) := 2F3 s /K 2/3 (0, 1) for K K0. From now on we will work on K K0, which actually will give us a result slightly stronger than Proposition 1.3. As a consequence, Proposition 1.3 will be concluded by selecting K = K0. Thanks to Lemmas 2.4 and 2.1, the flow S(t) satisfies, for t [t1, +), |S(t)fkH3([0,L]) F3 s t 3/2 1 kfk K 2 kfk, |S(t)fkH6([0,L]) F6 s t3 1 kfk K2 F6 s 4(F3 s )2 kfk, |AS(t)fkL2(0,L) K(1 + p E1 3 )kfk 2 < e Kkfk, |AS(t)fkH3(0,L) K2 F6 s (1 + p E4 6 )kfk 4(F3 s )2 < e Kkfk, |A2 S(t)fkL2(0,L) K2 F6 s (1 + 2 p E4 6 ) 4(F3 s )2 + K p E2 3 2 ! kfk =: e Kkfk. Observation (ii). Another important observation is that the L2 norm of the flow stays close to its initial value, thanks to (2.2): kf(t)k = kf0k + O | R T 0 f2 x(t, 0) dt| kf0k , t [0, T]. For instance, if kf0k = 1 and the flux | R T 0 f2 x(t, 0) dt| < a, then the energy of the flow stays close to 1: kS(t)fk [1 - a, 1]. Observation (iii). Owning to the strong regularity of S(t)u, we are able to estimate S(s)u-S(t)u. More precisely, for any (0, 1/2) and any t [t1, +), direct calculation implies, S(t + )f - S(t)f = Z t+ t S0 (s)fds = S0 (t)f + Z t+ t Z s t S00 (r) drds, thus S(t + )f - S(t)f = AS(t)f + e KOL2 ()kfk, t [t1, +). (3.12) COST FOR A CONTROLLED LINEAR KDV EQUATION 15 Observation (iv). Thanks to the relation (3.12), we can estimate the flux of Af(t). Assuming | R T 0 f2 x(t, 0) dt| < a < 1/2 and (0, t1), we have that, for any t [t1, T - t1 - ], Z T -t-t1 0 S(s) AS(t)f 2 x (0) ds, Z T -t- 0 S(s) AS(t)f 2 x (0) ds, = Z T -t- 0 S(s) S(t + )f - S(t)f - e KOL2 ()kfk 2 x (0) ds, = Z T -t- 0 1 S(s) S(t + )f - 1 S(s) S(t)f - S(s) e KOL2 ()kfk 2 x (0) ds, 3 Z T -t- 0 1 S(s) S(t + )f 2 x (0) + 1 S(s) S(t)f 2 x (0) + S(s) e KOL2 ()kfk 2 x (0) ds, 3 e K2 2 kfk2 + 6 2 Z T t S(t0 )f 2 x (0) dt0 , 3 e K2 2 kfk2 + 6a 2 . Observation (v). Observe that Af(t) and f(t) are orthogonal, provided the null flux, i.e. R T 0 f2 x(t, 0) dt = 0, hAf(t), f(t)i = h-fx - fxxx, fi = - f2 x(t, 0) 2 , hAf(t), f(t)i = h d dt f(t), f(t)i = 1 2 d dt kf(t)k2 = 0. Thus it is natural to have a perturbed version, Af(t) and f(t) are "almost" orthogonal when the flux is small. Suppose that | R T 0 f2 x(t, 0) dt| < a, then for any t [t1, T - ], any (0, 1/2), we have hS(t)f, AS(t)fi, = S(t)f, S(t + )f - S(t)f - e KOL2 ()kfk , = e KO()kfk2 + S(t)f, S(t + )f - S(t)f , = e KO()kfk2 + 1 2 S(t)f - S(t + )f, S(t + )f - S(t)f + S(t)f + S(t + )f, S(t + )f - S(t)f , = e KO()kfk2 + O() 2 AS(t)f + e KOL2 ()kfk 2 + 1 2 S(t + )f 2 - S(t)f 2 , = e KO()kfk2 + O() e K2 + e K2 2 kfk2 + O(a) 2 , = O 4 e K2 kfk2 + a 2 which is small provided that a 1. 16 J. KRIEGER AND S. XIANG Observation (vi). If two small flux flows are orthogonal at the beginning, then they are "almost" orthogonal along the flow. Indeed, suppose that for some a < 1/2 we have Z T 0 S(t)f 2 x (0) dt < a, Z T 0 S(t)g 2 x (0) dt < a, then direct integration by parts shows hS(t)f, S(t)gi - hf, gi = Z t 0 d dt hS(s)f, S(s)gi ds, = Z t 0 hAS(s)f, S(s)gi + hS(s)f, AS(s)gi ds, = - Z t 0 S(s)f x (0) S(s)g x (0)ds, = O(a), t [0, T]. Observation (vii). Let V be a subspace of L2 . Then k S(t)V S(t)fk k V fk. Indeed, there exists a f1 V such that f = f1 + g, g = V f, in view of the linearity of the flow, we have S(t)f = S(t)f1 + S(t)g. Because S(t)f1 S(t)V , the projection satisfies, k S(t)V S(t)fk kS(t)gk kgk = k V fk. Remark 3.1. If the flux small condition is replaced by the null flux condition, then all these observations become even better. From now on we will present the constructive approach to find orthonormal functions {gj} verifying condi- tions (3.7)­(3.11), following the commutative diagram in Figure 1. Step 1: In particular, we have that u and g11 := S(t1)u satisfy kg11kH3 K 2 , kg11kL2([0,L]) = 1 + O(), S(t1 + )u - S(t1)u = Ag11 + e KOL2 (), (0, 1/2), (3.13) the second inequality on account of the flux assumption on u, and the third is small for some very small 1, which, however, is much larger than . COST FOR A CONTROLLED LINEAR KDV EQUATION 17 Figure 1. The constructive approach. If kAg11k < 2 , then stop. We further define g11(s) := S(t1 + s)u = S(s)g11, s [0, t1], which satisfies kg11(s)kH3 K 2 , kg11(s)kL2([0,L]) = 1 + O(), kAg11(s)k = kS(s)Ag11k kAg11k < 2 , Z t1 0 (g11(s)) 2 x (0)ds = Z 2t1 t1 (S(t)u)2 x(0) dt Z T 0 (S(t)u)2 x(0) dt , hence there exists s such that |g11(s)x(0)| (/t1)1/2 . Observe that if we set y11 := g11(s) kg11(s)k , V := span{y11}, then ky11k = 1, ky11kH3 K, y11(0) = y11(L) = (y11)x(L) = 0, |(y11)x(0)| < p B(L, K) , kAy11 - V Ay11k kAy11k < , V = span{y11}, if < 1/18 satisfies 3 2 r t1 < p B(L, K) . As a consequence, conditions (3.7)­(3.11) hold with p = 1, g1 = y11, which, as it is shown at the beginning, implies that B 6= . If on the other hand we have kAg11k 2 , 18 J. KRIEGER AND S. XIANG then proceed to the next step. Step 2: Now we have kAg11k 2 . For ease of notations, we define the following L2 ­normalization operator: L : f - f kfk , f 6= 0. It can be easily checked that L verifies LS(t) = LS(t)L, LA = LAL. Set y21 = LAy11 = LAg11. First we recall some properties of y11 and y21, thanks to the observations in Step 0: ky11k = ky21k = 1, 1 - kg11k 1, kAg11k 2 , |hg11, Ag11i| 4 e K2 + 2 , (0, 1/2), |hy11, y21i| 4 e K2 + 2 4 , (0, 1/2), (3.14) Z T -t1 0 S(s)y11 2 x (0) ds = 1 kg11k2 Z T t1 S(s)u 2 x (0) ds 2. (3.15) Despite that the normalized function y21 may have a poor H3 -bound, its boundary trace at x = 0 is small in the sense that, (0, t1), Z T -2t1 0 S(s)Ag11 2 x (0) ds = Z T -2t1 0 S(s) AS(t)u 2 x (0) ds 32 e K2 + 6 2 , Z T -2t1 0 S(s)y21 2 x (0) ds 1 kAg11k2 Z T -2t1 0 S(s)Ag11 2 x (0) ds 32 e K2 + 6 2 4 2 , (3.16) having taken advantage of observation (iv). For the sake of simplicity, we define an upper bound for (3.14), (3.15) and (3.16): C1 = C1(, 1, , e K) := 2 1 e K2 + 2 1 24 2 , 1 (0, min{ 1 2 , t1}), which can be sufficiently small for well chosen and 1. To make it clear, C1 is an upper bound for Z T -2t1 0 S(s)y11 2 x (0) ds, Z T -2t1 0 S(s)y21 2 x (0) ds, |hy11, y21i|. From the above inequality, we derive the existence of t1 [t1, 2t1] such that |(S(t1)y11)x(0)|, |(S(t1)y21)x(0)| (2C1/t1)1/2 . Set g12 := S(t1)y11, g22 := S(t1)y21. COST FOR A CONTROLLED LINEAR KDV EQUATION 19 They share similar properties as y11 and y21, thanks to the observations and the flux condition: kg12k2 , kg22k2 [1 - C1, 1], kg12kH3 , kg22kH3 K 2 ; |(g12)x(0)|, |(g22)x(0)| 2C1 t1 1/2 . Z T -4t1 0 S(s)g12 2 x (0) ds C1, Z T -4t1 0 S(s)g22 2 x (0) ds C1, |hg12, g22i| 2C1, Ag12 = AS(t1)Lg11 span{g22} Finally we can set y12 := Lg12, y22 := L y12 g22, y 32 := y12 Ay22, y32 := Ly 32. Notice that y22 is obtained from the Gram-Schmidt procedure, thus Ay12 span{g12, g22} = span{y12, y22}. It can be proved that {y12, y22, y32} share similar properties as {y11, y21}: "small" flux. It is easy to get for y12: Z T -4t1 0 S(s)y12 2 x (0) ds 2C1. As for y22, it can be written in the form of a "prepared" flow: y22 = 1 k y12 g22k g22 - g12 kg12k2 |hg12, g22i| , = 1 k y12 g22k S(t1) y21 - |hg12, g22i| |g12k2 y11 , =: S(t1)z22, for which we can successively get, k y12 g22k2 = g22 - g12 kg12k2 |hg12, g22i| 2 [1 - 5C1, 1], kz22k 1 + C1 (1 - C1) 1 - 5C1 2, Z T -2t1 0 S(s)z22 2 x (0) ds 8C1 1 - 5C1 16C1, Z T -5t1 0 S(s)AS(t1)z22 2 x (0) ds 12 e K2 2 + 96C1 2 , (0, t1). 20 J. KRIEGER AND S. XIANG Thanks to the above inequalities on z22, we can further get ky22kH3 k y12 g22kH3 k y12 g22k 1 1 - 5C1 · K 2 (1 + 2C1 1 - C1 ) K 1 + 4C1 2 1 - 5C1 3K 4 , kAy22kL2 e K 1 - 5C1 2 e K, Z T -4t1 0 S(s)y22 2 x (0) ds 8C1 1 - 5C1 16C1, if C1 1/18. About y 32 which is related to Ay22, we know from its definition that, y 32 = Ay22 - y12 Ay22 = Ay22 - ly12 with |l| 2 e K, thus the inner products are hy 32, y12i = 0, hy 32, y22i = hAy22, y22i = hAS(t1)z22, S(t1)z22i 16 e K2 + 8C1 , (0, 1 2 ). Moreover, the flux of y 32 can be estimated by Z T -5t1 0 S(s)y 32 2 x (0) ds = Z T -5t1 0 S(s)Ay22 - lS(s)y12 2 x (0) ds, 2 Z T -5t1 0 S(s)Ay22 2 x (0) ds + 2l2 Z T -5t1 0 (S(s)y12)2 x(0) ds, 242 e K2 + 192C1 2 + 16 e K2 C1, (0, t1). If ky 32k < 2 , then stop, and we verify that {y12, y22} satisfy conditions (3.7)­(3.11), thus B is not empty, provided that (2C1/t1)1/2 < / p B(L, K). If ky 32k 2 , then we define C2 = C2(C1, 2) = C2(, 1, 2, , e K) by C2 := 2 2 242 2 e K2 + 192C1 2 2 + 16 e K2 C1 , 2 (0, min{ 1 2 , t1}), which is an upper bound for Z T -5t1 0 S(s)yi2 2 x (0) ds i {1, 2, 3}, and |hy32, y22i|. COST FOR A CONTROLLED LINEAR KDV EQUATION 21 Now we proceed to the next step. Step 3: Here we assume ky 32k 2 , and set y32 := L y12 Ay22. Then proceeding as before, we can obtain boundary trace inequality. Observe that the projection onto y 12 introduces a flux term of size at most O() due to our earlier boundary flux estimation for y12. Since we have lost regularity for y32, we regain this by applying the flow S(t2) again, for some t2 [t1, 21], resulting in g13 = S(t2)y12, g23 = S(t2)y22, g33 = S(t2)y32. Then we apply the Gram-Schmidt procedure, by first orthogonalizing h13 = g13, h23 = h13 g23, h33 = {h13,h23}g33 = S(t2)z33, and then normalizing y13 = Lh13, y23 = Lh23, y33 = Lh33. As the demonstration in Step 2, we are able to estimate y13, y23, y33 and Ay33 in terms of some C3 which only depends on C2 and 3. Then if y 43 := {y13,y23}Ay33, ky 43k < 2 , we stop the process. Else we continue iteratively. We ignore detailed calculation in this step, as it will be covered by the next step for general cases. Step 4: General induction step. In this step we provide general iteration estimates. At first we prove the following lemma. Lemma 3.2. Let n 2. For any 0 < (n + 1)cn < min{1/18, 1/2n}, any Tn 3t1, any orthonormal functions {y1n, ..., ynn}, and any normal function y(n+1)n satisfying Ayin span{y1n, ..., ynn}, i {1, ..., n - 1}, (3.17) Aynn span{y1n, ..., y(n-1)n, y(n+1)n}, (3.18) hyin, y(n+1)ni = 0, i {1, ..., n - 1}, (3.19) |hynn, y(n+1)ni| cn, (3.20) Z Tn 0 S(s)yin 2 x (0) ds cn, i {1, ..., n + 1}, (3.21) we can find orthonormal functions {y1(n+1), ..., y(n+1)(n+1)} such that kyi(n+1)kH3 3K/4, |(yi(n+1))x(0)| 3 2 s (n + 1)cn t1 , (3.22) Z Tn-3t1 0 S(s)yi(n+1) 2 x (0) ds 2cn, i {1, ..., n + 1}, (3.23) Ayi(n+1) span{y1(n+1), ..., y(n+1)(n+1)}, i {1, ..., n}. (3.24) 22 J. KRIEGER AND S. XIANG Moreover, if we further project Ay(n+1)(n+1) on span{y1(n+1), ..., yn(n+1)}, y (n+2)(n+1) := y1,...,yn(n+1) Ay(n+1)(n+1), (3.25) then it satisfies, Z Tn-3t1 0 S(s)y (n+2)(n+1) 2 x (0) ds 2 2 cn+1, |hy(n+1)(n+1), y (n+2)(n+1)i| 2 2 cn+1, where cn+1 = cn+1 cn, n+1 is given by cn+1 := 2 2 (n + 1) 6 e K2 2 n+1 + 12cn 2 n+1 + 4 e K2 cn , n+1 (0, min{1/2, t1}). Proof. These functions are directly constructed via the Gram-Schmidt procedure. It follows from (3.21) that Z 2t1 t1 n+1 X i=1 (S(s)yin)2 x (0) ds (n + 1)cn, hence there exists tn [t1, 2t1] such that n+1 X i=1 (S(tn)yin)2 x(0) (n + 1)cn t1 . (3.26) For i {1, ..., n + 1}, we define gi(n+1) := S(tn)yin, (3.27) which, thanks to the boundary bound condition (3.26), the flux condition (3.21), Observation (i), and Observation (vi), satisfies 1 - cn kgi(n+1)k 1, kgi(n+1)kH3 K 2 , gi(n+1)(0) = gi(n+1)(L) = (gi(n+1))x(L) = 0, |(gi(n+1))x(0)| s (n + 1)cn t1 , Z Tn-2t1 0 S(s)gi(n+1) 2 x (0) ds cn, COST FOR A CONTROLLED LINEAR KDV EQUATION 23 hgi(n+1), gj(n+1)i cn, (j, i) 6= (n, n + 1), j < i, hgn(n+1), g(n+1)(n+1)i 2cn. Then we derive from (3.17)­(3.18) that Ag1(n+1), ..., Ag(n-1)(n+1) span{S(tn)y1n, ..., S(tn)ynn}, = span{g1(n+1), ..., gn(n+1)}, and that Agn(n+1) = S(tn)Aynn span{S(tn)y1n, ..., S(tn)y(n-1)n, S(tn)y(n+1)n}, = span{g1(n+1), ..., g(n-1)(n+1), g(n+1)(n+1)}, thus Agi(n+1) span{g1(n+1), ..., g(n+1)(n+1)}, i {1, ..., n}. Next, we orthogonalize {gi(n+1)} by {hi(n+1)}. More precisely, we find an upper triangular matrix An+1 = (aij)1i,jn+1 with aii = 1, such that the elements of (h1(n+1), h2(n+1), ..., h(n+1)(n+1)) := (g1(n+1), g2(n+1), ..., g(n+1)(n+1))An+1, are orthogonal. In such a case, the orthonormal functions {yi(n+1)} can be chosen by yi(n+1) := Lhi(n+1) = hi(n+1) khi(n+1)k . In the remaining part of the proof, we check that {yi(n+1)} verify the lemma. Now we need to fix the value of aij. From the construction of hi(n+1) we know that span{h1(n+1), ..., hi(n+1)} = span{g1(n+1), ..., gi(n+1)}, i {1, ..., n + 1}, which implies that Ahi(n+1), Agi(n+1) span{h1(n+1), ..., h(n+1)(n+1)}, i {1, ..., n}. (3.28) Moreover, by the definition of hi(n+1), hi(n+1) gj(n+1), 1 j < i n + 1, hence 0 = hhi(n+1), gj(n+1)i = i X k=1 akihgk(n+1), gj(n+1)i, 24 J. KRIEGER AND S. XIANG which implies |aji| cn X 1ki-1 |aki| + |hgi(n+1), gj(n+1)i|, thus X 1ki-1 |aki| icn 1 - (i - 1)cn (n + 1)cn 1 - ncn 2(n + 1)cn, 1 < i n + 1. Therefore, khi(n+1)k = k i X k=1 akigk(n+1)k [ 1 - cn - 2(n + 1)cn, 1 + 2(n + 1)cn] [1 - 3(n + 1)cn, 1 + 2(n + 1)cn], khi(n+1)k-1 [1 - 2(n + 1)cn, 1 + 4(n + 1)cn], 1 i n + 1. Many informations about the orthonormal basis {yi(n+1)}i{1,...,n+1} can be obtained from such explicit formulas. At first condition (3.24) is guaranteed by (3.28) and the definition of yi(n+1). Then, kyi(n+1)kH3 khi(n+1)k-1 khi(n+1)kH3 , (1 + 4(n + 1)cn) i X k=1 akikgk(n+1)kH3 , K 2 (1 + 4(n + 1)cn)(1 + 2(n + 1)cn), 3K 4 . Similarly, we have |(yi(n+1))x(0)| 3 2 s (n + 1)cn t1 , Z Tn-2t1 0 S(s)yi(n+1) 2 x (0) ds (1 + 4(n + 1)cn)2 Z Tn-2t1 0 S(s)hi(n+1) 2 x (0) ds, (1 + 4(n + 1)cn)2 Z Tn-2t1 0 i X k=1 aki S(s)gk(n+1) x 2 (0) ds, (1 + 4(n + 1)cn)2 Z Tn-2t1 0 i X k=1 p |aki| · p |aki| S(s)gk(n+1) x (0) 2 ds, (1 + 4(n + 1)cn)2 Z Tn-2t1 0 (1 + 2(n + 1)cn) i X k=1 |aki| S(s)gk(n+1) 2 x (0) ds, (1 + 4(n + 1)cn)2 (1 + 2(n + 1)cn)2 cn, 2cn. COST FOR A CONTROLLED LINEAR KDV EQUATION 25 It remains to estimate y(n+2)(n+1) := Ay(n+1)(n+1). Instead of dealing with Ay(n+1)(n+1) directly, we con- sider some z(n+1)(n+1) such that S(tn)z(n+1)(n+1) = y(n+1)(n+1), therefore Ay(n+1)(n+1) can be estimated from observations:(i), (iv) and (v). In fact, y(n+1)(n+1) = 1 kh(n+1)(n+1)k n+1 X k=1 ak(n+1)gk(n+1), = 1 kh(n+1)(n+1)k n+1 X k=1 ak(n+1)S(tn)ykn, = S(tn) 1 kh(n+1)(n+1)k n+1 X k=1 ak(n+1)ykn ! , =: S(tn)z(n+1)(n+1), while z(n+1)(n+1) satisfies kz(n+1)(n+1)k (1 + 4(n + 1)cn)(1 + 2(n + 1)cn) 2, Z Tn 0 S(s)z(n+1)(n+1) 2 x (0) ds (1 + 4(n + 1)cn)2 (1 + 2(n + 1)cn)2 cn 2cn. Thanks to Observation (i), we have kAy(n+1)(n+1)k = kAS(tn)z(n+1)(n+1)k 2 e K. Thus, y (n+2)(n+1) = y1,...,yn(n+1) Ay(n+1)(n+1), = Ay(n+1)(n+1) + n X k=1 bkyk(n+1), = AS(tn)z(n+1)(n+1) + n X k=1 bkyk(n+1), with k n X k=1 bkyk(n+1)k kAS(tn)z(n+1)(n+1)k 2 e K, which, together with the orthonormal property of yi(n+1), implies that n X k=1 b2 k 2 e K2 . Finally, we are able to get |hy(n+1)(n+1), y (n+2)(n+1)i| = |hy(n+1)(n+1), AS(tn)z(n+1)(n+1) + n X k=1 bkyk(n+1)i|, 26 J. KRIEGER AND S. XIANG = |hS(tn)z(n+1)(n+1), AS(tn)z(n+1)(n+1)i|, 4 e K2 kz(n+1)(n+1)k2 + cn , 8 e K2 + cn , (0, 1/2), and Z Tn-3t1 0 S(s)y (n+2)(n+1) 2 x (0) ds, = Z Tn-3t1 0 S(s)AS(tn)z(n+1)(n+1) x (0) + n X k=1 bk S(s)yk(n+1) x (0) !2 ds, (n + 1) Z Tn-3t1 0 S(s)AS(tn)z(n+1)(n+1) 2 x (0) + n X k=1 b2 k S(s)yk(n+1) 2 x (0) ! ds, (n + 1) 6 e K2 2 + 12cn 2 + 4 e K2 cn , (0, t1). Notice that 2 n+1 < n+1 for any n+1 < min{1/2, t1}, we get the last two inequalities of Lemma 3.2 and complete its proof. Let us define y(n+2)(n+1) := Ly (n+2)(n+1). Then it satisfies hyi(n+1), y(n+2)(n+1)i = 0, i {1, ..., n}, (3.29) |hy(n+1)(n+1), y(n+2)(n+1)i| 2 2 cn+1 ky (n+2)(n+1)k , (3.30) Z Tn-3t1 0 S(s)y(n+2)(n+1) 2 x (0) ds 2 2 cn+1 ky (n+2)(n+1)k2 , (3.31) where 0 < min{1/2, t1}. Suppose that 3 2 s (n + 1)cn t1 < p B(L, K) . (3.32) If ky (n+2)(n+1)k < 2 , then the orthonormal basis {yi(n+1)}1in+1 satisfies conditions (3.7)­(3.11), thus B 6= . If ky (n+2)(n+1)k 2 , then from Lemma 3.2 and inequalities (3.29)­(3.31) we derive that the orthonormal functions {yi(n+1)}1in+1 and normal function y(n+2)(n+1) satisfy Ayi(n+1) span{y1(n+1), ..., y(n+1)(n+1)}, i {1, ..., n}, Ay(n+1)(n+1) span{y1(n+1), ..., yn(n+1), y(n+2)(n+1)}, hyi(n+1), y(n+2)(n+1)i = 0, i {1, ..., n}, |hy(n+1)(n+1), y(n+2)(n+1)i| cn+1, COST FOR A CONTROLLED LINEAR KDV EQUATION 27 Z Tn-3t1 0 S(s)yi(n+1) 2 x (0) ds cn+1, i {1, ..., n + 2}, this closes the induction loop as the above conditions have the same form of conditions (3.17)­(3.21). Step 5: Find the parameters. Let T 3B(L, K) - 1 t1(K). We find 0 < 0 1 2 ... B(L,K)+1 min{1/2, t1}, (3.33) such that the increasing sequence {Cn}, C1 = 2 1 e K2 + 0 2 1 24 2 , C2 = 2 2 242 2 e K2 + 192C1 2 2 + 16 e K2 C1 , Cn+1 = 2 2 (n + 1) 6 e K2 2 n+1 + 12Cn 2 n+1 + 4 e K2 Cn , 2 n B(L, K), satisfies 3 2 s (n + 1)Cn t1 < p B(L, K) , 1 n B(L, K), (n + 1)Cn min{1/18, 1/2n}, 1 n B(L, K). As Cn is increasing, it suffices to let CB(L,K) 1 2B(L, K) and 3 2 s (B(L, K) + 1)CB(L,K) t1 < p B(L, K) . (3.34) Suppose that the preceding conditions are fulfilled, then clearly we have Cn 2 n+1. For ease of computation, we assume for that moment Cn 2 n+1 and define a sequence Dn which is larger than Cn: D0 = 0, D1 = 24 2 2 1 e K2 + 0 2 1 , D2 = 768 2 2 2 e K2 + D1 2 2 , Dn+1 = 48(n + 1) 2 e K2 2 n+1 + Dn 2 n+1 , n 2. It suffices to let 3 2 s (B(L, K) + 1)DB(L,K) t1 < p B(L, K) . (3.35) 28 J. KRIEGER AND S. XIANG We try to find n from backward. It is rather easy to fix a constant, as e DB(L,K), that verifies (3.35). Then we choose n+1 and e Dn iteratively by making e K2 2 n+1 and e Dn/2 n+1 equivalent, e K2 2 n+1 = e Dn 2 n+1 = 2 96(n + 1) e Dn+1, n 2, as well as several similar relations for n = 0 and 1. Therefore, we conclude that n+1 = e Dn e K2 !1 4 , n 0. e Dn+1 = 96(n + 1) e K 2 e Dn 1 2 , n 2; e D2 = 1536 e K 2 e D1 1 2 , e D1 = 48 e K 2 e D0 1 2 , which gives the values of e D0 = 0 = 0(L, , K): 0 = e DB(L,K) 2B(L,K) B(L,K)-1 Y k=2 96(k + 1) e K 2 !-2k+1 1536 e K 2 !-4 48 e K 2 !-2 , (3.36) as well as the values of n and e Dn that verifies all the above conditions, the details of which we omitted. To conclude the proof of Proposition 1.3, it suffices to take K0 = 2F3 s , K = K0, K1(L) = K1(L, K), T0(K, L) = 3B(L, K) - 1, and = (L, ) := 0(L, , K). We say that 0 is the value of for Propo- sition 1.3. Indeed, suppose that procedure does not stop for 1 n B(L, K), therefore, we have constructed orthonormal functions {yiB(L,K)+1}1iB(L,K)+1 A, which is in contradiction with Corollary 2.7. It means that the procedure has to stop at a certain step, i.e. there exists 1 m B(L, K) such that, we have found orthonormal functions {yim}1im and function y (m+1)m satisfying ky (m+1)mk < 2 , then {yim}1im verifies conditions (3.7)­(3.11). It means that B(K1) is not empty. 4. Length critical cases Our method also gives the value of observability constant for the case L N. The subspace H is called the controllable part, thanks to the observability inequality: Z T 0 | S(t)u x (0)|2 dt ckuk2 L2(0,L), u H. Furthermore, both H and M are S(t) invariant. Therefore, if we replace L2 space by (H, k · kL2 ), then the same results hold, which yields a value of c(L). Proposition 4.1. Let K1 1, L N. There exists = (L, K1) > 0 effectively computable such that the set B(K1), B := u H3 (0, L; C) CH; kukL2 = 1, kukH3 K1, u(0) = u(L) = ux(L) = 0, |ux(0)| < , inf C ku - ux - uxxxkL2 < COST FOR A CONTROLLED LINEAR KDV EQUATION 29 is empty. Proposition 4.2. There exist K1(L) and T0(L) such that for any > 0 there is = (L, ) > 0 effectively computable with the property that, if there are u H\{0}, K1 K1(L), and T T0(L) satisfying Z T 0 S(t)u x (t, 0) 2 dt < kuk2 L2(0,L), then B is not empty. Let us comment on Proposition 4.1. Define a set of eigenfunctions SL := {; u = Au, u M}, then following the proof of Proposition 1.2 we are able to give some computable and small such that, if u B then we can find some u1 and some SL verifying u1(0) = u1(L) = (u1)x(L) = (u1)x(0) = 0, ku1 - (u1)x - (u1)xxxkL2 < 1, where 1 = 1() can be sufficiently small if is. This can be considered as perturbation of M, thus contradicts the fact that u H by assuming small than a certain computable value. 5. Further comments and questions This is a quantitative way of characterizing observability constant, mainly based on flux observations and strong smoothing effects of the initial boundary value problem, e.g. Observations (i)­(vii) and Lemma 2.4 in our case. We believe that this method can be applied to many other models. Moreover, it is also of great interests to consider the following further questions. 5.1. Observability constant behavior around critical points Let L0 N. Our method gives a finite constant c(L0) > 0, while it also provides a vanishing sequence {c(L)}LL0 : c(L) 0+ . Apparently, this difference, e.g. the "jump" of the observability, comes from the uncontrollable subspace M. Thus when the length is not critical it is natural to ask for the existence of subspace, comparing to M, such that the observability constant of the quotient space is continuous. In other words, is that possible to find some finite dimensional space ML for L near L0 such that the related "observability" of L2 /ML, which is denoted by cHL (L), satisfies cHL (L) c(L0). 5.2. Optimal estimates According to the duality between controllability and observability, the sharp observability inequality constant is also the optimal control cost. This optimal estimate is of both mathematical and engineering interests, as stated in Introduction, it is the fundamental result for many other studies upon this model. However, it does not seem that the value that we obtain in this paper is optimal. Therefore, it would be interesting to further get sharp estimates of c(L). 5.3. Observability inequality for small time On account of to the smoothing procedure S(t1(L, K1)) and Lemma 2.6, our constructive approach and quantitative result only apply for large time, i.e. T bigger than some T0(L, K1). It is not clear whether some modifications and minimizations on our method can make the time small. What is the behavior of the constant when T tends to 0, and what is the sharp asymptotic estimate? Should the cost be like Ce- c T , as it is the case for many other models [9, 18]? 30 J. KRIEGER AND S. XIANG 5.4. Is backstepping another option? Originally introduced to stabilize system exponentially [15, 20], recently it is further developed as a tool for null control and small-time stabilization problems [14, 17, 28­31], the so called piecewise backstepping, which shares the advantage that the feedback (control) is well formulated. It consists in stabilizing system with arbitrary exponential decay rate (rapid stabilization) with explicit computable estimates, and splitting the time interval into infinite many parts such that on each part backstepping exponential stabilization is applied to make the energy divide at least by 2. Concerning our KdV case, at least for non-critical cases, rapid stabilization by backstepping is achieved in [13], where they used the controllability of KdV equation with control of the form b(t) = ux(t, 0) - ux(t, L) as an intermediate step. If it is possible to perform piecewise backstepping by obtaining Cec type estimates on each step, then we are able to get null controllability and small time stabilization with precise cost estimates. Appendix A. The L 4 case Let us consider the flow y(t) = S(t)y0. Integration by parts and (2.2) show T Z L 0 y2 0(x) dx Z T 0 Z L 0 y2 (t, x) dx dt + T Z T 0 y2 x(t, 0) dt, then Poincare's inequality leads to T Z L 0 y2 0(x) dx L2 2 Z T 0 Z L 0 y2 x(t, x) dx dt + T Z T 0 y2 x(t, 0) dt, which combines with (2.1) yield T - L2 (T + L) 32 Z L 0 y2 0(x) dx T Z T 0 y2 x(t, 0) dt. Consequently, when T and L satisifes L3 3T 2 + L2 32 < 1, the observability constant can be 3T 2 3T 2-L3-T L2 . Appendix B. Sobolev estimates and some properties of the flow We start from giving some quantitative Sobolev embedding and interpolation estimates. In the literature these bounds are usually simply provided by some unknown constant C, for example Brezis [4] and Adams [1], though ideas of getting which are well illustrated. For any (0, L/3), (2L/3, L), there exists (, ) such that |f0 ()| = f() - f() - 3 L (|f()| + |f()|) . Therefore, x (0, L), |f0 (x)| = f0 () + Z x f00 (t) dt 3 L (|f()| + |f()|) + Z L 0 |f00 (t)| dt, COST FOR A CONTROLLED LINEAR KDV EQUATION 31 then we integrate on (0, L/3) and on (2L/3, L) to get |f0 (x)| 9 L2 Z L 0 |f(t)| dt + Z L 0 |f00 (t)| dt. Hence, Z L 0 |f0 (x)|2 dx 162 L2 Z L 0 |f(t)|2 dt + 2L2 Z L 0 |f00 (t)|2 dt. (B.1) Because for any (0, L2 ] there exists n N such that L/n [1/2 /2, 1/2 ], we can split [0, L] by n parts. By performing (B.1) on each part and combining them together, we get Z L 0 |f0 (x)|2 dx 2 Z L 0 |f00 (t)|2 dt + 648 Z L 0 |f(t)|2 dt, (0, 16], thus Z L 0 |f0 (x)|2 dx 42 Z L 0 |f00 (t)|2 dt + 1 Z L 0 |f(t)|2 dt ! , (0, 1]. (B.2) Notice that (B.2) also holds for complex valued functions. By replacing f by f(n) , we also get Z L 0 |f(n+1) (x)|2 dx 42 Z L 0 |f(n+2) (t)|2 dt + 1 Z L 0 |f(n) (t)|2 dt ! , (0, 1]. (B.3) Moreover, we are able to find a constant En m which only depends on m and n such that Z L 0 |f(n) (x)|2 dx En m m-n Z L 0 |f(m) (t)|2 dt + -n Z L 0 |f(t)|2 dt ! , (0, 1], (B.4) while, more precisely, En m can be calculated by E1 2 = 42, (B.5) Em m+1 = 2m 42m (Em-1 m )m , (B.6) Ek-1 m = Ek-1 k (Ek m + 1). (B.7) For ease of notations, we denote an := Z L 0 |f(n) (x)|2 dx = kfk2 Hn and kfk2 Hn = kfk2 Hn +kfk2 L2 . In fact, E1 2 = 42 as shown in (B.2), further estimated are obtained from a reduction procedure on m. Suppose that En i with i m are known, then from (B.3) and (B.4) we derive am-1 Em-1 m 1am + -(m-1) 1 a0 , 1 (0, 1], am E1 2 am+1 + -1 am-1 , (0, 1]. 32 J. KRIEGER AND S. XIANG By taking 1 := /(2E1 2 Em-1 m ), we obtain am 2E1 2 am+1 + 2m-1 (E1 2 )m-1 (Em-1 m )m -m a0 , which concludes (B.6). As for (B.7), we perform (B.3) and (B.4) once again to get, for k m, ak-1 Ek-1 k ak + -(k-1) a0 , Ek-1 k Ek m+1(m+1-k am+1 + -k a0) + -(k-1) a0 , Ek-1 k (Ek m+1 + 1) m+2-k am+1 + -(k-1) a0 . By taking := (a0/(a0 + am))1/m in (B.3), we get an En m ( a0 a0 + am ) m-n m am + ( a0 + am a0 ) n m a0 , 2En ma m-n m 0 (a0 + am) n m . This implies that kfk2 Hn 2En mkfk 2(m-n) m L2 kfk 2n m Hm , thus kfk2 Hn (2En m + 1)kfk 2(m-n) m L2 kfk 2n m Hm , 0 < n < m. Now we turn to the proof of Lemma 2.6 and Corollary 2.7. Actually, assuming Lemma 2.6, for any gi there exists fni such that kgi - fni kL2 < 2 2 . Suppose that P > R, then there exists i 6= j such that fni = fnj , contradiction, as 2 = kgi - gjkL2 kgi - fni kL2 + kgj - fni kL2 < 2. It remains to prove Lemma 2.6 which is of course a direct consequence of Rellich's theorem. In fact, as the injection H3 , L2 is compact, it suffices to find a finite open cover composed by the union of balls with radius 2/2. By this way, fi can be chosen in A. However, one does not know the exact value of covering balls. Instead, we present a constructive proof, which explicitly characterizes the value of B. Notice that if f A then f satisfies, f H and f(0) = f(L) = 0, which means f = X nN an sin nx L in H1 , COST FOR A CONTROLLED LINEAR KDV EQUATION 33 with its H1 norm given by kfk2 H1 = X nN a2 n n2 2 2L , kfk2 L2 = X nN a2 n L 2 . Thanks to (B.4) and the definition of A, kfk2 H1 E1 3 Z L 0 kf(3) (x)k2 + kf(x)k2 dx ! = E1 3 kfk2 H3 , then X nN a2 n n2 2 2L E1 3 K2 , (B.8) thus an K p 2LE1 3 n . Next, we pick up all the functions of the following form, which are denoted by {fm}, fm = Nc-1 X n=1 am n sin nx L , |am n | ( 0, K p 2LE1 3 n · 1 Mc , K p 2LE1 3 n · 2 Mc , ..., K p 2LE1 3 n · Mc Mc ) , where Nc and Mc are some integers only depend on L to be chosen later on. It can be proved that with a good choice of Nc and Mc the above sequence {fm} satisfies Lemma 2.6. Clearly, fm C H3 ([0, L]). Let f A. On the one hand, thanks to the above construction, there exists a function fm such that |am n - an| < K p 2LE1 3 n · 1 2Mc , n {1, 2, ..., Nc - 1}. Hence Nc-1 X n=1 (am n - an)2 · L 2 < L 2 Nc-1 X n=1 LE1 3 K2 2M2 c n22 L2 4 · 1 6 · E1 3 K2 M2 c . On the other hand, we know from (B.8) that X nNc (am n - an)2 · L 2 E1 3 K2 L2 N2 c . 34 J. KRIEGER AND S. XIANG Therefore, we can choose Mc and Nc as Mc = Mc(L) = l KL r E1 3 6 m , Nc = Nc(L) = l 2KL q E1 3 m , which yields kfm - fk2 L2 = X nN (an - am n )2 L 2 < 1 2 . In such a case, the value of B(L, K) is given by (2Mc + 1)Nc-1 . Proof of Lemma 2.2. (i). Case k = 0. It is a straightforward consequence of (2.1)­(2.2) that F0 0 = 1, F0 1 = p 5L/3. (ii). Case k = 3. Suppose that f0 L2 , then as f satisfies ft(t, x) = Af(t, x), t (0, T), x (0, L) f(t, 0) = f(t, L) = fx(t, L) = 0, t (0, T) f(0, x) = f0(x), x (0, L), we know that g := ft = Af is the solution of gt(t, x) = Ag(t, x), t (0, T), x (0, L) g(t, 0) = g(t, L) = gx(t, L) = 0, t (0, T) g(0, x) = g0(x) := (Af0)(x), x (0, L). Since kuxk2 L2 E1 3 (2 kuxxxk2 L2 + -1 kuk2 L2 ), we have kuxkL2 q E1 3 (kuxxxkL2 + -1/2 kukL2 ). Therefore, by choosing := 1/ p 4E1 3 we get kuxkL2 1 2 kuxxxkL2 + (4E1 3 )3/4 kukL2 , which implies kuxxxkL2 kAukL2 + kuxkL2 , kAukL2 + 1 2 kuxxxkL2 + (4E1 3 )3/4 kukL2 , 2kAukL2 + 2(4E1 3 )3/4 kukL2 COST FOR A CONTROLLED LINEAR KDV EQUATION 35 and kAukL2 kuxxxkL2 + kuxkL2 (1 + q E1 3 )kukH3 . (B.9) Thanks to the result in the case k = 0, we have kfxkL2(0,T ;L2) r 2L 3 kf0kL2 , kf(t)kL2 kf0kL2 , and, by replacing f by g, Z T 0 Z L 0 g2 x(t, x) dx dt 2L 3 Z L 0 g2 0(x) dx, Z L 0 g2 (t, x) dx Z L 0 g2 0(x) dx, which implies kAf(t)kL2 kAf0kL2 (1 + q E1 3 )kf0kH3 , kA(fx)kL2(0,T ;L2) = k(Af)xkL2(0,T ;L2) r 2L 3 (1 + q E1 3 )kf0kH3 . Hence, kfxxx(t)kL2 2(1 + q E1 3 )kf0kH3 + 2(4E1 3 )3/4 kf(t)kL2 , 2(1 + q E1 3 )kf0kH3 + 2(4E1 3 )3/4 kf0kL2 , kfx(t)kL2 1 2 kfxxx(t)kL2 + (4E1 3 )3/4 kf(t)kL2 , (1 + q E1 3 )kf0kH3 + 2(4E1 3 )3/4 kf0kL2 , kfxxxxkL2(0,T ;L2) 2kA(fx)kL2(0,T ;L2) + 2(4E1 3 )3/4 kfxkL2(0,T ;L2), 2 r 2L 3 (1 + q E1 3 )kf0kH3 + 2 r 2L 3 (4E1 3 )3/4 kf0kL2 , thus kS(t)f0kC([0,T ];H3 (0) (0,L)) 2(1 + q E1 3 ) + 2(4E1 3 )3/4 + 1 kf0kH3 (0) (0,L), kS(t)f0kL2(0,T ;H4 (0) (0,L)) 2 r 2L 3 (1 + q E1 3 ) + 2 r 2L 3 (4E1 3 )3/4 + L ! kfkH3 (0) , which gives the value of F3 i : F3 0 = 2(1 + q E1 3 ) + 2(4E1 3 )3/4 + 1, (B.10) 36 J. KRIEGER AND S. XIANG F3 1 = 2 r 2L 3 (1 + q E1 3 ) + 2 r 2L 3 (4E1 3 )3/4 + L. (B.11) (iii). Case k = 6. Suppose that f0 H6 (0)(0, L), then g := ft = Af satisfies gt(t, x) = Ag(t, x), t (0, T), x (0, L) g(t, 0) = g(t, L) = gx(t, L) = 0, t (0, T) g(0, x) = g0(x) := (Af0)(x), x (0, L), and h := gt = Ag = A2 f satisfies ht(t, x) = Ah(t, x), t (0, T), x (0, L) h(t, 0) = h(t, L) = hx(t, L) = 0, t (0, T) h(0, x) = h0(x) := (Af0)(x) = (A2 f0)(x), x (0, L). Simple embedding estimate shows ku(4) kL2 1 4 ku(6) kL2 + 16(E4 6 )3/2 kukL2 , ku(4) kL2 q E4 6 kukH6 , ku(2) kL2 1 4 ku(6) kL2 + 2(E2 6 )3/4 kukL2 , ku(2) kL2 q E2 6 kukH6 . It is known from the case k = 0 that kf(6) (t) + 2f(4) (t) + f(2) (t)kL2 = kh(t)kL2 kh0kL2 , kf(7) + 2f(5) + f(3) kL2(0,T ;L2) = khxkL2(0,T ;L2) r 2L 3 kh0kL2 , which, combined with the preceding embedding estimates, yields kf(6) (t)kL2 4kh0kL2 + 128(E4 6 )3/2 kfkL2 + 8(E2 6 )3/4 kfkL2 , 8 q E4 6 + 4 q E2 6 + 128(E4 6 )3/2 + 8(E2 6 )3/4 kf0kH6 , and kf(7) kL2(0,T ;L2) 4khxkL2(0,T ;L2) + 128(E4 6 )3/2 kfxkL2(0,T ;L2) + 8(E2 6 )3/4 kfxkL2(0,T ;L2), 4 r 2L 3 kh0kL2 + r 2L 3 128(E4 6 )3/2 + 8(E2 6 )3/4 kf0kL2 , 8 r 2L 3 (E4 6 )1/2 + 4 r 2L 3 (E2 6 )1/2 + 128 r 2L 3 (E4 6 )3/2 + 8 r 2L 3 (E2 6 )3/4 ! . Thus, the value of F6 i can be chosen as F6 0 = 8 q E4 6 + 4 q E2 6 + 128(E4 6 )3/2 + 8(E2 6 )3/4 + 1, COST FOR A CONTROLLED LINEAR KDV EQUATION 37 F6 1 = 8 r 2L 3 (E4 6 )1/2 + 4 r 2L 3 (E2 6 )1/2 + 128 r 2L 3 (E4 6 )3/2 + 8 r 2L 3 (E2 6 )3/4 + L. (iii). Case k = 1, 2, 4, 5. It can be achived by the (real) interpolation of Sobolev spaces. To avoiding getting too much involved into this classical theory, we directly use some quantitative results in [8], and following several related notations there. kuk2 Hm(R) := X m m k uk2 L2(R), kuk2 Hm(0,L) := X m m k uk2 L2(0,L), the interpolation spaces as well as their norms are given by K-method, H1 = H0 (0, L), H3 (0, L) 1 3 , H2 = H0 (0, L), H3 (0, L) 2 3 , H4 = H3 (0, L), H6 (0, L) 1 3 , H5 = H3 (0, L), H6 (0, L) 2 3 , L2H2 := L2 (0, T; H1 (0, L)), L2 (0, T; H4 (0, L) 1 3 , L2H3 := L2 (0, T; H1 (0, L)), L2 (0, T; H4 (0, L) 2 3 , L2H5 := L2 (0, T; H4 (0, L)), L2 (0, T; H7 (0, L) 1 3 , L2H6 := L2 (0, T; H4 (0, L)), L2 (0, T; H7 (0, L) 2 3 . Then we have the following lemma concerning these interpolation spaces. Lemma B.1. (I) There exists an extension E and constants m = m(L) such that E : Hm (0, L) Hm (R), kukHm(0,L) kEukHm(R) mkukHm(0,L), m {0, 1, 2, 3, 4, 5, 6, 7}. (II) The norms Hm (0, L) and Hm are equivalent: (0 )- 2 3 (3 )- 1 3 kukH1(0,L) kukH1 kukH1(0,L), (B.12) moreover, (1 )- 2 3 (4 )- 1 3 kukL2(0,T ;H2(0,L)) kukL2H2 kukL2(0,T ;H2(0,L)). (B.13) Similar results hold for Hm and L2Hm+1 when m {2, 4, 5}. (III) There exist constants Gm such that kukHm(0,L) kukHm(0,L) Gm kukHm(0,L), m {0, 1, 2, 3, 4, 5, 6, 7}. Proof of Lemma B.1. (I) This is a classical extension problem, we recall Stein ([26], p. 182, Thm. 50 ) for a precise construction. In fact the same type of results also exists for many other spaces, like Besov space etc. (II) Inequality (B.12) is exactly ([8], Lem. 4.2), and the same method also leads to (B.13). 38 J. KRIEGER AND S. XIANG (III) The first inequality is obvious. It suffices to prove the second one. If m = 0 or 1, then Gm = 1. Else, we get from the definition that kuk2 Hm(0,L) = X m m k uk2 L2(0,L), = kuk2 Hm(0,L) + X 0<<m m k uk2 L2(0,L), kuk2 Hm(0,L) + X 0<<m m E mkuk2 Hm(0,L), = kuk2 Hm(0,L) 1 + X 0<<m m E m ! , which gives the value of Gm : Gm := 1 + X 0<<m m E m. Armed with the preceding lemma, we can apply the interpolation theory on cases k = 1, 2, 4 and 5. Here we only prove the case k = 1, while the other cases can be treated in the same way. Since we are dealing with the KdV flow, we add the natural compatibility conditions on interpolation spaces, for example H1 (0) which is endowed with the same norm as H1. For any t (0, T], we define a mapping operator Lt 0 : f 7- S(t)f. We also define L1 : f 7- S(·)f, t [0, T]. From the preceding part we know that, for m {0, 3} the linear operators Lt 0 : Hm (0)(0, L) Hm (0)(0, L), L1 : Hm (0)(0, L) L2 (0, T; Hm+1 (0) (0, L)), are bounded. Indeed, these bounds are given by kLt 0kHm,Hm Fm 0 Gm , kL1kHm,L2Hm+1 Fm 1 Gm+1 . Therefore, thanks to the interpolation theory, we get kLt 0kH1,H1 kLt 0k 2 3 H0,H0 kLt 0k 1 3 H3,H3 (F0 0 G0 ) 2 3 (F3 0 G3 ) 1 3 , kL1kH1,L2H2 kL1k 2 3 H0,L2H1 kL1k 1 3 H3,L2H4 (F0 1 G1 ) 2 3 (F3 1 G4 ) 1 3 . COST FOR A CONTROLLED LINEAR KDV EQUATION 39 Thus kLt 0kH1,H1 (0 ) 2 3 (3 ) 1 3 kLt 0kH1,H1 (0 F0 0 G0 ) 2 3 (3 F3 0 G3 ) 1 3 , kL1kH1,L2H2 (1 ) 2 3 (4 ) 1 3 kLt 0kH1,L2H2 (1 F0 1 G1 ) 2 3 (4 F3 1 G4 ) 1 3 , hence kLt 0kH1,H1 G1 kLt 0kH1,H1 G1 (0 F0 0 G0 ) 2 3 (3 F3 0 G3 ) 1 3 , kL1kH1,L2H2 G1 kL1kH1,L2H2 G1 (1 F0 1 G1 ) 2 3 (4 F3 1 G4 ) 1 3 , Hence we get kS(t)f0kL([0,T ];H1 (0) (0,L)) F1 0 kfkH1 (0) (0,L), kS(t)f0kL2(0,T ;H2 (0) (0,L)) F1 1 kfkH1 (0) (0,L), with F1 0 , F1 1 defined by F1 0 := G1 (0 F0 0 G0 ) 2 3 (3 F3 0 G3 ) 1 3 , F1 1 := G1 (1 F0 1 G1 ) 2 3 (4 F3 1 G4 ) 1 3 . As the flow conserves the Sobolev regularity, we know that kS(t)f0kC0([0,T ];H1 (0) (0,L)) F1 0 kfkH1 (0) (0,L). (B.14) Similar calculation provides F2 0 := G2 (0 F0 0 G0 ) 1 3 (3 F3 0 G3 ) 2 3 , F2 1 := G2 (1 F0 1 G1 ) 1 3 (4 F3 1 G4 ) 2 3 , F4 0 := G4 (3 F3 0 G3 ) 2 3 (6 F6 0 G6 ) 1 3 , F4 1 := G4 (4 F3 1 G4 ) 2 3 (7 F6 1 G7 ) 1 3 , F5 0 := G5 (3 F3 0 G3 ) 1 3 (6 F6 0 G6 ) 2 3 , F5 1 := G5 (4 F3 1 G4 ) 1 3 (7 F6 1 G7 ) 2 3 . Proof of Lemma 2.4. Since the L2 energy of the flow decays, it suffices to prove (2.5). For any t (0, T], there exists a unique n Z such that t (2n , 2n+1 ]. Then, thanks to Lemma 2.2, we can find some t0 (2n-1 , 2n ] satisfies kS(t0 )f0kHk+1 (0) Fk 1 2-(n-1)/2 kf0kHk (0) . (B.15) Otherwise, we have Z T 0 kS(t0 )f0)k2 Hk+1 (0) dt Z 2n 2n-1 kS(t0 )f0)k2 Hk+1 0 dt, > Z 2n 2n-1 (Fk 1 )2 2-(n-1) kf0k2 Hk (0) dt, 40 J. KRIEGER AND S. XIANG = Fk 1 kf0kHk (0) 2 , which is in contradiction with (2.4). Thanks to inequality (2.3), we get kS(t)f0kHk+1 (0) = kS(t - t0 ) S(t0 )f0 kHk+1 (0) , Fk+1 0 kS(t0 )f0kHk+1 (0) , 2-(n-1)/2 Fk 1 Fk+1 0 kf0kHk (0) , 2t-1/2 Fk 1 Fk+1 0 kf0kHk (0) , t (0, T], T L. (B.16) By applying (B.16) with k = 0, 1, ..., k respectively, we are able to get kS(t)f0kHn (0) = S t n n f0 Hn (0) , 2 t n -1/2 Fn-1 1 Fn 0 S t n n-1 f0 Hn-1 (0) , 4 t n -1 Fn-1 1 Fn 0 Fn-2 1 Fn-1 0 S t n n-2 f0 Hn-2 (0) , 2n nn/2 tn/2 n-1 Y i=0 Fi 1Fi+1 0 ! kf0kL2 , t (0, T], T L. Hence, we conclude the proof of Lemma 2.4 by selecting Fk s := 2k kk/2 k-1 Y i=0 Fi 1Fi+1 0 ! , k {1, 2, 3, 4, 5, 6}. (B.17) References [1] R.A. Adams, Sobolev Spaces. Vol. 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). [2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024­1065. [3] J.L. Bona, S.M. Sun and B.-Y, Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. Comm. Partial Differential Equations 28 (2003) 1391­1436 [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). [5] E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optim. 46 (2007) 877­899. [6] E. Cerpa, Control of a Korteweg-de Vries equation: a tutorial. Math. Control Relat. Fields 4 (2014) 45­99. [7] E. Cerpa and E. Crepeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain. Ann. Inst. Henri Poincare Anal. Non Lineaire 26 (2009) 457­475 [8] S.N. Chandler-Wilde, D.P. Hewett and A. Moiola, Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples. Mathematika 61 (2015) 414­443. [9] F.W. Chaves-Silva and G. 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Zhang, Finite-time internal stabilization of a linear 1-D transport equation. Systems Control Lett. 133 (2019) 104529. [1] R. A. Adams, Sobolev Spaces. Vol. 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). [2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. [3] J. L. Bona, S. M. Sun and B.-Y, Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. Comm. Partial Differential Equations 28 (2003) 1391–1436. [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). [5] E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optim. 46 (2007) 877–899. [6] E. Cerpa, Control of a Korteweg-de Vries equation: a tutorial. Math. Control Relat. Fields 4 (2014) 45–99. [7] E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009) 457–475. [8] S. N. Chandler-Wilde, D. P. Hewett and A. Moiola, Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples. Mathematika 61 (2015) 414–443. [9] F. W. Chaves-Silva and G. Lebeau, Spectral inequality and optimal cost of controllability for the Stokes system. ESAIM: COCV 22 (2016) 1137–1162. [10] J. Chu, J.-M. Coron and P. Shang, Asymptotic stability of a nonlinear Korteweg–de Vries equation with critical lengths. J. Differential Equations 259 (2015) 4045–4085. [11] J.-M. Coron, Control And , Vol. 136. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). [12] J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc. (JEMS) 6 (2004) 367–398. [13] J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right. J. Math. Pures Appl. 102 (2014) 1080–1120. [14] J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach. Arch. Ration. Mech. Anal. 225 (2017) 993–1023. [15] J.-M. Coron and L. Praly, Adding an integrator for the stabilization problem. Systems Control Lett. 17 (1991) 89–104. [16] J.-M. Coron, I. Rivas and S. Xiang, Local exponential stabilization for a class of Korteweg–de Vries equations by means of time-varying feedback laws. Anal. Partial Differ. Equ. 10 (2017) 1089–1122. [17] J.-M. Coron and S. Xiang, Small-time global stabilization of the viscous Burgers equation with three scalar controls. 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Xiang, Null controllability of a linearized Korteweg-de Vries equation by backstepping approach. SIAM J. Control Optim. 57 (2019) 1493–1515. [30] C. Zhang, Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Preprint (2019). [31] C. Zhang, Finite-time internal stabilization of a linear 1-D transport equation. Systems Control Lett. 133 (2019) 104529. COCV_2021__27_S1_A23_04bb99ec9-1e39-41d7-9aeb-c7a2ba093452cocv19019110.1051/cocv/202008010.1051/cocv/2020080 Weighted Sobolev inequalities in $\mathrm{CD}(0,N)$spaces 0000-0002-7619-9795 Tewodrose David * CY Cergy Paris University, 95000 Cergy, France. *Corresponding author: [e-mail] SupplementS22 © The authors. Published by EDP Sciences, SMAI 2021 2021 The authors. Published by EDP Sciences, SMAI This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Full (PDF)Full (DJVU)

In this note, we prove global weighted Sobolev inequalities on non-compact CD(0, N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result from [V. Minerbe, G.A.F.A. 18 (2009) 1696–1749] stated for Riemannian manifolds with non-negative Ricci curvature. We use this result in the context of RCD(0, N) spaces to get a uniform bound of the corresponding weighted heat kernel via a weighted Nash inequality.

Sobolev inequalities metric measure spaces#curvature-dimension conditions heat kernel 46E36 53C23 35K08 51K10 idline ESAIM: COCV 27 (2021) S22 open-access yes cover_date 2021 first_year 2021 last_year 2021 transformative_agreement national-agreement-fr_2020 ESAIM: COCV 27 (2021) S22 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020080 www.esaim-cocv.org WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES David Tewodrose* Abstract. In this note, we prove global weighted Sobolev inequalities on non-compact CD(0, N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result from [V. Minerbe, G.A.F.A. 18 (2009) 1696­1749] stated for Riemannian manifolds with non-negative Ricci curvature. We use this result in the context of RCD(0, N) spaces to get a uniform bound of the corresponding weighted heat kernel via a weighted Nash inequality. Mathematics Subject Classification. 46E36, 53C23, 35K08, 51K10. Received November 8, 2019. Accepted November 18, 2020. 1. Introduction Riemannian manifolds with non-negative Ricci curvature have strong analytic properties. Indeed, the doubling condition and the local L2 -Poincare inequality are satisfied on such spaces, and they imply many important results, like the well-known Li-Yau Gaussian estimates for a class of Green functions including the heat kernel [19] or powerful local Sobolev inequalities and parabolic Harnack inequalities (see e.g. [28]). In the recent years, several classes of possibly non-smooth metric measure spaces containing the collection of Riemannian manifolds with non-negative Ricci curvature have been under investigation, both from a geometric and an analytic point of view. For instance, in the context of measure spaces endowed with a suitable Dirichlet form, Sturm proved existence and uniqueness of the fundamental solution of parabolic operators along with Gaussian estimates and parabolic Harnack inequalities [29, 30], provided the doubling and Poincare properties hold. Afterwards, general doubling spaces with Poincare-type inequalities were studied at length by Hajlasz and Koskela [15] who proved local Sobolev-type inequalities, a Trudinger inequality, a Rellich-Kondrachov theorem, and many related results. Approximately a decade ago, Sturm [31] and Lott and Villani [20] independently proposed the curvature- dimension condition CD(0, N), for N [1, +), as an extension of non-negativity of the Ricci curvature and bound from above by N of the dimension for possibly non-smooth metric measure spaces. Coupled with the infinitesimal Hilbertiannity introduced later on by Ambrosio, Gigli and Savare [2] to rule out non-Riemannian structures, the CD(0, N) condition leads to the stronger RCD(0, N) condition, where R stands for Riemannian. The classes of CD(0, N) and RCD(0, N) spaces have been extensively studied over the past few years, and it is by now well-known that they both contain the measured Gromov-Hausdorff closure of the class of Riemannian manifolds with non-negative Ricci curvature and dimension lower than N, as well as Alexandrov spaces with non-negative generalized sectional curvature and locally finite and non-zero n-dimensional Hausdorff measure, Keywords and phrases: Sobolev inequalities, metric measure spaces, curvature-dimension conditions, heat kernel. CY Cergy Paris University, 95000 Cergy, France. * Corresponding author: [e-mail] c The authors. Published by EDP Sciences, SMAI 2021 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 D. TEWODROSE n being lower that N. Moreover, CD(0, N) spaces satisfy the doubling and Poincare properties, and RCD(0, N) spaces are, in addition, endowed with a regular and strongly local Dirichlet form called Cheeger energy (see Sect. 2). Therefore, the works of Sturm [29, 30] imply existence and uniqueness of an heat kernel, which by the way satisfies Gaussian estimates, on RCD(0, N) spaces. One of the interest of the CD(0, N) and RCD(0, N) conditions, and of the more general CD(K, N) and RCD(K, N) conditions for arbitrary K R, is the possibility of proving classical functional inequalities on spaces with rather loose structure thanks to optimal transport or gradient flow arguments. In this regard, Lott and Villani obtained in ([21], Thm. 5.29) a global Sobolev-type inequality for CD(K, N) spaces with K > 0 and N (2, +). Later on, in their striking work ([8], Thm. 1.11), Cavaletti and Mondino proved a global Sobolev- type inequality with sharp constant for bounded essentially non-branching CD (K, N) spaces with K R and N (1, +); in case K > 0 and N > 2, they get the classical Sobolev inequality with sharp constant. This last inequality had been previously justified on RCD (K, N) spaces with K > 0 and N > 2 by Profeta [26]. The aim of this note is to provide a new related analytic result, namely a global weighted Sobolev inequality, for certain non-compact CD(0, N) spaces with N > 2. It is worth underlying that our result does not require the Riemannian synthetic condition RCD(0, N). Here and throughout the paper, if (X, d, m) is a metric measure space, we write Br(x) for the ball of radius r > 0 centered at x X, and V (x, r) for m(Br(x)). Theorem 1.1 (Weighted Sobolev inequalities). Let (X, d, m) be a CD(0, N) space with N > 1. Assume that there exists 1 < N such that 0 < inf := lim inf r+ V (o, r) r sup := lim sup r+ V (o, r) r < + (1.1) for some o X. Then for any 1 p < , there exists a constant C > 0, depending only on N, , inf , sup and p, such that for any continuous function u : X R admitting an upper gradient g Lp (X, m), ^ X |u|p dµ 1 p C ^ X gp dm 1 p where p = Np/(N - p) and µ is the measure absolutely continuous with respect to m with density wo = V (o, d(o, ·))p/(N-p) d(o, ·)-Np/(N-p) . Theorem 1.1 extends a result by Minerbe stated for p = 2 on n-dimensional Riemannian manifolds with non- negative Ricci curvature ([23], Thm. 0.1). The motivation there was that the classical L2 -Sobolev inequality does not hold on those manifolds which satisfy (1.1) with < N = n, see ([23], Prop. 2.21). This phenomenon also holds on some metric measure spaces including Finsler manifolds, see the forthcoming [32] for related results. Our proof is an adaptation of Minerbe's proof to the setting of CD(0, N) spaces and is based upon ideas of Grigor'yan and Saloff-Coste introduced in the smooth category [14] which extend easily to the setting of metric measure spaces. More precisely, we apply an abstract process (Thm. 2.15) which permits to patch local inequalities into a global one by means of an appropriate discrete Poincare inequality. In the broader context of metric measure spaces with a global doubling condition, a local Poincare inequality, and a reverse doubling condition weaker than (1.1), this method provides "adimensional" weighted Sobolev inequalities, as explained in the recent work [33]. After that, we follow a classical approach (see e.g. [4]) which was neither considered in [23] nor in the subsequent related work [17] to deduce a weighted Nash inequality (Thm. 4.1) for CD(0, N) spaces satisfying the growth assumption (1.1), provided > 2. Let us mention that in the context of non-reversible Finsler manifolds, Ohta put forward an unweighted Nash inequality [24] and that Bakry, Bolley, Gentil and Maheux introduced weighted Nash inequalities in the study of possibly non-ultracontractive Markov semigroups [3], but these inequalities seem presently unrelated to our. WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 3 We conclude this note with a natural consequence in the setting of RCD(0, N) spaces satisfying a uniform local Ahlfors regularity property, namely a uniform bound for the weighted heat kernel associated with a suitable modification of the Cheeger energy. To the best knowledge of the author, this is the first appearance of this weighted heat kernel whose properties would require a deeper investigation. The paper is organized as follows. In Section 2, we introduce the tools of non-smooth analysis that we shall use throughout the article. We also define the CD(0, N) and RCD(0, N) conditions, and present the aforementioned patching process. Section 3 is devoted to the proof of Theorem 1.1. Section 4 deals with the weighted Nash inequality and the uniform bound on the weighted heat kernel we mentioned earlier. The final Section 5 provides a non-trivial non-smooth space to which our main theorem applies. 2. Preliminaries Unless otherwise mentioned, in the whole article (X, d, m) denotes a triple where (X, d) is a proper, complete and separable metric space and m is a Borel measure, positive and finite on balls with finite and non-zero radius, such that supp(m) = X. We use the standard notations for function spaces: C(X) for the space of d-continuous functions, Lip(X) for the space of d-Lipschitz functions and Lp (X, m) (respectively Lp loc(X, m)) for the space of p-integrable (respectively locally p-integrable) functions, for any 1 p +. If U is an open subset of X, we denote by Cc(U) the space of continuous functions on X compactly supported in U. We also write L0 (X, m) (respectively L0 +(X, m)) for the space of m-measurable (respectively non-negative m-measurable) functions. If A is a subset of X, we denote by A its closure. For any x X and r > 0, we write Sr(x) for Br(x)\Br(x). For any > 0, if B denotes a ball of radius r > 0, we write B for the ball with same center as B and of radius r. If A is a bounded Borel subset of X, then for any locally integrable function u : X R, we write uA or ffl A u dm for the mean value 1 m(A) ´ A u dm, and huiA for the mean value 1 µ(A) ´ A u dµ, where µ is as in Theorem 1.1. Several constants appear in this work. For better readability, if a constant C depends only on parameters a1, a2, · · · we always write C = C(a1, a2, . . .) for its first occurrence, and then write more simply C if there is no ambiguity. 2.1. Non-smooth analysis Let us recall that a continuous function : [0, L] X is called a rectifiable curve if its length L() := sup ( n X i=1 d((xi), (xi-1)) : 0 = x0 < · · · < xn = L, n N\{0} ) is finite. If : [0, L] X is rectifiable then so is its restriction [t,s] to any subinterval [t, s] of [0, L]; moreover, there exists a continuous function : [0, L()] X, called arc-length parametrization of , such that L( [t,s] ) = |t - s| for all 0 t s L(), and a non-decreasing continuous map : [0, L] [0, L()], such that = (see e.g. [6], Prop. 2.5.9). When = , we say that is parametrized by arc-length. In the context of metric analysis, a weak notion of norm of the gradient of a function is available and due to Heinonen and Koskela [18]. Definition 2.1 (Upper gradients). Let u : X [-, +] be an extended real-valued function. A Borel function g : X [0, +] is called upper gradient of u if for any rectifiable curve : [0, L] X parametrized by arc-length, |u((L)) - u((0))| ^ L 0 g((s)) ds. Building on this, one can introduce the so-called Cheeger energies and the associated Sobolev spaces H1,p (X, d, m), where p [1, +), in the following way: 4 D. TEWODROSE Definition 2.2 (Cheeger energies and Sobolev spaces). Let 1 p < +. The p-Cheeger energy of a function u Lp (X, m) is set as Chp(u) := inf lim inf i kgikp Lp . where the infimum is taken over all the sequences (ui)i Lp (X, m) and (gi)i L0 +(X, m) such that gi is an upper gradient of ui and kui - ukLp 0. The Sobolev space H1,p (X, d, m) is then defined as the closure of Lip(X) Lp (X, m) with respect to the norm kukH1,p := (kukp Lp + Chp(u)) 1/p . Remark 2.3. Following a classical convention, we call Cheeger energy the 2-Cheeger energy and write Ch instead of Ch2. The above relaxation process can be performed with slopes of bounded Lipschitz functions instead of upper gradients, see Lemma 4.2. Recall that the slope of a Lipschitz function f is defined as |f|(x) := lim sup yx |f(x)-f(y)| d(x,y) if x X is not isolated, 0 otherwise, and that it satisfies the chain rule, namely |(fg)| f|g| + g|f| for any f, g Lip(X). Let us recall that (X, d, m) is called doubling if there exists CD 1 such that V (x, 2r) CDV (x, r) x X, r > 0, (2.1) and that it satisfies a uniform weak local Lp -Poincare inequality, where p [1, +), if there exists > 1 and CP > 0 such that ^ B |u - uB|p dm CP rp ^ B gp dm (2.2) holds for any ball B of arbitrary radius r > 0, any u L1 loc(X, m) and any upper gradient g Lp (X, m) of u. If (2.2) holds with = 1, we say that a uniform strong local Lp -Poincare inequality holds. The next notion serves to turn weak inequalities into strong inequalities, see e.g. ([15], Sect. 9). Definition 2.4 (John domain). A bounded open set X is called a John domain if there exists xo and CJ > 0 such that for every x , there exists a Lipschitz curve : [0, L] parametrized by arc-length such that (0) = x, (L) = xo and t-1 d((t), X\) CJ for any t [0, L]. Finally let us introduce a technical property taken from [15]. For any v L0 (X, m) and 0 < t1 < t2 < +, we denote by vt2 t1 the truncated function min(max(0, v - t1), t2 - t1) + t1. We write A for the characteristic function of a set A X. Definition 2.5 (Truncation property). We say that a pair of m-measurable functions (u, g) such that for some p [1, +), CP > 0 and > 1, the inequality (2.2) holds for any ball B of arbitrary radius r > 0, has the truncation property if for any 0 < t1 < t2 < +, b R and {-1, 1}, there exists C > 0 such that (2.2) holds for any ball B of arbitrary radius r > 0 with u, g and CP replaced by ((u - b))t2 t1 , g{t1<u<t2} and C respectively. The next proposition is a particular case of ([15], Thm. 10.3). WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 5 Proposition 2.6. If (X, d, m) satisfies a uniform weak local L1 -Poincare inequality, any pair (u, g) where u C(X) and g L1 loc(X, m) is an upper gradient of u has the truncation property. 2.2. The CD(0, N) and RCD(0, N) conditions Let us give the definition of the curvature-dimension conditions CD(0, N) and RCD(0, N). For the general condition CD(K, N) with K R, we refer to ([34], Chap. 29 & 30). Recall that a curve : [0, 1] X is called a geodesic if d((s), (t)) = |t - s|d((0), (1)) for any s, t [0, 1]. The space (X, d) is called geodesic if for any couple of points (x0, x1) X2 there exists a geodesic such that (0) = x0 and (1) = x1. We denote by P(X) the set of probability measures on X and by P2(X) the set of probability measures µ on X with finite second moment, i.e. such that there exists xo X for which ´ X d2 (xo, x) dµ(x) < +. The Wasserstein distance between two measures µ0, µ1 P2(X) is by definition W2(µ0, µ1) := inf ^ X×X d(x0, x1)2 d(x0, x1) 1/2 where the infimum is taken among all the probability measures on X × X with first marginal equal to µ0 and second marginal equal to µ1. A standard result of optimal transport theory states that if the space (X, d) is geodesic, then the metric space (P2, W2) is geodesic too. Let us introduce the Renyi entropies. Definition 2.7 (Renyi entropies). Given N (1, +), the N-Renyi entropy relative to m, denoted by SN (·|m), is defined as follows: SN (µ|m) := - ^ X 1- 1 N dm µ P(X), where µ = m + µsing is the Lebesgue decomposition of µ with respect to m. We are now in a position to introduce the CD(0, N) condition, which could be summarized as weak geodesical convexity of all the N0 -Renyi entropies with N0 N. Definition 2.8 (CD(0, N) condition). Given N (1, +), a complete, separable, geodesic metric measure space (X, d, m) satisfies the CD(0, N) condition if for any N0 N, the N0 -Renyi entropy is weakly geodesically convex, meaning that for any couple of measures (µ0, µ1) P2(X)2 , there exists a W2-geodesic (µt)t[0,1] between µ0 and µ1 such that for any t [0, 1], SN (µt|m) (1 - t)SN (µ0|m) + tSN (µ1|m). Any space satisfying the CD(0, N) condition is called a CD(0, N) space. The Bishop-Gromov theorem holds on CD(0, N) spaces ([34], Thm. 30.11), and as a direct consequence, the doubling condition (2.1) holds too, with CD = 2N . Moreover, Rajala proved the following uniform weak local L1 -Poincare inequality ([27], Thm. 1.1). Proposition 2.9. Assume that (X, d, m) is a CD(0, N) space. Then for any function u C(X) and any upper gradient g L1 loc(X, m) of u, for any ball B X of arbitrary radius r > 0, ^ B |u - uB| dm 4r ^ 2B g dm. The CD(0, N) condition does not distinguish between Riemannian-like and non-Riemannian-like structures: for instance, Rn equipped with the distance induced by the L -norm and the Lebesgue measure satisfies the 6 D. TEWODROSE CD(0, N) condition (see the last theorem in [34]), though it is not a Riemannian structure because the L -norm is not induced by any scalar product. To focus on Riemannian-like structures, Ambrosio, Gigli and Savare added to the theory the notion of infinitesimal Hilbertianity, leading to the so-called RCD condition, R standing for Riemannian [2]. Definition 2.10 (RCD(0, N) condition). (X, d, m) is called infinitesimally Hilbertian if Ch is a quadratic form. If in addition (X, d, m) is a CD(0, N) space, it is said to satisfy the RCD(0, N) condition, or more simply it is called a RCD(0, N) space. Let us provide some standard facts taken from [2, 12]. First, note that (X, d, m) is infinitesimally Hilber- tian if and only if H1,2 (X, d, m) is a Hilbert space, whence the terminology. Moreover, for infinitesimally Hilbertian spaces, a suitable diagonal argument justifies for any f H1,2 (X, d, m) the existence of a func- tion |f| L2 (X, m), called minimal relaxed slope or minimal generalized upper gradient of f, which gives integral representation of Ch, meaning: Ch(f) = ^ X |f|2 dm f H1,2 (X, d, m). The minimal relaxed slope is a local object, meaning that |f| = |g| m-a.e. on {f = g} for any f, g H1,2 (X, d, m), and it satisfies the chain rule, namely |(fg)| f|g| + g|f| m-a.e. on X for all f, g H1,2 (X, d, m). In addition, the function hf1, f2i := lim 0 |(f1 + f2)|2 - |f1|2 2 provides a symmetric bilinear form on H1,2 (X, d, m) × H1,2 (X, d, m) with values in L1 (X, m), and Ch(f1, f2) := ^ X hf1, f2i dm f1, f2 H1,2 (X, d, m), defines a strongly local, regular and symmetric Dirichlet form. Finally, the infinitesimally Hilbertian condition allows to apply the general theory of gradient flows on Hilbert spaces, ensuring the existence of the L2 -gradient flow (ht)t0 of the convex and lower semicontinuous functional Ch, called heat flow of (X, d, m). This heat flow is a linear, continuous, self-adjoint and Markovian contraction semigroup in L2 (X, m). The terminology `heat flow' comes from the characterization of (ht)t0 as the only semigroup of operators such that t 7 htf is locally absolutely continuous in (0, +) with values in L2 (X, m) and d dt htf = htf for L1 -a.e. t (0, +) holds for any f L2 (X, m), the Laplace operator being defined in this context by: f D() h := f L2 (X, m) s.t. Ch(f, g) = - ^ X hg dm g H1,2 (X, d, m). 2.3. Patching process Let us present now the patching process [14, 23] that we shall apply to get Theorem 1.1. In the whole paragraph, (X, d) is a metric space equipped with two Borel measures m1 and m2 both finite and nonzero on balls with finite and nonzero radius and such that supp(m1) = supp(m2) = X. For any bounded Borel set A X and any locally m2-integrable function u : X R, we denote by {u}A the mean value 1 m2(A) ´ A u dm2. For any given set S, we denote by Card(S) its cardinality. WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 7 Definition 2.11 (Good covering). Let A A# X be two Borel sets. A countable family (Ui, U i , U# i )iI of triples of Borel subsets of X with finite mj-measure for any j {1, 2} is called a good covering of (A, A# ) with respect to (m1, m2) if: 1. for every i I, Ui U i U# i ; 2. there exists a Borel set E A such that A\E S i Ui S i U# i A# and m1(E) = m2(E) = 0; 3. there exists Q1 > 0 such that Card({i I : U# i0 U# i 6= }) Q1 for any i0 I; 4. for any (i, j) I × I such that Ui Uj 6= , there exists k(i, j) I such that Ui Uj U k(i,j); 5. there exists Q2 > 0 such that for any (i, j) I × I satisfying Ui Uj 6= , m2(U k(i,j)) Q2 min(m2(Ui), m2(Uj)). When A = A# = X, we say that (Ui, U i , U# i )iI is a good covering of (X, d) with respect to (m1, m2). For the sake of clarity, we call condition 3. the overlapping condition, condition 4. the embracing condition and condition 5. the measure control condition of the good covering. Note that in [23] the measure control condition was required also for m1 though never used in the proofs. From now on, we consider two numbers p, q [1, +) and two Borel sets A A# X. We assume that a good covering (Ui, U i , U# i )iI of (A, A# ) with respect to (m1, m2) exists. Let us explain how to define from (Ui, U i , U# i )iI a canonical weighted graph (V, E, ), where V is the set of vertices of the graph, E is the set of edges, and is a weight on the graph (i.e. a function : V t E R). We define V by associating to each Ui a vertex i (informally, we put a point i on each Ui). Then we set E := {(i, j) V × V : i 6= j and Ui Uj 6= }. Finally we weight the vertices of the graph by setting (i) := m2(Ui) for every i V and the edges by setting (i, j) := max((i), (j)) for every (i, j) E. The patching theorem (Thm. 2.15) states that if some local inequalities are true on the pieces of the good covering and if a discrete inequality holds on the associated canonical weighted graph, then the local inequalities can be patched into a global one. Let us give the precise definitions. Definition 2.12 (Local continuous Lq,p -Sobolev-Neumann inequalities). We say that the good covering (Ui, U i , U# i )iI satisfies local continuous Lq,p -Sobolev-Neumann inequalities if there exists a constant Sc > 0 such that for all i I, ^ Ui |u - {u}Ui |q dm2 1 q Sc ^ U i gp dm1 !1 p (2.3) for all u L1 (Ui, m2) and all upper gradients g Lp (U i , m1), and ^ U i |u - {u}U i |q dm2 !1 q Sc ^ U# i gp dm1 !1 p (2.4) for all u L1 (U i , m2) and all upper gradients g Lp (U# i , m1). Definition 2.13 (Discrete Lq -Poincare inequality). We say that the weighted graph (V, E, ) satisfies a discrete Lq -Poincare inequality if there exists Sd > 0 such that: X iV |f(i)|q (i) !1 q Sd X {i,j}E |f(i) - f(j)|q (i, j) 1 q f Lq (V, ). (2.5) 8 D. TEWODROSE Remark 2.14. Here we differ a bit from Minerbe's terminology. Indeed, in [23], the following discrete Lq Sobolev-Dirichlet inequalities of order k were introduced for any k (1, +] and any q [1, k): X iV |f(i)| qk k-q (i) !k-q qk Sd X {i,j}E |f(i) - f(j)|q (i, j) 1 q f Lq (V, ). In the present paper we only need the case k = +, in which we recover (2.5): here is why we have chosen the terminology "Poincare" which seems, in our setting, more appropriate. We are now in a position to state the patching theorem. Theorem 2.15 (Patching theorem). Let (X, d) be a metric space equipped with two Borel measures m1 and m2, both finite and nonzero on balls with finite and nonzero radius, such that supp(m1) = supp(m2) = X. Let A A# X be two Borel sets, and p, q [1, +) be such that q p. Assume that (A, A# ) admits a good covering (Ui, U i , U# i ) with respect to (m1, m2) which satisfies the local Lq,p -Sobolev-Neumann inequalities (2.3) and (2.4) and whose associated weighted graph (V, E, ) satisfies the discrete Lq -Poincare inequality (2.5). Then there exists a constant C = C(p, q, Q1, Q2, Sc, Sd) > 0 such that for any function u Cc(A# ) and any upper gradient g Lp (A# , m1) of u, ^ A |u|q dm2 1 q C ^ A# gp dm1 1 p . Although the proof of Theorem 2.15 is a straightforward adaptation of ([23], Thm. 1.8), we provide it for the reader's convenience. Proof. Let us consider u Cc(A# ). Then ^ A |u|q dm2 X iV ^ Ui |u|q dm2. From convexity of the function t 7 |t|q , we deduce |u|q 2q-1 (|u - {u}Ui |q + |{u}Ui |q ) m2-a.e. on each Ui, and then ^ A |u|q dm2 2q-1 X iV ^ Ui |u - {u}Ui |q dm2 + 2q-1 X iV |{u}Ui |q (i). (2.6) From (2.3) and the fact that P j x q/p j ( P j xj)q/p for any finite family of non-negative numbers {xj} (since q p), we get X iV ^ Ui |u - {u}Ui |q dm2 Sq/p c X iV ^ U i gp dm1 !q/p Sq/p c Q q/p 1 ^ A# gp dm1 q/p , (2.7) WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 9 this last inequality being a direct consequence of the overlapping condition 3. Now the discrete Lq -Poincare inequality (2.5) implies X iV |{u}Ui |q (i) Sd X (i,j)E |{u}Ui - {u}Uj |q (i, j). (2.8) For any (i, j) E, a double application of Holder's inequality yields to |{u}Ui - {u}Uj |q (i, j) (i, j) m2(Ui)m2(Uj) ^ Ui ^ Uj |u(x) - u(y)|q dm2(x) dm2(y), and as the measure control condition 5. ensures (i, j) = max(m2(Ui), m2(Uj)) Q2m2(U k(i,j)), the embracing condition 4. implies |{u}Ui - {u}Uj |q (i, j) Q2 m2(U k(i,j)) ^ U k(i,j) ^ U k(i,j) |u(x) - u(y)|q dm2(x) dm2(y) and then |{u}Ui - {u}Uj |q (i, j) Q22q ^ U k(i,j) |u - {u}U k(i,j) |q dm2 where we have used again the convexity of t 7 |t|q . Summing over (i, j) E, we get X (i,j)E |{u}Ui - {u}Uj |q (i, j) Q22q X (i,j)E ^ U k(i,j) |u - {u}U k(i,j) |q dm2. (2.9) Then (2.4) yields to X (i,j)E |{u}Ui - {u}Uj |q (i, j) Q22q Sq/p c X (i,j)E ^ U# k(i,j) gp dm1 q/p . (2.10) Finally, a simple counting argument shows that X (i,j)E ^ U# k(i,j) gp dm1 Q3 1 ^ A# gp dm. (2.11) The result follows from combining (2.6), (2.7), (2.8), (2.9), (2.10) and (2.11). A similar statement holds if we replace the discrete Lq -Poincare inequality by a discrete "Lq -Poincare- Neumann" version: X iV |f(i) - (f)|q (i) !1 q Sd X {i,j}E |f(i) - f(j)|q (i, j) 1 q (2.12) for all compactly supported f : V R, where (f) = P i : f(i)6=0 (i) -1 P i f(i)(i). The terminology "Poincare-Neumann" comes from the mean-value in the left-hand side of (2.12) and the analogy with the 10 D. TEWODROSE local Poincare inequality used in the study of the Laplacian on bounded Euclidean domains with Neumann boundary conditions, see ([28], Sect. 1.5.2). Theorem 2.16 (Patching theorem - Neumann version). Let (X, d) be a metric space equipped with two Borel measures m1 and m2, both finite and nonzero on balls with finite and nonzero radius, such that supp(m1) = supp(m2) = X. Let A A# X be two Borel sets such that 0 < m(A) < + and p, q [1, +) such that q p. Assume that (A, A# ) admits a good covering (Ui, U i , U# i ) with respect to (m1, m2) which satisfies the local Lq,p -Sobolev-Neumann inequalities (2.3) and (2.4) and whose associated weighted graph (V, E, ) satisfies the discrete Lq -Poincare-Neumann inequality (2.12). Then there exists a constant C = C(p, q, Q1, Q2, Sc, Sd) > 0 such that for any u Cc(A# ) and any upper gradient g Lp (A# , m1), ^ A |u - {u}A|q dm2 1 q C ^ A# gp dm1 1 p . The proof of Theorem 2.16 is similar to the proof of Theorem 2.15 and writes exactly as ([23], Thm. 1.10) with upper gradients instead of norms of gradients, so we skip it. 3. Proof of the main result In this section, we prove Theorem 1.1 after a few preliminary results. As already pointed out in [23], the local continuous L2 ,2 -Sobolev-Neumann inequalities on Riemannian manifolds (where 2 = 2n/(n - 2) and n is the dimension of the manifold) can be derived from the doubling condition and the uniform strong local L2 -Poincare inequality which are both implied by non-negativity of the Ricci curvature. However, the discrete L2 -Poincare inequality requires an additional reverse doubling condition which is an immediate consequence of the growth condition (1.1), as shown in the next lemma. Lemma 3.1. Let (Y, dY , mY ) be a metric measure space such that 0 < inf := lim inf r+ mY (Br(yo)) r sup := lim sup r+ mY (Br(yo)) r < + (3.1) for some yo Y and > 0. Then there exists A > 0 and CRD = CRD(inf , sup) > 0 such that mY (BR(yo)) mY (Br(yo)) CRD R r A < r R. (3.2) Proof. The growth condition (3.1) implies the existence of A > 0 such that for any R r > A, inf /2 r- mY (Br(yo)) 2sup and R- mY (BR(yo)) inf /2, whence (3.2) with CRD = inf /(4sup). Remark 3.2. Note that the doubling condition (2.1) easily implies (3.2): see for instance ([13], p. 9) for a proof giving CRD = (1 + C-4 D )-1 and = log2(1 + C-4 D ). But in this case, > 1 if and only if CD < 1 which is impossible. So we emphasize that in our context, in which we want the segment (1, ) to be non-empty, doubling and reverse doubling must be thought as complementary hypotheses. The next result, a strong local Lp -Sobolev inequality for CD(0, N) spaces, is an important technical tool for our purposes. In the context of Riemannian manifolds, it was proved by Maheux and Saloff-Coste [22]. WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 11 Lemma 3.3. Let (Y, dY , mY ) be a CD(0, N) space. Then for any p [1, N) there exists C = C(N, p) > 0 such that for any u C(Y ), any upper gradient g L1 loc(Y, mY ), and any ball B with arbitrary radius r > 0, ^ B |u - uB|p dmY 1 p C r mY (B)1/N ^ B gp dmY 1 p , (3.3) where p = Np/(N - p). Proof. Let u be a continuous function on Y , g L1 loc(Y, mY ) be an upper gradient of u, B be a ball with arbitrary radius r > 0, and p [1, N). In this proof uB stands for mY (B)-1 ´ B u dmY . Thanks to Holder's inequality and the doubling property, Proposition 2.9 implies B |u - uB| dmY 2N+2 r 2B gp dmY 1/p . Let x0, x1 Y and r0, r1 > 0 be such that x1 Br0 (x0) and r1 r0. Then mY (Br1 (x1)) mY (Br0 (x0)) mY (Br1 (x1)) mY (Br0+dY (x0,x1)(x1)) 2-N r1 r0 + dY (x0, x1) N 2-2N r1 r0 N by the doubling condition. Moreover, we know from Proposition 2.6 that (u, g) satisfies the truncation property, so that ([15], Thm. 5.1, 1) applies and gives B |u - uB|p dmY 1/p Cr 10B gp dmY 1/p where C depends only on p and the doubling and Poincare constants of (Y, dY , mY ) which depend only on N. As (Y, dY , mY ) is a CD(0, N) space, the metric structure (Y, dY ) is proper and geodesic, so it follows from ([15], Cor. 9.5) that all the balls in Y are John domains with a universal constant CJ > 0. Then ([15], Thm. 9.7) applies and yields to the result since 1/p - 1/p = 1/N. Finally, let us state a result whose proof - omitted here - can be deduced from ([23], Prop. 2.8) by using Proposition 2.9. Note that even if Proposition 2.9 provides only a weak inequality, one can harmlessly substitute it to the strong one used in the proof of ([23], Prop. 2.8), because it is applied there to a function f which is Lipschitz on a ball B and extended by 0 outside of B. Note also that Proposition 2.9 being a L1 -Poincare inequality, we can assume > 1 (a L2 -Poincare inequality would have only permit > 2). Proposition 3.4. Let (Y, dY , mY ) be a CD(0, N) space satisfying the growth condition (3.1) with > 1. Then there exists 0 = 0(N, ) > 1 such that for any R > 0 such that SR(yo) is non-empty, for any couple of points (x, x0 ) SR(yo)2 , there exists a rectifiable curve from x to x0 that remains inside BR(yo)\B-1 0 R(yo). Let us prove now Theorem 1.1. Let (X, d, m) be a non-compact CD(0, N) space with N 3 satisfying the growth condition (1.1) with parameter (1, N], and p [1, ). We recall that µ is the measure absolutely continuous with respect to m with density wo = V (o, d(o, ·))p/(N-p) d(o, ·)-Np/(N-p) , and that p = Np/(N - p). Note that Lemma 3.1 applied to (X, d, m), assuming with no loss of generality that A = 1, implies: V (o, R) V (o, r) CRD R r 1 < r < R. (3.4) 12 D. TEWODROSE Figure 1. for simplicity assume a0 = a; if U0 i+1,a Si+1 (o) = , then we glue the small piece U0 i+1,a to the adjacent piece U0 i,a to form Ui,a. Step 1: The good covering. Let us briefly explain how to construct a good covering on (X, d, m), referring to ([23], Sect. 2.3.1) for addi- tional details. Define as the square-root of the constant 0 given by Proposition 3.4. Then for any R > 0, two connected components X1 and X2 of BR(o)\BR(o) are always contained in one component of BR(o)\B-1R(o): otherwise, linking x X1 SR(o) and x0 X2 SR(o) by a curve remaining inside BR(o)\B-1R(o) would not be possible. Every point in a complete geodesic metric space of infinite diameter is the origin of some geodesic ray: see e.g. ([25], Prop. 10.1.1). Therefore, there exists a geodesic ray starting from o. For any i N, let us write Ai = Bi (o)\Bi-1 (o) and denote by (U0 i,a)0ah0 i the connected components of Ai, U0 i,0 being set as the one intersecting . The next simple result was used without a proof in [23]. Claim 1. There exists a constant h = h(N, ) < such that supi h0 i h. Proof. Take i N. For every 0 a h0 i, pick xa in Ui,a S(i+i-1)/2(o). As the balls (Ba := B(i-i-1)/4(xa))0ah0 i are disjoint and all included in Bi (o), we have h0 i min 0ah0 i m(Ba) X 0ah0 i m(Ba) V (o, i ). With no loss of generality, we can assume that min 0ah0 i m(Ba) = m(B0). Notice that d(o, x0) i . Then h0 i V (o, i ) m(B0) V (x0, i + d(o, x0)) m(B0) 8i i - i-1 N by the doubling condition. This yields to the result with h := 8 -1 N . Define then the covering (U0 i,a, U0 i,a, U0# i,a)iN,0ah0 i where U0 i,a is by definition the union of the sets U0 j,b such that U0 j,b U0 i,a 6= , and U0# i,a is by definition the union of the sets U0 j,b such that U0 j,b U0 i,a 6= . Note that (U0 i,a, U0 i,a, U0# i,a)iN,0ah0 i is not necessarily a good covering, as pieces U0 i,a might be arbitrary small compared to their neighbors: in this case, the measure control condition 5. would not be true. So whenever U0 i+1,a Si+1 (o) = (this condition being satisfied by all "small" pieces), we set Ui,a := U0 i+1,a U0 i,a0 where a0 is the integer such that U0 i+1,a Ui,a0 6= ; otherwise we set Ui+1,a := U0 i+1,a. WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 13 We define U i,a and U# i,a in a similar way from U0 i,a and U0# i,a respectively. Using the doubling condition, one can easily show that (Ui,a, U i,a, U# i,a)iN,0ahi is a good covering of (X, d) with respect to (µ, m), with constants Q1 and Q2 depending only on N. Step 2: The discrete Lp -Poincare inequality. Let (V, E, ) be the weighted graph obtained from (Ui,a, U i,a, U# i,a)iN,0ahi . Define the degree deg(i, a) of a vertex (i, a) as the number of vertices (j, b) such that Ui,a Uj,b = . As a consequence of Claim 1, sup{deg(i, a) : (i, a) V} 2h. Moreover: Claim 2. There exists C 1 such that C-1 (j, b)/(i, a) C for any (i, a), (j, b) E. Proof. Take (i, a), (j, b) E. With no loss of generality we can assume j = i + 1. Take x Ui,a S(i+i-1)/2(o) and set r = (i - i-1 )/4, R = 2i+1 , so that Br(x) Ui,a and Ui+1,b BR(x). Then the doubling condition implies (i + 1, b) m2(BR(x)) CD(R/r)log2 CD m2(Br(x)) C(i, a) where C = CD(82 /( - 1))log2(CD) 1. A similar reasoning starting from x Ui+1,b S(i+1+i)/2(o) provides the existence of C0 1 such that (i, a) C0 (i + 1, b). Set C = max(C, C0 ) to conclude. We are now in a position to apply ([23], Prop. 1.12) which ensures that the discrete L1 -Poincare inequality implies the Lq one for any given q 1. But the discrete L1 -Poincare inequality is equivalent to the isoperimetric inequality ([23], Prop. 1.14): there exists a constant I > 0 such that for any V with finite measure, () () I where := {(i, a), (j, b) E : (i, a) , (j, b) / }. The only ingredients to prove this isoperimetric inequal- ity are the doubling and reverse doubling conditions, see Section 2.3.3 in [23]. Then the discrete Lq -Poincare inequality holds for any q 1, with a constant Sd depending only on q, , inf , sup and on the doubling and Poincare constants of (X, d, m), i.e. on N. In case q = p , we have Sd = Sd(N, , p, inf , sup). Step 3: The local continuous Lp ,p -Sobolev-Neumann inequalities. Let us explain how to get the local continuous Lp ,p -Sobolev-Neumann inequalities. We start by deriving from the strong local Lp -Sobolev inequality (3.3) a Lp -Sobolev-type inequality on connected Borel subsets of annuli. Claim 3. Let R > 0 and > 1. Let A be a connected Borel subset of BR(o)\BR(o). For 0 < < 1, denote by [A] the -neighborhood of A, i.e. [A] = S xA B(x). Then there exists a constant C = C(N, , , p) > 0 such that for any function u C(X) and any upper gradient g Lp ([A], m) of u, ^ A |u - uA|p dm 1/p C Rp V (o, R)p/N ^ [A] gp dm !1/p . Proof. Define s = R and choose an s-lattice of A (i.e. a maximal set of points whose distance between two of them is at least s) (xj)jJ . Set Vi = B(xi, s) and V i = V # i = B(xi, 3s). Using the doubling condition, there is no difficulty in proving that (Vi, V i , V # i ) is a good covering of (A, [A]) with respect to (m, m). A discrete Lp - Poincare inequality holds on the associated weighted graph, as one can easily check following the lines of ([23], 14 D. TEWODROSE Lem. 2.10). The local continuous Lp ,p -Sobolev-Neumann inequalities stem from the proof of ([23], Lem. 2.11), where we replace (14) there by (3.3). Then Theorem 2.16 gives the result. Let us prove that Claim 3 implies the local continuous Lp ,p -Sobolev-Neumann inequalities with a constant Sc depending only on N, and p. Take a piece of the good covering Ui,a. Choose = (1 - -1 )/2 so that [Ui,a] U i,a. Take a function u C(X) and an upper gradient g Lp ([Ui,a], m) of u. Since |u - huiUi,a | |u - c| + |c - huiUi,a | for any c R, convexity of t 7 |t|p and Holder's inequality imply ^ Ui,a |u - huiUi,a |p dµ 2p -1 ^ Ui,a (|u - c|p + |c - huiUi,a |p ) dµ 2p inf cR ^ Ui,a |u - c|p dµ 2p ^ Ui,a |u - uUi,a |p wo dm. As wo is a radial function, we can set wo(r) := wo(x) for any r > 0 and any x X such that d(o, x) = r. Note that by the Bishop-Gromov theorem, wo is a decreasing function, so ^ Ui,a |u - huiUi,a |p dµ 2p wo(i-1 ) ^ Ui,a |u - uUi,a |p dm. Applying Claim 3 with A = Ui,a, R = i-1 and = 2 yields to ^ Ui,a |u - huiUi,a |p dµ 2p wo(i-1 ) C pp (i-1) V (o, i-1)pp/N ^ U i,a gp dm !p /p C ^ U i,a gp dm !p /p where we used the same letter C to denote different constants depending only on N, p, and . As depends only on N, and p, we get the result. An analogous argument implies the inequalities between levels 2 and 3. Step 4: Conclusion. Apply Theorem 2.15 to get the result. 4. Weighted nash inequality and bound of the corresponding heat kernel In this section, we deduce from Theorem 1.1 a weighted Nash inequality. We use this result in the context of RCD(0, N) spaces to get a uniform bound on a corresponding weighted heat kernel. Theorem 4.1 (Weighted Nash inequality). Let (X, d, m) be a CD(0, N) space with N > 2 satisfying (1.1) with > 2. Then there exists a constant CNa = CNa(N, , inf , sup) > 0 such that: kuk 2+ 4 N L2(X,µ) CNakuk 4 N L1(X,µ)Ch(u) u L1 (X, µ) H1,2 (X, d, m), where µ m has density wo = V (o, d(o, ·))2/(N-2) d(o, ·)-2N/(N-2) . WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 15 To prove this theorem, we need a standard lemma which states that the relaxation procedure defining Ch can be performed with slopes of Lipschitz functions with bounded support (we write Lipbs(X) in the sequel for the space of such functions) instead of upper gradients of L2 -functions. We omit the proof for brevity and refer to the paragraph after Propositon 4.2 in [1] for a discussion on this result. Note that here and until the end of this section we write Lp (m), Lp (µ) instead of Lp (X, m), Lp (X, µ) respectively for any 1 p +. Lemma 4.2. Let (X, d, m) be a complete and separable metric measure space, and u H1,2 (X, d, m). Then Ch(u) = inf lim inf n ^ X |un|2 dm : (un)n Lipbs(X), kun - ukL2(m) 0 . In particular, for any u H1,2 (X, d, m), there exists a sequence (un)n Lipbs(X) such that ku - unkL2(m) 0 and k|un|k2 L2(m) Ch(u) when n +. We are now in a position to prove Theorem 4.1. Proof. By the previous lemma it is sufficient to prove the result for u Lipbs(X). By Holder's inequality, kukL2(µ) kuk L1(µ)kuk1- L2 (µ) where 1 2 = 1 + 1- 2 i.e. = 2 N+2 . Then by Theorem 1.1 applied in the case p = 2 < , kukL2(µ) Ckuk 2 N+2 L1(µ)k|u|k N N+2 L2(m). It follows from the identification between slopes and minimal relaxed gradients established in ([9], Thm. 5.1) that Ch(u) = k|u|k2 L2(m), so the result follows by raising the previous inequality to the power 2(N + 2)/N. Let us consider now a RCD(0, N) space (X, d, m) satisfying the growth condition (1.1) for some > 2 and the uniform local N-Ahlfors regularity property: C-1 o V (x, r) rN Co x X, 0 < r < ro (4.1) for some Co > 1 and ro > 0. Such spaces are called weakly non-collapsed according to the terminology introduced by Gigli and De Philippis in [11]. Note that it follows from [5] that N is an integer which coincides with the essential dimension of (X, d, m). We take the weight wo = V (o, d(o, ·))2/(N-2) d(o, ·)-2N/(N-2) which corresponds to the case p = 2 in Theorem 1.1. Note that (4.1) together with Bishop-Gromov's theorem implies that wo is bounded from above by C 2/(N-2) o , thus L2 (m) L2 (µ). Set H1,2 loc (X, d, m) = {f L2 loc(m) : f H1,2 (X, d, m) Lipbs(X)} and note that as an immediate consequence of (4.1) combined with Bishop-Gromov's theorem, wo is bounded from above and below by positive constants on any compact subsets of X, thus f L2 loc(m) if and only if f L2 loc(µ). Define a Dirichlet form Q on L2 (µ) as follows. Set D(Q) := {f L2 (µ) H1,2 loc (X, d, m) : |f| L2 (m)} and Q(f) = (´ X |f|2 dm if f D(Q), + otherwise. 16 D. TEWODROSE Q is easily seen to be convex. Let us show that it is a L2 (µ)-lower semicontinuous functional on L2 (µ). Let {fn}n D(Q) and f L2 (µ) be such that kfn - fkL2(µ) 0. Let K X be a compact set. For any i N\{0}, set i(·) = max(0, 1 - (1/i)d(·, K)) and note that i Lipbs(X), 0 i 1, i 1 on K and |i| (1/i). Then for any i, the sequence {ifn}n converges to if in L2 (m). The L2 (m)-lower semicontinuity of the Cheeger energy and the chain rule for the slope imply ^ K |f|2 dm ^ X |(if)|2 dm lim inf n ^ X |(ifn)|2 dm lim inf n ^ X |fn|2 dm + 2 i lim inf n ^ X fn|fn| dm + 1 i2 lim inf n ^ X f2 n dm. Letting i tend to +, then letting K tend to X, yields the result. Then we can apply the general theory of gradient flows to define the semigroup (hµ t )t>0 associated to Q which is characterized by the property that for any f L2 (X, µ), t hµ t f is locally absolutely continuous on (0, +) with values in L2 (X, µ), and d dt hµ t f = -Ahµ t f for L1 -a.e. t (0, +), where the self-adjoint operator -A associated to Q is defined on a dense subset D(A) of D(Q) = {Q < +} and characterized by: Q(f, g) = ^ X (Af)g dµ f D(A), g D(Q). Be aware that although Q is defined by integration with respect to m, it is a Dirichlet form on L2 (µ), whence the involvement of µ in the above characterization. Note that by the Markov property, each hµ t can be uniquely extended from L2 (X, µ) L1 (X, µ) to a contraction from L1 (X, µ) to itself. We start with a preliminary lemma stating that a weighted Nash inequality also holds on the appropriate functional space when Ch is replaced by Q. Lemma 4.3. Let (X, d, m) be a RCD(0, N) space with N > 3 satisfying (1.1) and (4.1) for some > 2, Co > 1 and ro > 0. Then there exists a constant C = C(N, , inf , sup) > 0 such that: kuk 2+ 4 N L2(µ) Ckuk 4 N L1(µ)Q(u) u L1 (µ) D(Q). Proof. Let u L1 (µ) D(Q). Then u L2 loc(m), u H1,2 (X, d, m) for any Lipbs(X) and |u| L2 (µ). In particular, if we take (n)n as in the proof of Lemma 4.2, for any n N we get that nu H1,2 (X, d, m) and consequently there exists a sequence (un,k)k Lipbs(X) such that un,k nu in L2 (m) and ´ X |un,k|2 dm ´ X |(nu)|2 dm. Apply Theorem 4.1 to the functions un,k to get kun,kk 2+ 4 N L2(µ) Ckun,kk 4 N L1(µ) ^ X |un,k|2 dm (4.2) WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 17 for any k N. As the un,k and nu have bounded support, and thanks to (4.1) which ensures boundedness of wo, the L2 (m) convergence un,k nu is equivalent to the L2 loc(m), L2 loc(µ), L2 (µ) and L1 (µ) convergences. Therefore, passing to the limit k + in (4.2), we get knuk 2+ 4 N L2(µ) Cknuk 4 N L1(µ) ^ X |(nu)|2 dm. By an argument similar to the proof of Lemma 4.2, we can show that lim sup n+ ^ X |(nu)|2 dm ^ X |u|2 dm. And monotone convergence ensures that knukL2(µ) kukL2(µ) and knukL1(µ) kukL1(µ), whence the result. Let us apply Lemma 4.3 to get a bound on the heat kernel of Q. Theorem 4.4 (Bound of the weighted heat kernel). Let (X, d, m) be a RCD(0, N) space with N > 3 satisfying the growth condition (1.1) for some > 2 and the uniform local N-Ahlfors regular property (4.1) for some Co > 1 and ro > 0. Then there exists C = C(N, , inf , sup) > 0 such that khµ t kL1(µ)L(µ) C tN/2 , t > 0. (4.3) Moreover, for any t > 0, hµ t admits a kernel pµ t with respect to µ such that for some C = C(N, , inf , sup) > 0, pµ t (x, y) C tN/2 x, y X. (4.4) To prove this theorem we follow closely the lines of ([28], Thm. 4.1.1). The constant C may differ from line to line, note however that it will always depend only on , N, inf and sup. Proof. Let u L1 (µ) be such that kukL1(µ) = 1. Let us show that khµ t ukL2(µ) Ct-N/4 for any t > 0. First of all, by density of Lipbs(X) in L1 (µ), we can assume u Lipbs(X) with kukL1(µ) = 1. Furthermore, since for any t > 0, the Markov property ensures that the operator hµ t : L1 (µ) L2 (µ) D(Q) extends uniquely to a contraction oerator from L1 (µ) to itself, we have hµ t u L1 (µ) ( D(Q)) and khµ t ukL1(µ) 1. Therefore, we can apply Lemma 4.3 to get: khµ t uk 2+ 4 N L2(µ) CQ(hµ t u) t > 0. As ^ X |hµ t u|2 dm = ^ X (Ahµ t u)hµ t u dµ = - ^ X d dt hµ t u hµ t u dµ = - 1 2 d dt khµ t uk2 L2(µ), we finally end up with the following differential inequality: khµ t uk 2+4/N L2(µ) - C 2 d dt khµ t uk2 L2(µ) t > 0. Writing (t) = khµ t uk2 L2(µ) and (t) = N 2 (t)-2/N for any t > 0, we get 2 C 0 (t) and thus 2 C t (t) - (0). As (0) = N 2 kuk -4/N L2(µ) 0, we obtain 2 C t (t), leading to khµ t ukL2(µ) C tN/4 . 18 D. TEWODROSE We have consequently khµ t kL1(µ)L2(µ) C tN/4 . Using the self-adjointness of hµ t , we deduce khµ t kL2(µ)L(µ) C tN/4 by duality. Finally the semigroup property khµ t kL1(µ)L(µ) khµ t/2kL1(µ)L2(µ)khµ t/2kL2(µ)L(µ) implies (4.3). Then the existence of a measurable kernel pµ t of hµ t for any t > 0 together with the bound (4.4) is a direct consequence of Lemma 4.3, thanks to ([7], Thm. (3.25)). 5. A non-smooth example To conclude, let us provide an example beyond the scope of smooth Riemannian manifolds to which Theorem 1.1 applies. For any positive integer n, let 0n be the origin of Rn . In [16], Hattori built a complete four dimensional Ricci-flat manifold (M, g) satisfying (1.1) for some (3, 4) and whose set of isometry classes of tangent cones at infinity T (M, g) is homeomorphic to S1 . Of particular interest to us is one specific element of T (M, g), namely (R3 , d 0 , 03), where d 0 is the completion of the Riemannian metric fge defined on R3 \{03} as follows: ge is the Euclidean metric on R3 and for any x = (x1, x2, x3) R3 \{03}, f(x) = ^ 0 bx(t) dt with bx(t) = 1 p (x1 - t)2 + x2 2 + x2 3 , for some > 1. Since bx(t) t- when t + and bx(t) |x|-1 when t 0 for any x 6= 03, then f(x) has no singularity on R3 \{0}; however, b03 (t) = t- is not integrable on any neighborhood of 0, so f(x) has a singularity at x = 03. In particular, (R3 , d 0 , 03) is a singular space with a unique singularity at 03. Hattori proved that this space is neither a metric cone nor a polar metric space. Let dg, vg be the Riemannian distance and Riemannian volume measure associated to g, and o M such that (R3 , d 0 , 03) is a tangent cone at infinity of (M, dg, o). Following a classical method (see e.g. [10]), one can equip (R3 , d 0 , 03) with a limit measure µ such that for some infinitesimal sequence (i)i (0, +) the rescalings (M, dgi , vgi , o), where gi = 2 i g and vgi = vgi (B1/i (o))-1 vgi , converge in the pointed measured Gromov-Hausdorff sense to (R3 , d 0 , µ, 03). As (M, g) is Ricci-flat, so are any of its rescalings, in particular they are all RCD(0, 4) spaces. The stability of the RCD(0, 4) condition with respect to pointed measured Gromov-Hausdorff convergence implies that (R3 , d 0 , µ, 03) is RCD(0, 4) too. Let us prove that (R3 , d 0 , µ, 03) also satisfies (1.1). Set inf (M, g) := lim inf r+ vg(Br(o)) r and sup(M, g) := lim sup r+ vg(Br(o)) r · Then for any r > 0, µ(Br(03)) r = lim i+ vgi (Bi r(o)) r = lim i+ vgi (Bi r(o)) vgi (Bi 1(o))r = lim i+ vg(Br/i (o)) vg(B1/i (o))r = lim i+ vg(Br/i (o)) (r/i) (1/i) vg(B1/i (o)) , so := inf (M, g) sup(M, g) µ(Br(03)) r j -1 from which (1.1) follows with inf and sup -1 . WEIGHTED SOBOLEV INEQUALITIES IN CD(0, N) SPACES 19 Acknowledgements. I warmly thank T. Coulhon who gave the initial impetus to this work. I am also greatly indebted towards L. Ambrosio for many relevant remarks at different stages of the work. Finally, I would like to thank V. Minerbe for useful comments, G. Carron and N. Gigli for helpful final conversations, and the anonymous referees for precious suggestions. References [1] L. Ambrosio, M. Colombo and S. Di Marino, Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope. Adv. Stud. Pure Math. 67 (2015) 1­58. [2] L. Ambrosio, N. Gigli and G. Savare, Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163 (2014) 1405­1490. [3] D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities. Rev. Matem. Iberoamericana 28 (2012) 879­906. [4] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-coste, Sobolev inequalities in disguise. Indiana Univ. Math. J. 44 (1995) 1033­1074. [5] E. Brue and D. Semola, Constancy of the dimension for RCD(K, N) spaces via regularity of Lagrangian flows. Commun. Pure Appl. Math. 73 (2020) 1141­1204. [6] D. Burago, Y. Burago and S. Ivanov, A course in metric geometry. In Vol. 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001). [7] E. Carlen, S. Kusuoka and D.W. Stroock, Upper Bounds for symmetric Markov transition functions. Ann. Inst. Henri Poincare Probabilites statistiques 23 (1987) 245­287. [8] F. Cavalletti and A. Mondino, Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds. Geom. Topol. 21 (2017) 603­645. [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999) 428­517. [10] J. Cheeger and T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46 (1997) 406­480. [11] G. De Philippis and N. Gigli, Non-collapsed spaces with Ricci curvature bounded from below. J. Ecol. polytech. Math. 5 (2018) 613­650. [12] N. Gigli, Nonsmooth differential geometry ­ an approach tailored for spaces with Ricci curvature bounded from below. Mem. Am. Math. Soc. 251 (2018) 1­161. [13] A. Grigor'yan, J. Hu and K.-S. Lau, Heat kernels on metric spaces with doubling measure. Fract. Geom. Stochastics IV, Progr. Prob. 61 (2009) 3­44. [14] A. Grigor'yan and L. Saloff-Coste, Stability results for Harnack inequalities. Ann. Inst. Fourier 55 (2005) 825­890. [15] P. Hajlasz, and P. Koskela, Sobolev met Poincare. Me. Am. Math. Soc. 145 (2000) 1­101. [16] K. Hattori, The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds. Geom. Topol. 21 (2017) 2683­2723. [17] H.-J. Hein, Weighted Sobolev inequalities under lower Ricci curvature bounds. Proc. Am. Math. Soc. 139 (2011) 2943­2955. [18] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998) 1­101. [19] P. Li and S.-T. Yau, On the parabolic kernel of the Schrodinger operator. Acta Math. 156 (1986) 153-201. [20] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169 (2009) 903­991. [21] J. Lott and C. Villani, Weak curvature conditions and functional inequalities. J. Funct. Anal. 245 (2007) 311­333. [22] P. Maheux and L. Saloff-Coste, Analyse sur les boules d'un operateur sous-elliptique. Math. Ann. 303 (1995) 713-740. [23] V. Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds. G.A.F.A. 18 (2009) 1696­1749. [24] S.-I. Ohta, Some functional inequalities on non-reversible Finsler manifolds. Proc. Indian Acad. Sci. Math. Sci. 127 (2017) 833­855. [25] A. Papadopoulos, Metric spaces, convexity and non-positive curvature, Second edition. In Vol. of IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society (EMS), Zurich (2014). [26] A. Profeta, The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds. Pot. Anal. 43 (2015) 513­529. [27] T. Rajala, local Poincare inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differ. Equ. 44 (2012) 477­494. [28] L. Saloff-Coste, Aspect of Sobolev-type inequalities, London Mathematical Society Lecture Note Series (No. 289). Cambridge University Press (2002). [29] K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 (1995) 275­312. [30] K.-T. Sturm, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75 (1996) 273­297. [31] K.-T. Sturm, On the geometry of metric measure spaces, I and II. Acta Math. 196 (2006) 65­131 and 133­177. [32] D. Tewodrose, Weighted Sobolev inequalities and volume growth on metric measure spaces. Preprint Hal available from https://hal.archives-ouvertes.fr/hal-01477215v2/document (2020). [33] D. Tewodrose, Adimensional weighted Sobolev inequalities in PI spaces. Preprint ArXiV 2006.10493 (2020). [34] C. Villani, Optimal transport. Old and new. Vol. 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (2009). [1] L. Ambrosio, M. Colombo and S. Di Marino, Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope. Adv. Stud. Pure Math. 67 (2015) 1–58. [2] L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163 (2014) 1405–1490. [3] D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities. Rev. Matem. Iberoamericana 28 (2012) 879–906. [4] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise. Indiana Univ. Math. J. 44 (1995) 1033–1074. [5] E. Brué and D. Semola, Constancy of the dimension for RCD(K, N) spaces via regularity of Lagrangian flows. Commun. Pure Appl. Math. 73 (2020) 1141–1204. [6] D. Burago, Y. Burago and S. Ivanov, A course in metric geometry. In Vol. 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001). [7] E. Carlen, S. Kusuoka and D. W. Stroock, Upper Bounds for symmetric Markov transition functions. Ann. Inst. Henri Poincaré Probabilités statistiques 23 (1987) 245–287. [8] F. Cavalletti and A. Mondino, Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds. Geom. Topol. 21 (2017) 603–645. [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999) 428–517. [10] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46 (1997) 406–480. [11] G. De Philippis and N. Gigli, Non-collapsed spaces with Ricci curvature bounded from below. J. Écol. polytech. Math. 5 (2018) 613–650. [12] N. Gigli, Nonsmooth differential geometry – an approach tailored for spaces with Ricci curvature bounded from below. Mem. Am. Math. Soc. 251 (2018) 1–161. [13] A. Grigor’Yan, J. Hu and K.-S. Lau, Heat kernels on metric spaces with doubling measure. Fract. Geom. Stochastics IV, Progr. Prob. 61 (2009) 3–44. [14] A. Grigor’Yan and L. Saloff-Coste, Stability results for Harnack inequalities. Ann. Inst. Fourier 55 (2005) 825–890. [15] P. Hajlasz, and P. Koskela, Sobolev met Poincaré. Me. Am. Math. Soc. 145 (2000) 1–101. [16] K. Hattori, The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds. Geom. Topol. 21 (2017) 2683–2723. [17] H.-J. Hein, Weighted Sobolev inequalities under lower Ricci curvature bounds. Proc. Am. Math. Soc. 139 (2011) 2943–2955. [18] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998) 1–101. [19] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986) 153–201. [20] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169 (2009) 903–991. [21] J. Lott and C. Villani, Weak curvature conditions and functional inequalities. J. Funct. Anal. 245 (2007) 311–333. [22] P. Maheux and L. Saloff-Coste, Analyse sur les boules d’un opérateur sous-elliptique. Math. Ann. 303 (1995) 713–40. [23] V. Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds. G.A.F.A. 18 (2009) 1696–1749. [24] S.-I. Ohta, Some functional inequalities on non-reversible Finsler manifolds. Proc. Indian Acad. Sci. Math. Sci. 127 (2017) 833–855. [25] A. Papadopoulos, Metric spaces, convexity and non-positive curvature, Second edition. In Vol. of IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society (EMS), Zürich (2014). [26] A. Profeta, The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds. Pot. Anal. 43 (2015) 513–529. [27] T. Rajala, local Poincare inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differ. Equ. 44 (2012) 477–494. [28] L. Saloff-Coste, Aspect of Sobolev-type inequalities, London Mathematical Society Lecture Note Series (No. 289). Cambridge University Press (2002). [29] K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 (1995) 275–312. [30] K.-T. Sturm, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75 (1996) 273–297. [31] K.-T. Sturm, On the geometry of metric measure spaces, I and II. Acta Math. 196 (2006) 65–131 and 133–177. [32] D. Tewodrose, Weighted Sobolev inequalities and volume growth on metric measure spaces. Preprint Hal available from (2020). [33] D. Tewodrose, Adimensional weighted Sobolev inequalities in PI spaces. Preprint ArXiV (2020). [34] C. Villani, Optimal transport. Old and new. Vol. 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (2009). COCV_2021__27_S1_A24_0243eaa40-1a96-484b-ab04-72f58f877f18cocv20012310.1051/cocv/202007910.1051/cocv/2020079 A quantitative reverse Faber-Krahn inequality for the first Robin eigenvalue with negative boundary parameter Cito Simone 1* La Manna Domenico Angelo 2 1 Universitá del Salento, Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Via per Arnesano, 73100 Lecce, Italy. 2 University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box 35 (MaD), 40014, Jyväskylä, Finland. *Corresponding author: [e-mail]; [e-mail] SupplementS23 © EDP Sciences, SMAI 2021 2021 EDP Sciences, SMAI Full (PDF)Full (DJVU)

The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalue λ$$ with negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer for λ$$ and the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem.

Robin eigenvalue quantitative isoperimetric inequality convex sets 35P15 35B35 49Q10 49R05 Academy of Finland http://dx.doi.org/10.13039/501100002341 314227 idline ESAIM: COCV 27 (2021) S23 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S23 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020079 www.esaim-cocv.org A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY FOR THE FIRST ROBIN EIGENVALUE WITH NEGATIVE BOUNDARY PARAMETER Simone Cito1,* and Domenico Angelo La Manna2 Abstract. The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalue with negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer for and the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem. Mathematics Subject Classification. 35P15, 35B35, 49Q10, 49R05. Received May 26, 2020. Accepted November 12, 2020. 1. Introduction Let Rn be a bounded Lipschitz domain and > 0. A number R is an eigenvalue of the Robin- Laplacian with boundary parameter - in if there exists a non-zero function u H1 () solving the problem -u = u in u - u = 0 on (1.1) (here is the outer normal on ), i.e., in the weak sense: Z Du · Dv dx - Z uv dHn-1 = Z uv dx v H1 (). Keywords and phrases: Robin eigenvalue, quantitative isoperimetric inequality, convex sets. 1 Universita del Salento, Dipartimento di Matematica e Fisica "Ennio De Giorgi", Via per Arnesano, 73100 Lecce, Italy. 2 University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35 (MaD), 40014, Jyvaskyla, Finland. * Corresponding author: [e-mail]; [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 S. CITO AND D.A. LA MANNA In order to handle the first Robin eigenvalue with negative boundary parameter - in (for brevity denoted by ()), it is very useful to use the following variational representation () = min uH1()\{0} Z |Du|2 dx - Z u2 dHn-1 Z u2 dx . (1.2) Notice that, for any Lipschitz domain , () is negative. Indeed, using the characteristic function of as a test function in (1.2), one obtains () - Hn-1 () || < 0. In particular, the map 7 () is bounded from above and unbounded from below (it is sufficient to consider a sequence of Lipschitz domains (j)j having prescribed measure and rapidly oscillating boundaries such that Hn-1 () +). Then, in terms of shape optimization problems, it makes sense to look for maximizers of in suitable classes of sets. In this paper we deal with the maximization of among convex sets with prescribed perimeter, i.e. with problem sup () : Rn , open, convex, Hn-1 () = m (1.3) (the choice of the surface area constraint is more natural in our framework than the classical upper bound on the Lebesgue measure of the admissible sets, as it will be clarified in the following). In [10] authors prove that the ball is the only maximizer of problem (1.3); more precisely, they proved the reverse Faber-Krahn inequality (# ) (), (1.4) where Rn is an open bounded convex set, # Rn is the ball having the same surface area of and the equality holds if and only if itself is a ball. A natural question is whether inequality (1.4) can be proved in a quantitative form, in other words if it holds some inequality like (# ) - () A() , where > 0 the map A() "quantifies" in some sense how much is "far" from the ball. In this paper we answer positively to the question, and the main result is the following stability issue. Theorem 1.1. Let n 2, > 0 and > 0. Then, there exist two positive constants C(n, , ) > 0 and 0(n, , ) > 0, depending only on the dimension n, on the boundary parameter and on , such that, for all Rn with P() = nnn-1 and (# ) - () 0, it holds (# ) - () C(n, , )g(AH()) (1.5) A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY 3 where # is the only ball with P() = P(# ) and whose center coincides with the barycenter of , AH is the Hausdorff asymmetry defined in (2.2) and g is defined by g(s) = s2 if n = 2 f-1 (s2 ) if n = 3 s n+1 2 if n 4 where f(t) = q t log 1 t for 0 < t < e-1 . The interest of our stability result is also linked to the uncontrolled behaviour of the map () whenever the variable rescales (see, for instance, [11]) and to the lack of general monotonicity properties. Indeed, differently from the Steklov case (see [18]) or the Dirichlet case (see, for instance, [17]), we cannot reduce to some equivalent scale invariant problem. In [9] it has been proved that, for any > 0, a maximizing set for the k-th Robin eigenvalue exists in the class of sets of finite perimeter with prescribed measure; moreover, optimal sets have perimeter and diameters uniformly controlled by the parameters of the problem (the measure of the admissible sets, the dimension of the space, the order of the eigenvalue and the boundary parameter). Using a similar technique, a general existence result has been proved in [10] replacing the prescription on the measure by a constraint on the perimeter (that, in general, is not saturated). Notice that also the constant in (1.5) depends on the uniform parameters of the problem (the prescribed perimeter, the dimension of the space and the boundary parameter); we expected this dependence in view of the above mentioned results of existence of optimal shapes and geometric control of the Robin spectrum. Moreover, up to our knowledge, our stability result is the first proved for negative eigenvalues of the Laplace operator. Following the approach of [14], where a reverse Faber-Krahn inequality is proven for sets of prescribed measure, our strategy is based on the study of the stability of an auxiliary Steklov-type shape optimization problem. The paper is structured as follows. In Section 2 we present some mathematical tools needed in the following and recall some impotant results in the framework of the shape optimization of the Robin eigenvalues with negative boundary parameter; in particular, we explain why the "right" constraint to consider is on the perimeter and not on the volume of the admissible sets. In Section 3 we prove Theorem 1.1 after introducing a Steklov-type auxiliary problem and proving a stability result for it. Finally, in Section 4, we remark a sharpness issue of a related stability result and show some open problems arising from our analysis. 2. Notations and preliminary results Throughout the paper, the unit ball centered at the origin will be denoted by B and its boundary by Sn-1 ; moreover, we will denote by BR the ball centered at the origin of radius R and by BR(x) the ball centered at x of radius R. Let Rn be a bounded, open set and let E Rn be a measurable set. For the sake of completeness, we recall here the definition of the perimeter of E in : P(E; ) = sup Z E div dx : C c (; Rn ), |||| 1 . The perimeter of E in Rn will be denoted by P(E) and, if P(E) < , we say that E is a set of finite perimeter. Moreover, if E has Lipschitz boundary, we have that P(E) = Hn-1 (E), (2.1) 4 S. CITO AND D.A. LA MANNA where Hn-1 is the (n - 1)-dimensional Hausdorff measure in Rn . We briefly recall the notion of Hausdorff distance between compact or open bounded sets (see for instance [19]). Definition 2.1 (Hausdorff distance and convergence). The Hausdorff distance between two non-empty compact sets E, F Rn is defined by: dH (E, F) = inf { > 0 : E F + B, F E + B} . If D Rn is a compact set and E, F D are two bounded open sets, we define the Hausdorff distance between the two open sets E and F by dH(E, F) := dH (D \ E, D \ F). This last definition is independent of the "big compact box" D. We say that a sequence of compact (respectively bounded open) sets (Ej)j converges to the compact (respectively bounded open) set E in the sense of Hausdorff if dH (Ej, E) 0 (respectively dH(Ej, E) 0). Notice that, if E and F are open convex sets, we have dH(E, F) = dH (E, F) = dH (E, F) and the following rescaling property holds dH(tE, tF) = t dH(E, F), t > 0. Let Rn be a bounded, open, convex set. We consider the following Hausdorff asymmetry functional related to : AH() = min xRn {dH (, BR(x)) , P() = P(BR(x))} . (2.2) Now, let us recall another useful notion of convergence of sets. Definition 2.2 (convergence in measure). Let Rn be a bounded, open set, let (Ej)j be a sequence of measurable sets in Rn and let E Rn be a measurable set. We say that (Ej)j converges in measure in to E, and we write Ej E, if Ej E in L1 (), or in other words, if limj V ((EjE) ) = 0. We recall also that the perimeter is lower semicontinous with respect to the local convergence in measure (see [2], Prop. 3.38), i.e. if the sequence of sets (Ej)j converges in measure in to E, then P(E; ) lim inf j P(Ej; ). As a consequence of the Rellich-Kondrachov theorem, the following compactness result holds; for a reference see for instance ([2], Thm. 3.39). Proposition 2.3. Let Rn be a bounded, open set and let (Ej) be a sequence of measurable sets of Rn , such that supj P(Ej; ) < . Then, there exists a subsequence (Ejk ) converging in measure in to a set E, such that P(E; ) lim inf k P(Ejk ; ). A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY 5 Another useful property concerning the sets of finite perimeter is stated in the next approximation result, see ([2], Thm. 3.42). Proposition 2.4. Let Rn be a bounded, open set and let E be a set of finite perimeter in . Then, there exists a sequence of smooth, bounded open sets (Ej) converging in measure in and such that limj P(Ej; ) = P(E; ). In the particular case of convex sets, the following lemma holds. Lemma 2.5. Let (Ej)j be a sequence of convex sets in Rn such that Ej B in measure, then limj P(Ej) = P(B). Proof. Since, in the case of convex sets, the convergence in measure implies the Hausdorff convergence, we have that limj dH(Ej, B) = 0 (see for instance [13]). Thus, for j large enough, there exists j, such that (1 - j)B Ej (1 + j)B. Being the perimeter monotone with respect to the inclusion of convex sets then (1 - j)n-1 P(B) P(Ej) (1 + j)n-1 P(B). When j goes to infinity, we have the thesis. Now we recall two useful results, whose proof can be found in [16, 17]. Lemma 2.6. If v W1, (Sn-1 ) and Z Sn-1 v dHn-1 = 0, then ||v||n-1 L(Sn-1) kD vkL2(Sn-1) n = 2 4||D v||2 L2(Sn-1) log 8e||D v||n-1 L(Sn-1) ||D v||2 L2(Sn-1) n = 3 C(n)||D v||2 L2(Sn-1)||D v||n-3 L(Sn-1) n 4 (2.3) For this second lemma see for instance [17]. Lemma 2.7. Let n 2. There exists a positive universal constant 0 < 1 2 such that, if E is a convex, nearly spherical set with V (E) = V (B) and ||v||W 1, 0, then ||D v||2 L 8||v||L . (2.4) 2.1. Nearly spherical sets In this section we give the definition of nearly spherical sets and we recall some of their basic properties (see for instance [6, 16, 17]). Definition 2.8. Let n 2. An open, bounded set E Rn is said a nearly spherical set parametrized by v if the barycenter of E is at the origin and there exists v W1, (Sn-1 ) such that E = y Rn : y = x(1 + v(x)), x Sn-1 , (2.5) with ||v||W 1, 1 2 . 6 S. CITO AND D.A. LA MANNA 2.2. The Robin eigenvalues with negative boundary parameter In this short paragraph we briefly recall some well known properties and shape optimization results for Robin eigenvalues with negative boundary parameter. For more details see, for instance, [11]. As already highlighted, for any Lipschitz domain , () is negative; moreover, if is connected, () is simple. In addiction, for any scale factor t 0, one has (t) = 1 t2 t(); notice that this scaling formula gives no scale invariance of the functional and no monotonicity properties, since also the boundary parameter rescales. This uncontrolled behaviour when the set rescales does not allow to reduce to equivalent scale invariant problems (for instance, to reduce ourselves to study problem (1.3) only in the class of convex sets with unitary perimeter). For many years it has been studied the maximization of among sets of prescribed measure, inspired by a conjecture by M. Bareket of 1977 (see [4] for the original statement in dimension n = 2 and [7] for a more general statement in any dimension): it was conjectured that the ball was the only maximizer for among suficiently smooth sets of prescribed measure for any value of and any dimension n. Even though there was some evidence supporting the conjecture, in 2015 the conjecture was disproved in [15]. More precisely, authors prove that for any r > 0, there exist a spherical shell Ar1,r2 := x Rd : r1 < |x| < r2 with the same volume as the ball Br, such that (Br) < (Ar1,r2 ) for every sufficiently large value of . On the other hand, for smooth domains in the plane, a reverse Faber- Krahn inequality holds for small negative values of (see Thm. 2 in [15]): for every m > 0, there exist 0 > 0 depending only on m such that, for any bounded domain of measure m with C2 boundary () (Bm) for every [0, 0], where Bm is the disk of measure m. In terms of existence of a maximizer for in any space dimension and for any value of , in [9] it is proved that among sets of prescribed measure a solution exists (also for higher eigenvalues and for more general functional), but nothing is known about the precise shape of the optimal sets. Since for a quantitative approach it is necessary to know what is the (unique) optimal set, it has been natural to replace the prescription of fixed measure with the prescription of fixed perimeter and to restrict the class of admissible sets. More precisely, in [10] authors prove the reverse Faber-Krahn inequality (1.4), i.e. that the ball is the only maximizer for among convex sets of fixed perimeter. In view of that uniqueness, the prescription on the perimeter and the convexity constraint on the admissible sets turn out to be two reasonable requirements to study the stability of inequality (1.4). In the next section we focus on this quantitative result, the main goal of our work. 3. A stability result for a Steklov-type problem and Proof of Theorem 1.1 This section is the core of the paper. We first introduce the auxiliary problem (3.3) and then present a stability inequality for it. Our approach follows the ideas in [18]. Finally, we prove Theorem 1.1 as a consequence of the quantitative reverse Faber-Krahn type inequality for (3.3). A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY 7 Let n 2, Rn and consider the following variational problem () = inf uH1\{0} Z |Du|2 dx + Z u2 dx Z u2 dHn+1 . (3.1) It is well known that if is an open, bounded set with Lipschitz boundary the infimum in (3.1) is achieved and it solves the following Steklov-type eigenvalue problem -v + v = 0 in v = ()v on . (3.2) In [14] it is highlighted that the shape optimization problem sup () : Rn , open, convex, Hn-1 () = m (3.3) is linked with the maximization of in a particular class of open sets with fixed measure; the proof that the ball maximizes is obtained via means of quantitative isoperimetric inequalities. Here we state some well known facts for (). If is the unit ball, the only solution of the problem (3.1), up to a multiplicative constant, is given by z(x) = |x|1- n 2 In 2 -1(|x|), where with I we indicate the modified Bessel function of index for every R. Following the notations in [14], we now introduce the functions h(t) = (t)1- n 2 In 2 -1(t) 2 ; f(t) = h0 (t) 2 = (t) 2-n In 2 -1(t)In 2 (t). (3.4) Since the minimizer of (3.1) is known to be z(x) in case is any ball centered at the origin, the idea is to test () against z(x). This leads to prove that the only solution to problem (3.3) is the ball, as stated in the following Lemma 3.1. Let n 2 and Rn be open and convex. Then () Z z z dHn-1 Z z2 =: N() D() =: I() and I(# ) = (# ) (3.5) where z(x) = |x|1- n 2 In 2 -1(|x|). The proof of this lemma is an easy consequence of the divergence theorem, the variational formulation of and the fact that -z + z = 0. In order to write the ratio N()/V () for nearly spherical sets, we recall that the tangential Jacobian of the map : x Sn-1 y = x(1 + u(x)) is given by J(x) = n-1 (1 + u)n-2 p (1 + u)2 + |D u|2 8 S. CITO AND D.A. LA MANNA where with D u we indicate the tangential gradient of u, while the exterior normal to at y = x(1 + u(x)) can be written as (x) = x(1 + u(x)) - D u(x) p (1 + u(x))2 + |D u(x)|2 . Hence recalling the explicit formula for z, i.e. z(x) = |x|1- n 2 In 2 -1(|x|), and using the area formula we can write the ratio N()/D() for nearly spherical sets as N() D() = Z Sn-1 f(1 + u(x))(1 + u(x))n-1 dHn-1 Z Sn-1 h(1 + u(x))(1 + u(x))n-1 p (1 + u(x))2 + |D u(x)|2dHn-1 . Proposition 3.2. Let n 2 and > 0. There exist 0 > 0 and a C1(n, ) > 0 such that, for any nearly spherical set as in the definition 2.5 with kvk 0, the following stability inequality holds true N(# )D() - D(# )N() nn C1(n, )kD vk2 L2(Sn-1). (3.6) Proof. Let the idea is to estimate from below the left hand side of (3.6) by performing a suitable "Taylor expansion" in terms of the deformation of compared to Sn-1 . For that reason, it is convenient to replace the polar representation v of by tu where t > 0 is sufficiently small. We thus obtain N(# )D() - D(# )N() nn = f(1) Z Sn-1 h(1 + tu)(1 + tu)n-1 s 1 + t2|D u|2 (1 + tu)2 d - h(1) Z Sn-1 f(1 + tu)(1 + tu)n-1 d. By a Taylor expansion up to the second order around t = 0 of the two integrals, we get N(# )D() - D(# )N() nn Z Sn-1 tu[f(1)h0 (1) - f0 (1)h(1)] d + Z Sn-1 t2 u2 2 [2(n - 1)(f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1)] d + Z Sn-1 f(1)h(1) t2 |D u|2 2 d - C1t2 kD uk2 L2(Sn-1) t2 n 2 (f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1) 2 kuk2 L2(Sn-1) + t2 f(1)h(1) 2 - f(1)h0 (1) - f0 (1)h(1) 2(n - 1) kD uk2 L2(Sn-1) - C2t2 kD uk2 L2(Sn-1) (3.7) Now we make use of the hypothesis that P() = P(# ). In term of integrals, this means Z Sn-1 (1 + u)n-2 p (1 + u)2 + |D u|2dHn-1 = nn A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY 9 Using the fact that t < , we can Taylor expand the LHS and find t Z Sn-1 udHn-1 -t2 n - 2 2 Z Sn-1 u2 dHn-1 - t2 1 2(n - 1) Z Sn-1 |D u|2 dHn-1 - C3(n)o(t2 ) (3.8) Notice that f(1)h(1) 2 - f(1)h0 (1) - f0 (1)h(1) 2(n - 1) = (I2 n 2 -1() - I2 n 2 ()) 2(n - 1)I2 n 2 () + In 2 -1() 2In 2 () > 0 (3.9) Then, since C2 can be taken arbitrarily small (the quantity C2 does not depend on ), if n 2 (f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1) 2 0, the proof is concluded (the positive constant in the statement is provided by (3.9)). Otherwise, we have to make a careful estimate in order to understand which term is the dominant one. To do that, let us recall that the space L2 (Sn-1 ) admits the set of spherical harmonics {Yk,i, 1 i Nk, k N} as orthonormal basis. To be more precise, for any k N, the eigenvalue problem -Sn-1 Yk,i = n(n - k - 2)Yk,i, where Sn-1 is the Laplace Beltrami operator, admits Nk independent solutions, called the spherical harmonics of order k, and we will indicate them as {Yk,i, 1 i Nk}. Moreover, it is well known that the spherical harmonics of order k are nothing else than homogeneous polynomials of degree k. Hence, we will write u as u = X k=1 Nk X i=1 ak,iYk,i where ak,i = R Sn-1 uYk,i. Since {Yk,i, 1 i Nk, k N} is an orthonormal basis we have kuk2 L2(Sn-1) = a2 0 + X k=0 Nk X i=1 a2 k,i and using the properties of Yk,i it holds kD uk2 L2(Sn-1) = X k=1 Nk X i=1 k(n + k - 2)a2 k,i. Since Y0 is a constant and Y1,i = xi for 1 i n, we now estimate a0 and a1,i using the geometric hypothesis on . Indeed, using (3.8) and the fact that t 0 we infer |a0| = Z Sn-1 u t n - 2 2 kuk2 L2(Sn-1) + t 1 2(n - 2) kD uk2 L2(Sn-1). 10 S. CITO AND D.A. LA MANNA Moreover, since B() = 0, we have Z Sn-1 xi(1 + tu(x))n+1 dHn-1 = 0 and then |a1,i| = Z Sn-1 xiu(x)dHn-1 nkuk2 L2(Sn-1). Thus, kuk2 L2(Sn-1) = a2 0 + n X i=1 a2 1,i + X k=2 Nk X i=2 a2 1,i X k=0 Nk X i=1 a2 k,i Cn2 (||u||2 W 1,2(Sn-1)) + X k=2 Nk X i=2 a2 k,i while kD uk2 L2(Sn-1) = X k=1 i=Nk X i=1 a2 2n X k=2 Nk X i=2 a2 k,i. If we combine the above two inequalities we get that kD uk2 L2(Sn-1) 2n - 1 - kuk2 L2(Sn-1) = 2n - 2 1 - kuk2 L2(Sn-1). (3.10) Using (3.7) and (3.10) we have N(# )D() - D(# )N() nn t2 2n n 2 (f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1) 2 kD uk2 L2(Sn-1) + t2 f(1)h(1) 2 - f(1)h0 (1) - f0 (1)h(1) 2(n - 1) kD uk2 L2(Sn-1) - C2t2 kD uk2 L2(Sn-1) t2 2n " n2 - 3n 2(n - 1) (f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1) 2 + n(f(1)h(1)) # kuk2 L2(Sn-1) - C2t2 kD uk2 L2(Sn-1). (3.11) To conclude the proof we are left to show that n2 - 3n n - 1 (f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1) + 2n(f(1)h(1)) > 0. (3.12) Now, in [14] authors prove that C(n, ) = (n - 1)(f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1) + 2n(f(1)h(1)) > 0. A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY 11 for any n 2 and > 0. Since it holds n2 - 3n n - 1 (f(1)h0 (1) - f0 (1)h(1)) + f(1)h00 (1) - f00 (1)h(1) + 2n(f(1)h(1)) = C(n, ) - n + 1 n - 1 (f(1)h0 (1) - f0 (1)h(1)), to prove (3.12) it is sufficient to show that (f(1)h0 (1) - f0 (1)h(1)) 0 for any n 2 and > 0. Now, in view of the definition of f and h, it results that f(1)h0 (1) - f0 (1)h(1) = f(1) = f2 (1) d dt h(t) f(t) t=1 = f2 (1) · d dt In 2 -1(t) In 2 (t) t=1 . In [3] it has been proved that the last derivative is negative; then (3.12) holds and the proof is concluded. As an immediate application of Lemma 3.1 and Proposition 3.2, we get the following Corollary 3.3. Let n 2 and a nearly spherical set parametrized by u W1, (Sn-1 ) with perimeter P() = nnn-1 . Then (# ) - () I(# ) - I() C1(n, )kD uk2 L2(Sn-1). (3.13) Moreover, the constant C1(n, ) depends continuously (actually analytically) on . A second corollary, actually the one we are interested in, regards the stability of the first eigenvalue for the Robin problem. Corollary 3.4. Let n 2 and > 0. There exists a constant C(n, , , ) such that if is a nearly spherical set parametrized by u W1, (Sn-1 ) with (# ) - () < , then (# ) - () C(n, , , )kD uk2 L2(Sn-1). (3.14) Proof. Let > 0 be fixed and let be chosen as in the statement. The map 7 () is continuous and monotonically decreasing from (0, +) onto (-, 0). Then, there exists such that () = (# ). Hence, (# ) - () = (# ) - (# ). Moreover, if = p |()|, for the rescaled sets and # it holds that Z |w|2 dx + Z w2 dx Z () w2 dHn-1 w H1 () 12 S. CITO AND D.A. LA MANNA and Z # |z|2 dx + Z # z2 dx Z (#) z2 dHn-1 z H1 (# ), with equality holding if w and z are, respectively, the eigenfunctions for the Robin problem on with parameter and on # with boundary parameter . In that case, the infimum for the Steklov problem (3.1) is achieved on and # , obtaining () = , (# ) = . Then, it follows that - p |()| = (# ) - (). Using the variational characterization of (# ) and (# ), if with u we denote the optimal function of # for the Robin problem with parameter , we have that (# ) - () = (# ) - (# ) ( - ) R # u2 dHn-1 R # u2 dx = C(n, , )( - ) C(n, , )((# ) - ()) C(n, , )C1(n, )kD uk2 L2(Sn-1), where C1 is the constant found in Proposition 3.2. Now we are able to prove our main stability result. Since the constant C1 continuously depends on we find that Lemma 3.5. Let m > 0. If is a convex set with with P() = m and () > 2(# ), then, there exists a positive constant C(m, n, ) such that diam() < C(m, n, ). Proof. Let us argue by contradiction and suppose that there exists a sequence (j)j of convex sets with P(j) = m such that (j) > 2(# ) and that diam(j) +. In view of the convexity of j and of the constraint P(j) = m, the sequence (j)j of the inradii of (j)j is necessarily vanishing. Recalling that, for any convex set A with inradius it holds |A| P(A) (see, for instance, [8], Prop. 2.4.3), we deduce that |j| vanishes as j goes to +. Now, using j as a test function for (j), we obtain (j) - P(j) |j| -, in contradiction with the lower bound on (j). The previous result can be even obtained as a particular case of the isodiametric control of the Robin spectrum proved in [9] also for higher eigenvalues in a wider class of sets. Another important result about the functional 7 () is the following upper semicontinuity issue. A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY 13 Lemma 3.6. If j, Rn are convex sets with j is measure, then lim sup j+ (j) (). Sketch of the proof. As shown in [9] and in [10], the upper semicontinuity of is based on the fact that, set u a (normalized) eigenfunction for (), the functional 7 Z u2 dHn-1 is lower semicontinuous (for a proof of this fact in a convex setting see [12]). Thus, () is an infimum of upper semicontinous functions, which is our thesis. In view of the upper semicontinuity of and of the isodiametric control in Lemma 3.5, since the ball is the unique maximizer of among convex sets with prescribed perimeter, it is immediate the following result. Lemma 3.7. If j is a sequence of convex sets with barycenter at the origin and (j) (B), then P(j) P(B) and dH(j, B) 0. Sketch of the proof. The diameters of the sets j are uniformly bounded. Up to subsequences, there exists an open bounded set such that dH(j, ) 0; in view of the maximality of the ball and of the upper semicontinuity of , we obtain that necessarily = B. In view of the previous lemma, we can restrict our main stability result to nearly spherical sets barycentered in the origin. Lemma 3.8. Let n 2 and > 0. There exists a positive constant 0 depending only on the dimension n, on the boundary parameter and on such that, if is a convex set with P() = nnn-1 and (# ) - () 0, then up to a translation is a nearly spherical set. Now we are able to prove our main result. Proof of Theorem 1.1. Lemma 3.8 ensures us that, if 0 is small enough, then we can suppose without loss of generality that our set is a nearly spherical set with barycenter at the origin. Thus, we can suppose = {x(1 + u(x)), x Sn-1 } and P() = nnn-1 = P(B). Let 1 such that |B1 | = ||. Since and B1 have the same measure, we have that n n Z Sn-1 (1 + u)n dHn-1 = nn 1 . (3.15) Define the function h := ((1 + u))n - n 1 . Expanding the integrand of the latter formula leads to n - n 1 = - 1 n Z Sn-1 n X 1 n k uk dHn-1 which immediately implies | - 1| < C(n)kuk. Hence it holds C3khk kuk C4khk, 14 S. CITO AND D.A. LA MANNA where C3 and C4 are constant depending only on the dimension. More over, the Leibnitz rule yelds D h = nn (1 + u)n-1 D u, hence to the control C5kD ukL2(Sn-1) kD hkL2(Sn-1) C6kD ukL2(Sn-1) where C5, C6 depend on the dimension and on . From (3.15), we know that h has zero integral, thus we can apply Lemma 2.6 to h and use the norm controls given above to infer ||u||n-1 L(Sn-1) kD ukL2(Sn-1) n = 2 4||D u||2 L2(Sn-1) log 8e||D u||n-1 L(Sn-1) ||D u||2 L2(Sn-1) n = 3 C(n)||D u||2 L2(Sn-1)||D u||n-3 L(Sn-1) n 4. We now prove our result only for n 4, since for n = 2, 3 the argument is the same. The above inequality and Lemma 2.7 lead to kukn-1 L(Sn-1) C(n)kD uk2 L2(Sn-1)kuk n-3 2 L(Sn-1), hence kuk n+1 2 L(Sn-1) C(n)kD uk2 L2(Sn-1). (3.16) Since kukL(Sn-1) = dH(, # ) = AH(), inequalities (3.16) and (3.14) give the result for n 4. Notice that, as expected, the final constant depends not only on the dimension, but also on the boundary parameter and on the size of the admissible sets for the maximization problem. 4. Further remarks and open problems In the same spirit of [18], it is possible to prove the sharpness of the quantitative "weighted" isoperimetric inequality considered in the previous section for the functional I(·). Theorem 4.1. Let n 2. There exists a family of convex sets (E) such that I(E# ) - I(E) 0, when 0 and I(E# ) - I(E) ' g(AH(E)) (4.1) where C is a suitable positive constant independent of . We do not write the proof of this last theorem since, due to the analytic and non decreasing nature of the Bessel functions, it follows straightforward as in [18]. However, for the reader convenience, we explicitly recall the sets which give the sharpness of the exponents. In the two dimensional case, to find the optimality of the exponent, one may use the "stadium" shaped sets (i.e. the convex hull of two balls with the same radius whose centers is connected by a segment passing through the origin) already considered in [1, 5]. A QUANTITATIVE REVERSE FABER-KRAHN INEQUALITY 15 If n 4, let ]0, /2[ and p Rn such that |p| = 1 cos . Then, we can define the set E as the convex hull of B {-p, p}. In the three dimensional case, the shape of the sets is quite different, due to the peculiar form of the function g. Let ]0, /2[ and consider the following function = () defined over S2 and depending only on the spherical distance , with [0, ], from a prescribed north pole S2 : = () = - sin2 log (sin ) + sin (sin - sin ) for sin sin - sin2 () log (sin ) for sin sin . (4.2) Let h := - , with the mean value of , i.e. = Z /2 0 () sin d = (1 - log 2) 2 + O(3 ), when goes to 0, and let R := (1 + 3h) 1/3 . The C1 function R = R() determines in polar coordinates (R, ) a planar curve. By rotating this curve about the line R, we determine the boundary of a convex and bounded set depending on the parameter . We denote that set by E. Remark 4.2. The result of Theorem 4.1 suggests that sharpness of inequality (1.5) can be expected, but at the moment it seems hard to be proved. An interesting question is whether the constraint on the perimeter is sufficient to ensure the ball to be the only solution for problem (1.3), even if the admissible sets are not necessarily convex (for instance, one can take the class of finite perimeter sets having the same perimeter). If one is able to prove in this more general framework the analogous of (1.4), in its quantitative version it should be suitable to replace the asymmetry function AH by some Fraenkel-type asymmetry function (for more details on the Fraenkel asymmetry, see [17]). Acknowledgements. The first author would thank both Professor Nicola Fusco and the second author for the hospitality and the support during the research visit at the University Federico II of Naples that inspired this work. The second author was partially supported by the Academy of Finland grant 314227. References [1] A. Alvino, V. Ferone and C. Nitsch, A sharp isoperimetric inequality in the plane. J. Eur. Math. Soc. 13 (2010) 185­206. [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000). [3] D.E. Amos, Computation of modified Bessel functions and their ratios. Math. Comput. 28 (1974) 239­251. [4] M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem. SIAM J. Math. Anal. 8 (1977) 280­287. [5] T. Bonnesen, Uber das isoperimetrische Defizit ebener Figuren. Math. Ann. 91 (1924) 252­268. [6] L. Brasco and G. De Philippis, 7 Spectral inequalities in quantitative form. In Shape optimization and spectral theory. Sciendo Migration (2017) 201­281. [7] F. Brock and D. Daners, Conjecture concerning a Faber-Krahn inequality for Robin problems. Vol. 4 of Mini-Workshop: Shape Analysis for Eigenvalues (Organized by: D. Bucur, G. Buttazzo, A. Henrot), Oberwolfach Rep (2007) 1022­1023. 16 S. CITO AND D.A. LA MANNA [8] D. Bucur and G. Buttazzo, Vol. 65 of Variational methods in shape optimization problems. Springer-Progress in Nonlinear Differential Equations and Their Applications (2004). [9] D. Bucur and S. Cito, Geometric control of the robin Laplacian eigenvalues: the case of negative boundary parameter. J. Geometr. Anal. 30 (2020) 4356­4385. [10] D. Bucur, V. Ferone, C. Nitsch and C. Trombetti, A sharp estimate for the first Robin-Laplacian eigenvalue with negative boundary parameter. Preprint arXiv:1810.06108 (2018). [11] D. Bucur, P. Freitas and J. Kennedy, Shape optimization and spectral theory, in The Robin Problem, edited by A. Henrot. De Gruyter Open, Warsaw (2017) 78­119. [12] S. Cito, Existence and regularity of optimal convex shapes for functionals involving the robin eigenvalues. J. Convex Anal. 26 (2019) 925­943. [13] L. Esposito, N. Fusco and C. Trombetti, A quantitative version of the isoperimetric inequality: the anisotropic case. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 4 (2005) 619­651. [14] V. Ferone, C. Nitsch and C. Trombetti, On a conjectured reverse Faber-Krahn inequality for a Steklov­type Laplacian eigenvalue. Commun. Pure Appl. Anal. 14 (2015) 63­82. [15] P. Freitas and D. Krejcirik, The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280 (2015) 322­339. [16] B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in Rn. Trans. Am. Math. Soc. 314 (1989) 619­638. [17] N. Fusco, The quantitative isoperimetric inequality and related topics. Bull. Math. Sci. 5 (2015) 517­607. [18] N. Gavitone, D.A. La Manna, G. Paoli and L. Trani, A quantitative Weinstock inequality for convex sets. Calc. Var. Partial Differ. Equ. 59 (2020) 2. [19] R. Schneider, Convex bodies: the Brunn­Minkowski theory. Number 151. Cambridge University Press (2014). [1] A. Alvino, V. Ferone and C. Nitsch, A sharp isoperimetric inequality in the plane. J. Eur. Math. Soc. 13 (2010) 185–206. [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000). [3] D. E. Amos, Computation of modified Bessel functions and their ratios. Math. Comput. 28 (1974) 239–251. [4] M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem. SIAM J. Math. Anal. 8 (1977) 280–287. [5] T. Bonnesen, Über das isoperimetrische Defizit ebener Figuren. Math. Ann. 91 (1924) 252–268. [6] L. Brasco and G. De Philippis 7 Spectral inequalities in quantitative form. In Shape optimization and spectral theory. Sciendo Migration (2017) 201–281. [7] F. Brock and D. Daners, Conjecture concerning a Faber-Krahn inequality for Robin problems. Vol. 4 of Mini-Workshop: Shape Analysis for Eigenvalues (Organized by: D. Bucur, G. Buttazzo, A. Henrot), Oberwolfach Rep (2007) 1022–1023. [8] D. Bucur and G. Buttazzo, Vol. 65 of Variational methods in shape optimization problems. Springer-Progress in Nonlinear Differential Equations and Their Applications (2004). [9] D. Bucur and S. Cito, Geometric control of the robin Laplacian eigenvalues: the case of negative boundary parameter. J. Geometr. Anal. 30 (2020) 4356–4385. [10] D. Bucur, V. Ferone, C. Nitsch and C. Trombetti, A sharp estimate for the first Robin-Laplacian eigenvalue with negative boundary parameter. Preprint (2018). [11] D. Bucur, P. Freitas and J. Kennedy, Shape optimization and spectral theory, in The Robin Problem, edited by A. Henrot. De Gruyter Open, Warsaw (2017) 78–119. [12] S. Cito, Existence and regularity of optimal convex shapes for functionals involving the robin eigenvalues. J. Convex Anal. 26 (2019) 925–943. [13] L. Esposito, N. Fusco and C. Trombetti, A quantitative version of the isoperimetric inequality: the anisotropic case. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 4 (2005) 619–651. [14] V. Ferone, C. Nitsch and C. Trombetti, On a conjectured reverse Faber-Krahn inequality for a Steklov–type Laplacian eigenvalue. Commun. Pure Appl. Anal. 14 (2015) 63–82. [15] P. Freitas and D. Krejčiřík, The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280 (2015) 322–339. [16] B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in ${𝐑}^n$. Trans. Am. Math. Soc. 314 (1989) 619–638. [17] N. Fusco, The quantitative isoperimetric inequality and related topics. Bull. Math. Sci. 5 (2015) 517–607. [18] N. Gavitone, D. A. La Manna, G. Paoli and L. Trani, A quantitative Weinstock inequality for convex sets. Calc. Var. Partial Differ. Equ. 59 (2020) 2. [19] R. Schneider, Convex bodies: the Brunn–Minkowski theory. Number 151. Cambridge University Press (2014). COCV_2021__27_S1_A25_049251cb1-5df2-4e74-a826-351f02ba568d cocv200129 10.1051/cocv/202007510.1051/cocv/2020075 Regularity analysis for an abstract thermoelastic system with inertial term Kuang Zhaobin 1 Liu Zhuangyi 23* Fernández Sare Hugo D. 4 1 Computer Science Department, Stanford University, Stanford, CA 94305, USA. 2 Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-2496, USA. 3 School of Mathematics, Beijing Institute of Technology, P.R. China. 4 Department of Mathematics, Federal University of Juiz de Fora, CEP 36036-900, Juiz de Fora, MG, Brazil. *Corresponding author: [e-mail] 01 03 2021 01 03 2021 2021 cocv/2021/01 Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). 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In this paper, we provide a complete regularity analysis for the following abstract thermoelastic system with inertial term

$$

where A is a self-adjoint, positive definite operator on a complex Hilbert space H and

$$

It is regarded as the second part of Fernández Sare et al. [J. Diff. Eqs. 267 (2019) 7085–7134]. where the asymptotic stability of this model was investigated. We are able to decompose the region E into three parts where the associated semigroups are analytic, of Gevrey classes of specific order, and non-smoothing, respectively. Moreover, by a detailed spectral analysis, we will show that the orders of Gevrey class are sharp, under proper conditions. We also show that the orders of polynomial stability obtained in Fernández Sare et al. [J. Diff. Eqs. 267 (2019) 7085–7134] are optimal.

Hyperbolic-parabolic equations analytic semigroup Gevrey class semigroup polynomial stability 35B65 35K90 35L90 47D03 93D05 idline ESAIM: COCV 27 (2021) S24 cover_date 2021 first_year 2021 last_year 2021 ESAIM: COCV 27 (2021) S24 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2020075 www.esaim-cocv.org REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM Zhaobin Kuang1 , Zhuangyi Liu2,3,* and Hugo D. Fernandez Sare4 Abstract. In this paper, we provide a complete regularity analysis for the following abstract thermoelastic system with inertial term utt + lA utt + Au - mA = 0, ct + mA ut + kA = 0, u(0) = u0, ut(0) = v0, (0) = 0, where A is a self-adjoint, positive definite operator on a complex Hilbert space H and (, , ) E = [0, + 1 2 ] × [0, 1] × [0, 1]. It is regarded as the second part of Fernandez Sare et al. [J. Diff. Eqs. 267 (2019) 7085­7134]. where the asymptotic stability of this model was investigated. We are able to decompose the region E into three parts where the associated semigroups are analytic, of Gevrey classes of specific order, and non- smoothing, respectively. Moreover, by a detailed spectral analysis, we will show that the orders of Gevrey class are sharp, under proper conditions. We also show that the orders of polynomial stability obtained in Fernandez Sare et al. [J. Diff. Eqs. 267 (2019) 7085­7134] are optimal. Mathematics Subject Classification. 35B65, 35K90, 35L90, 47D03, 93D05. Received June 3, 2020. Accepted November 5, 2020. 1. Introduction Let H be a complex Hilbert space with the inner product h · , · i and the induced norm k · k. We consider the following abstract system of coupled hyperbolic and parabolic equations: Keywords and phrases: Hyperbolic-parabolic equations, analytic semigroup, Gevrey class semigroup, polynomial stability. 1 Computer Science Department, Stanford University, Stanford, CA 94305, USA. 2 Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-2496, USA. 3 School of Mathematics, Beijing Institute of Technology, P.R. China. 4 Department of Mathematics, Federal University of Juiz de Fora, CEP 36036-900, Juiz de Fora, MG, Brazil. * Corresponding author: [e-mail] Article published by EDP Sciences c EDP Sciences, SMAI 2021 2 Z. KUANG ET AL. utt + lA utt + Au - mA = 0, ct + mA ut + kA = 0, u(0) = u0, ut(0) = v0, (0) = 0, (1.1) where A is a self-adjoint, positive definite (unbounded) operator on the complex Hilbert space H, m 6= 0, k > 0 and (, , ) E = 0, + 1 2 × [0, 1] × [0, 1]. Our main interest is the regularity of the solution to this system in terms of the parameters , , . For this purpose, we reformulate system (1.1) in a semigroup setting on the state space H = D(A 1 2 ) × D(A 2 ) × H, where any element in H is denoted by U = (u, v, w)T . The inner product in H is defined by h U1, U2 iH = h A 1 2 u1, A 1 2 u2 i + h v1, v2 i +lhA 2 v1, A 2 v2i + c h 1, 2 i, for all Ui = (ui, vi, i)T H, i = 1, 2, under which H is a Hilbert space. By denoting v = ut and U0 = (u0, v0, w0)T , system (1.1) can be written as an abstract linear evolution equation on the space H, dU dt (t) = A,,U(t), t 0, U(0) = U0, (1.2) where the operator A,, : D(A,,) H H is defined for 1 2 by A,, u v = v -(I + l A )-1 A A1- u - m - m c A v - k c A (1.3) with domain D(A,,) = n (u, v, )T H | v D(A1/2 ), D(A ), u D(A1- ), A1- u - m D(A-/2 ) H o . For > 1 2 it is defined by A,, u v = v -(I + l A )-1 A A1- u - m -A- 1 2 m c A 1 2 v + k c A-(- 1 2 ) (1.4) REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 3 with domain D(A,,) = n (u, v, )T H | v D(A1/2 ), A1- u - m D(A-/2 ), A-(-1/2) H, A-1/2 h mA1/2 v + kA-(-1/2) i H o . It is known that A,, generates a C0-semigroup eA,, t of contractions on H, see [8]. Then the solution to the evolution equation (1.2) admits the following representation: U(t) = eA,, t U0, t 0, which leads to the well-posedness of (1.2). With this in hand, in general, regularity and stability are the main properties that attract the researchers' attention [3, 4, 13, 18] where these properties are associated with regularity/stability results of their corresponding C0 semigroups. Then, before going further, let us recall the definitions of these properties. Definition 1.1. Let eAt be a C0-semigroup on a Hilbert space H. 1. The semigroup eAt is said to be analytic if there exists an extension T() of eAt to the following set { C | arg | < } {0}, for some (0, 2 ) such that, for any x H, 7 T()x is continuous on satisfying the following semigroup property T(1 + 2) = T(1)T(2), 1, 2 , with 1 + 2 , and 7 T() is analytic over \ {0} in the uniform operator topology of L(H) (the space of all linear bounded operators from H to H). 2. Semigroup eAt is said to be of Gevrey class (with > 1) if it is infinitely differentiable and, for any compact set K (0, ) and any > 0, there exists a constant K = K(, K), such that kAn eAt kL(H) Kn (n!) , t K, n 0. (1.5) 3. Semigroup eAt is said to be differentiable if, for any x H, t 7 eAt x is differentiable on (0, ). 4. Semigroup eAt is said to be exponentially stable with decay rate > 0 if there exists a constant M 1 such that keAt k Me-t , t 0. 5. Semigroup eAt is said to be polynomially stable of order j > 0 if there exists a constant M > 0 such that keAt A-1 k Mt-j , t > 0. Note that, in Definition 1.1, the first three items are about the regularity of C0-semigroups ([5, 7, 14, 17, 19]) and the last two are about the asymptotic stability of C0-semigroups ([6, 7, 17]). Also note that, if = 1 in (1.5), the associated semigroup is analytic; see [19]. Returning to system (1.1), the case of = 0 was first considered in [1] for exponential stability. Then the region of C smoothness was founded in [16]. A complete picture of the stability and regularity of this model were given in [9, 10]. In the case > 0, the analysis become more demanding due to the following challenges: How to divide the parameter region E into subregions with specific stability and regularity properties due to 4 Z. KUANG ET AL. Figure 1. Regions S1, S2, and S3 for some 's. the additional parameter ? How to determine the order of polynomial stability and Gevrey class in terms of the parameters? How to prove these properties in each subregion? Recently, the stability analysis of the system (1.1) has been reported in [8]. For reader's convenience, we give a brief summary here. 1. The semigroup eA,, t is exponentially stable in S1, 2. The semigroup eA,, t is polynomially stable of order > - 1 µ(,,) in S2 S3 with µ(, , ) = 2 - - (1 - )/2 , (, , ) S2; 2 + - 1 (1 - )/2 , (, , ) S3. (1.6) where S1 = n (, , ) E : 1 2 + , 2 - 1 o , S2 = (, , ) E : 0 < + 2 , 1 - 2 1 , S3 = (, , ) E : 0 < 1 - 2 , 0 < 1 - 2 . Examples of regions S1, S2, and S3 are given in Figure 1. As mentioned previously, the aim of this paper is on conducting a regularity analysis of system (1.1). We organize the paper as follows. In Section 2, we will summarize the main results of the paper. Specifically, (a) We will decompose the parameter region E into three parts where the semigroup associated with the system is analytic, of specific order of Gevrey class, and non-smoothing, respectively. (b) By a detailed spectral analysis, we will show that the orders of Gevrey class is sharp, under proper conditions. (c) We also show that the orders of polynomial stability obtained in [8] are optimal. In Section 3, we provide a list frequency domain characterization of semigroup properties, useful to proof our results. Section 4 is devoted to the Proof of main result (a). Finally, proofs of results (b) and (c) are provided in Section 5 and Section 6. REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 5 Figure 2. Regions given in (2.1). The figures are plotted choosing = 1 8 . We would like to remark here that the method we introduce in Section 5-6 to analyze the asymptotic behavior of the eigenvalues of the system, which is a cubic equation with coefficients depending on three parameters , , in region E, is crucial to our success. Before ending this section, we give an example to illustrate the motivation of our investigation. Example: The thermoelastic plate equation utt(x, t) - lutt(x, t) = -E2 (x, t) + k(x, t), in × (0, ), ct(x, t) = -ut(x, t) + k(x, t), in × (0, ), u(x, t) = u(x, t) = (x, t) = 0, on × (0, ), u(x, 0) = u0(x), u(x, 0) = u1(x), (x, 0) = 0(x) in . (1.7) It is known that the semigroup associated with this system is analytic when l = 0, but is not analytic when l > 0, see for example [11­13]. This system falls into the framework of system (1.1) for A = -, (, , ) = (1 2 , 1 2 , 0 or 1). For the sake of simplicity and without loss of generality, in our analysis we assume l = = = m = c = k = 1. 2. Main results To state the main results, we first divide the parameter region E into the following subregions: R1 = {(, , ) | 1, -2 + + 1 > 0, 2 - 2 - 0, 4 - 2 - - 1 0, 0 1} , R2 = (, , ) | 1- 2 1, 4 - 2 - - 1 < 0, 2 - - > 0, 0 < 1 , R3 = (, , ) | < 1 2 , < 1- 2 , 2 + - 1 > 0, 0 < 1 , R4 = (, , ) | 1 2 , 2 - 2 - > 0, -2 + + 1 > 0, 0 < 1 , R5 = E \ (R1 R2 R3 R4). (2.1) The regions defined in (2.1) are visualized in Figure 2. Additionally, Figure 3 shows how these regions change at various 's. 6 Z. KUANG ET AL. Figure 3. Regions defined by (2.1) at various 's. With the regions defined above, we can decompose E into three parts where the semigroup associated with (1.1) is analytic, of specific order of Gevrey classes, and non-smoothing, respectively via Theorem 2.1. Theorem 2.1. The semigroup eA,, t has the following regularity properties: (i) It is analytic in R1; (ii) It is of Gevrey class > 1 µ(,,) in R2 R3 R4 with µ(, , ) = 2-- (1-)/2 , (, , ) R2; 2+-1 (1-)/2 , (, , ) R3; -/2 , (, , ) R4. (2.2) (iii) It is not differentiable in R5. REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 7 It should be noticed that when = 1, the analytic region R1 degenerates to the line -2 + + 1 = 0. Furthermore, the sharpness of the orders of Gevrey class corresponding to (2.2) of Theorem 2.1 as well as the sharpness of the orders of polynomial stability corresponding to (1.6) can be characterized by Theorem 2.2: Theorem 2.2. If A admits a sequence of eigenvalues µn R such that limn µn = , then the Gevrey class orders in (2.2) of Theorem 2.1 are sharp in the following sense: For any > 0, the semigroup is not Gevrey class of order 1 µ(,,)+ . Furthermore, the orders of polynomial stability in (1.6) are also sharp: for any > 0, the semigroup is not polynomially stable of order - 1 µ(,,)+ . The significance of the above results includes the following: ­ System (1.1) covers a variety of partial differential equations. Within this framework, the stability and regu- larity problem of the system are now completely resolved. It is possible to extend the results to the case when operators in the system are not in the form of fractional power of A, but equivalent to it in some sense. See [14] for a simpler system. ­ The semigroup properties obtained in [8] and this paper provides important information for the study of nonlinear evolution equations related to the linear system (1.1). ­ The asymptotic analysis of eigenvalues for system (1.1) is crucial to precisely divide the parameter region E into subregions corresponding to the semigroup properties. The method designed presented in Section 5-6 provides a powerful mean to fulfill this formidable task, which can be applied to other type of general systems with parameters. 3. Frequency domain characterization of semigroup's properties In this section, we will present the frequency domain characterization of semigroup properties, which are our tools of proving the regularity results in Theorem 2.1. Lemma 3.1. Let A : D(A) H H generate a C0-semigroup eAt on H such that keAt k M, t 0, (3.1) for some M 1 and i (A), R, || large enough. (3.2) Then the following hold: 1. Semigroup eAt is analytic if and only if for some a R and b, C > 0 such that (A) (a, b) n C Re > a - b|Im | o , (3.3) and k(i - A)-1 k C 1 + || , (a, b). (3.4) This is the case if and only if lim R, || || k(i - A)-1 k < . (3.5) 8 Z. KUANG ET AL. 2. Semigroup eAt is of Gevrey class > 1 if and only if for any b, > 0, there are constants a R and C > 0 depending on b, , such that (A) b() n C Re > a - b|Im | 1 o , (3.6) and k(i - A)-1 k C e-Re + 1 , b(). (3.7) This is the case, in particular, if for some µ (-1 , 1), lim R, || ||µ k(i - A)-1 k < . (3.8) 3. Semigroup eAt is differentiable if and only if for any b > 0, there are constants ab R and Cb > 0 such that (A) b n C Re > ab - b log |Im | o , (3.9) and k(i - A)-1 k Cb|Im |, b, Re 0. (3.10) This is the case, in particular, if lim R, || log ||k(i - A)-1 k = 0. (3.11) 4. Semigroup eAt is exponentially stable if and only if i (A), R, (3.12) and lim R,|| k(i - A)-1 k < . (3.13) 5. Semigroup eAt is polynomially stable of order j > 0 if and only if (3.5) holds and lim R,|| ||- 1 j k(i - A)-1 k < . (3.14) For notational simplicity, hereafter, we write i - A instead of iI - A, omitting I. In the above result, the regularity and stability properties of the semigroup eAt are deliberately related to the spectral/resolvent of the generator A. Practically, we will use the limit relations (3.5), (3.8) and (3.11) to establish the regularity property of the semigroup, and use the spectrum relations (3.3), (3.6) and (3.9) to show that the relevant indices are sharp. The following corollary will be useful below. See [10]. REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 9 Corollary 3.2. 1. Suppose (A) contains a sequence n such that lim n Re n = a, lim n |n| = , (3.15) for some a R, then the semigroup eAt is not differentiable. 2. Suppose there exists a sequence n (A) such that lim n Re n |Im n| 1 = 0, (3.16) then eAt is not of Gevrey class . 4. Proof of Theorem 2.1 In this section, we will prove Theorem 2.1. It is technical and lengthy. Let us now make some preparations. First of all, in our proof, the following interpolation theorem will play a crucial role. Lemma 4.1. Let A : D(A) H be self-adjoint and positive definite. Then kAp xk kAq xk p-r q-r kAr xk q-p q-r , 0 r p q, x D(Aq ). (4.1) In particular, for any [0, 1 2 ], one has (with r = 0, p = , and q = 1 2 ) kA xk kA 1 2 xk2 kxk1-2 , x D(A 1 2 ), (4.2) and for any [1 2 , 1] (with r = 1 2 , p = , and q = 1) kA xk kAxk2-1 kA 1 2 xk2(1-) , x D(A). (4.3) Next, for any R, and any U (u, v, w)T D(A,,), (i - A,,)U = iu - v iv + (I + A ) -1 A (A1- un - n) i + T(vn, n) , (4.4) where T(vn, n) := ( A vn + A n , for 1/2 A- 1 2 (A 1 2 vn + A-+ 1 2 n) , for > 1/2. (4.5) Our proof for Theorem 2.1 will be based on contradiction arguments. Suppose for some given (, , ) S and µ [0, 1], without having any specific relations among them, that the following is not true: lim R, || ||µ k(i - A,,)-1 k < . 10 Z. KUANG ET AL. Then there exists a sequence {(n, Un) n 1} R × D(A,,) with Un (un, vn, n)T , and lim n |n| = , kUnk2 H = kA 1 2 unk2 + kvnk2 + kA 2 vnk2 + knk2 = 1, n 1, (4.6) such that lim n |n|-µ k(in - A,,)UnkH = 0, (4.7) i.e. (note (4.4)), without loss of generality, n > 0, i1-µ n A 1 2 un - -µ n A 1 2 vn = o(1), in H, (4.8a) i1-µ n (I + A )vn + -µ n A (A1- un - n) - 0, in D(A 2 )0 , (4.8b) i1-µ n n + -µ n T(vn, n) = o(1) in H, (4.8c) where (4.8b) is derived by acting (I + A ) on the second component of (4.4). Hereafter, if not specified, o(1) stands for a vector in H (or a quantity in R) which goes to zero as n , and O(1) a bounded vector in H (or uniformly in n 1, in R). The advantage of using such a notation is that (4.8a)­(4.8c) can be regarded as a system of equations, which will be convenient to the analysis later on. Furthermore, acting A- 2 on (4.8b) yields, i1-µ n A- 2 + A 2 vn + -µ n A- 2 (A1- un - n) = o(1), (4.9) where the l.h.s. of (4.9) H. For the sequence {(n, un, vn, n)} satisfying (4.8a)­(4.8c), we have the following result. Lemma 4.2. For the sequence {(n, un, vn, n)} satisfying (4.6)­(4.7), the following convergences are true: - µ 2 n A 2 n = o(1), (4.10a) ikvnk2 + ikA 2 vnk2 - ikA 1 2 unk2 - h -1+ µ 2 n A 2 n, - µ 2 n A- 2 vni = o(1). (4.10b) -1 n A- 2 (A1- un - n) = O(1). (4.10c) i1-µ n knk2 + -µ n hA 1 2 vn, A- 1 2 ni = o(1). (4.10d) Proof. To show (4.10a) is true, consider: Reh-µ n (in - A,,) Un, UniH = -Reh-µ n A,,Un, UniH = - -µ n Re hA 1 2 vn, A 1 2 uni + hA-/2 (A1- un - n), A/2 vni + hT(vn, n), ni =k - µ 2 n A 2 nk2 . (4.11) REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 11 For the analysis of the last term, see reference [8]. Then, by (4.6) and (4.7), we obtain k - µ 2 n A 2 nk2 = o(1). To show (4.10b) is true, we see that since µ [0, 1], µ-1 n A 2 vn is bounded in H because of (4.6). Taking the inner product of (4.9) with µ-1 n A 2 vn in H yields: hi (I + A ) vn, vni + h-1 n A- 1 2 (A1- un - n), A 1 2 vni = ikvnk2 + ikA 2 vnk2 + hA 1 2 un, -1 n A 1 2 vni - -1 n hA- 1 2 n, A 1 2 vni = ikvnk2 + ikA 2 vnk2 - ikA 1 2 unk2 - h -1+ µ 2 n A 2 n, - µ 2 n A- 2 vni + o(1) = o(1), where A- 1 2 n H because - 1 2 2 in S1. Also, the penultimate equality can be obtained by multiplying (4.8a) with the bounded µ-1 n to yield iA 1 2 un = -1 n A 1 2 vn + o(1). (4.10c) follows from multiplying (4.9) by the bounded µ-1 n and the fact that kA 2 vnk 1. Finally, (4.10d) is obtained from the inner product of (4.8c) with n in H and using (4.10a). Theorem 4.3. Let µ(, , ) = 1, (, , ) R1 2(2--) 1- , (, , ) R2 2(2+-1) 1- , (, , ) R3 2 2- , (, , ) R4 (4.12) where Ri, i = 1, 2, 3, 4 are defined by (2.1). Then lim R, || ||µ(,,) k(i - A,,)-1 k < . (4.13) Proof. In order to prove the result, let us argue by contradiction arguments. In fact, assuming that (4.13) is false, then the convergence (4.6)­(4.9) and also Lemma 4.2 hold. We will show that kUnkH = o(1), which is a contradiction with (4.6). We divide our analysis by cases, depending on the region Ri, i = 1, 2, 3, 4. Case 1. Let (, , ) R1 where R1 = {(, , ) | 1, -2 + + 1 0, 2 - 2 - 0, 4 - 2 - - 1 0, 0 1} , and is shown in Figure 4 where is 1 4 , 1 2 , and 3 4 , respectively. Take µ = 1. The inner product of (4.8c) with A- vn in H, which is bounded since - 2 in R1 and kA 2 vnk is bounded, gives hin, A- vni + k - 1 2 n A- 2 vnk2 + -1 n hA- 1 2 n, A 1 2 vni = o(1). (4.14) It follows from (4.10d) and µ = 1 that -1 n (A- 1 2 n, A 1 2 vn) is bounded. Therefore, we obtain from (4.14) that k - 1 2 n A- 2 vnk2 = O(1), (4.15) because hin, A- vni is also bounded. Consequently, (4.15) implies that 12 Z. KUANG ET AL. Figure 4. Visualization of R1 at various 's. -1 n hA- 1 2 n, A 1 2 vni = h - 1 2 n A 2 n, - 1 2 n A- 2 vni k - 1 2 n A 2 nkk - 1 2 n A- 2 vnk = o(1), (4.16) where we have used also (4.10a). Now, returning to (4.10d), the estimate (4.16) implies that knk = o(1). (4.17) On the other hand, from (4.8c) and (4.17) we have -1 n A- 1 2 A 1 2 vn + A-+ 1 2 n = o(1). (4.18) Since 1 2 in R1, kA1- unk = O(1) by (4.6). Then, taking the inner product of (4.18) with A1- un in H, we obtain o(1) = h-1 n A- 1 2 (A 1 2 vn + A-+ 1 2 n), A1- uni = h-1 n (A 1 2 vn + A-+ 1 2 n), A 1 2 uni = ikA 1 2 unk2 + o(1) + -1 n hA-+ 1 2 n, A 1 2 un - A- 1 2 ni + -1 n hA-+ 1 2 n, A- 1 2 ni = - ikA 1 2 unk2 + o(1) + h - 1 2 n A 2 n, - 1 2 n A 2 (A1- un - n)i + -1 n kA 2 nk2 , (4.19) where the penultimate equality is due to (4.8a). It follows from (4.9) that k-1 n A- 2 (A1- un - n)k = O(1). (4.20) By taking the inner product of (4.20) with A1-2++ 2 un in H, which is bounded due to 1 - 2 + + 2 1 2 in R1, we arrive at h-1 n A- 2 (A1- un - n), A1-2++ 2 uni = h - 1 2 n A- 2 (A1- un - n), - 1 2 n (A1-2++ 2 un - A-++ 2 ni + h - 1 2 n A- 2 (A1- un - n), A-++ 2 ni = k - 1 2 n A 2 (A1- un - n)k2 - h - 1 2 n A 2 (A1- un - n), - 1 2 n A 2 ni = O(1). (4.21) REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 13 Figure 5. Visualization of R2 at various 's. Therefore, by (4.10a) and (4.21), we have that k - 1 2 n A 2 (A1- un - n)k2 = O(1). (4.22) It follows from (4.10a), (4.19), and (4.22) that kA 1 2 unk = o(1), (4.23) which further implies in reference of (4.10a), (4.10b), and (4.15) that kvnk2 + kA 2 vnk2 = o(1). (4.24) From (4.17), (4.23), and (4.24), we have proved kUnkH = 0, which is a contradiction to (4.6). Case 2. Let (, , ) R2, where R2 = (, , ) | 1 - 2 1, 4 - 2 - - 1 < 0, 2 - - > 0, 0 < 1 , and is shown in Figure 5 where is 1 4 , 1 2 , and 3 4 , respectively. Taking µ = 2(2 - - ) 1 - . Note that in R2, 2 < - 2 < 1 2 , where we have used (0, 1). Then, using the interpolation's inequality (4.1), k - µ 2 n A- 2 vnk - µ 2 n kA 1 2 vnk - 2 - 2 1 2 - 2 kA 2 vnk 1 2 -+ 2 1 2 - 2 = - µ 2 n kA 1 2 vnk 2-- 1- kA 2 vnk 1-2+ 1- = - µ 2 n kA 1 2 vnk 2-- 1- kA 2 vnk1- 2-- 1- = k-1 n A 1 2 vnk µ 2 kA 2 vnk1- µ 2 , (4.25) where µ 2 (0, 1) in S2, Hence, by (4.6) and (4.8a), k - µ 2 n A- 2 vnk = O(1), (4.26) 14 Z. KUANG ET AL. which, combined with (4.10a), further leads to -µ n hA 1 2 vn, A- 1 2 ni = h - µ 2 n A- 2 vn, - µ 2 n A 2 ni = o(1). (4.27) It follows from (4.10d) and (4.27) that k 1-µ 2 n nk = o(1) knk = o(1). (4.28) Dividing (4.8c) by 1-µ 2 n , and reference to (4.28), we have - 1+µ 2 n A- 1 2 A 1 2 vn + A-+ 1 2 n = o(1). (4.29) On the other hand, when 1 2 , take b = 1-µ 2 which is in (0, 1). It is clear that k-b n A1- unk = o(1) by (4.6). When < 1 2 , let b = 1-2 1- . Then b (0, 1) because in R2, 2 - - > 0 2 - > 0 1 - 2 < 1 - b (0, 1). Furthermore, when < 1 2 , we have that - 1 2 < 0 < - 2 . Then, by interpolation we have -b n kA1- un - nk k-1 n A- 2 (A1- un - n)kb kA- 1 2 (A1- un - n)k1-b = O(1), (4.30) since k-1 n A- 2 (A1- un - n)k is bounded due to (4.9) and kA- 1 2 (A1- un - n)k = kA 1 2 un - A- 1 2 nk kA 1 2 unk + kA- 1 2 nk is also bounded because we are considering < 1 2 . In this case we also obtain the boundedness of k-b n A1- unk because k-b n nk is bounded. For both 1 2 and < 1 2 , the inner product of (4.29) with -b n A1- un gives - 1+µ 2 -b n hA 1 2 vn, A 1 2 uni + - 1+µ 2 -b n hA 2 n, A1-+ 2 uni = o(1). (4.31) We can multiple (4.31) by -1+ 1+µ 2 +b n to get h-1 n A 1 2 vn, A 1 2 uni + -1 n hA 2 n, A1-+ 2 uni = o(1) ikA 1 2 unk2 + h - µ 2 n A 2 n, -1+ µ 2 n A1-+ 2 uni = o(1), (4.32) where we have used the fact that for 1 2 , 1+µ 2 + b = 1; and for < 1 2 , 1+µ 2 + b = 3 2 - 1- 1 because 1- 2 in R2. On the other hand, k -1+ µ 2 n A1-+ 2 unk = k -1+ µ 2 n A 2 (A1- un - n) + -1+ µ 2 n A 2 nk k -1+ µ 2 n A 2 (A1- un - n)k + k -1+ µ 2 n A 2 nk = k -1+ µ 2 n A 2 (A1- un - n)k + o(1), (4.33) where for the last equality we have used (4.10a) and the fact that -1 + µ 2 < -µ 2 because µ < 1 by the fact that 4 - 2 - - 1 < 0 in R2. Furthermore, let c = 1-2+ 1- , by - 1 2 < 2 < - 2 in R2 and (4.30), -1+ µ 2 n kA 2 (A1- un - n)k -1+ µ 2 +c n k-1 n A- 2 (A1- un - n)kc kA- 1 2 (A1- un - n)k1-c = O(1), (4.34) where -1 + µ 2 + c = 0 and hence c (0, 1). REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 15 Figure 6. Visualization of R3 at various 's. Applying (4.10a), (4.33), and (4.34) to (4.32), we conclude that kA 1 2 unk = o(1). (4.35) With(4.10a), (4.10b), (4.26), (4.28), and (4.35), and by an argument similar to that presented in Case 1, we have proved kUnkH = 0, which is a contradiction to (4.6). Case 3. Let (, , ) R3, where R3 = (, , ) | < 1 2 , < 1 - 2 , 2 + - 1 > 0, 0 < 1 , and is shown in Figure 6 where is 1 4 , 1 2 , and 3 4 , respectively. Take µ = 2(2+-1) 1- . Then µ (0, 1). µ > 0 is due to 2 + - 1 > 0 and < 1 in R3. To see µ < 1, consider < 1- 2 1 > 2 1- 1 > 2+-1 > 2(2+-1) 1- = µ, where 1 > 2+-1 is due to < 1 2 in R3. Note that 2 < - 2 < 1 2 in R3, where the first inequality holds because > 1- 2 > + 2 due to the fact that 2 + - 1 > 0 and < 1- 2 in R3, and the second inequality is due to < 1 2 in R3. Let a = 2-- 1- . Then a (0, 1) because the foregoing inequality 2 < - 2 < 1 2 . -a n kA- 2 vnk k-1 n A 1 2 vnka kA 2 vnk1-a = O(1). (4.36) Note that a - µ 2 = 1 - 2 1- > 0 because < 1- 2 , and 1 - µ 2 - a = 2-4 1- > 0 because < 1 2 and < 1. Since a - µ 2 > 0, dividing (4.10d) by a- µ 2 n yields, o(1) = 1- µ 2 -a n knk2 + - µ 2 -a n hA 1 2 vn, A- 1 2 ni = 1- µ 2 -a n knk2 + h-a n A- 2 vn, - µ 2 n A 2 ni = 1- µ 2 -a n knk2 + o(1) k 1 2 - µ 4 - a 2 n nk2 = o(1) knk2 = o(1), (4.37) where we have used (4.10a) and (4.36) to show h-a n A- 2 vn, - µ 2 n A 2 ni = o(1). 16 Z. KUANG ET AL. Figure 7. Visualization of R4 at various 's. Note that 1 2 - 3µ 4 + a 2 = 2-2-2- 1- > 0 because < 1 and 1 - - 2 > 0 > 2 - 1 in R3. Dividing (4.8c) by 1 2 - 3µ 4 + a 2 n and in reference of (4.37), we obtain i 1 2 - µ 4 - a 2 n n + - 1 2 - µ 4 - a 2 n A- 1 2 A 1 2 vn + A-+ 1 2 n = o(1) - 1 2 - µ 4 - a 2 n A vn + A n = o(1). (4.38) Note that -1 2 + µ 4 + a 2 = 2-1 1- = -b in (4.30). Because - 1 2 < 0 < - 2 still holds, where we have used 2 < - 2 for to show 0 < - 2 , (4.30) is still true for R3. Then, k - 1 2 + µ 4 + a 2 n A1- unk = O(1) by (4.30). Taking the inner product of (4.38) with - 1 2 + µ 4 + a 2 n A1- un in H yields, hA 1 2 un, -1 n A 1 2 vni + h -1+ µ 2 n A1-+ 2 un, - µ 2 n A 2 ni = o(1) ikA 1 2 unk2 + h -1+ µ 2 n A1-+ 2 un, - µ 2 n A 2 ni = o(1), (4.39) where we have used (4.10a) and µ (0, 1). Again, we only need to show the boundedness of k -1+ µ 2 n A1-+ 2 unk to get kA 1 2 unk = o(1). By (4.34), -1+ µ 2 n (A1-+ 2 un - A 2 n) = o(1) because -1 + µ 2 + c = -1 + 2 1- < 0. Therefore, -1+ µ 2 n A1-+ 2 un = -1+ µ 2 n A 2 n = o(1), where we have used the fact that -1 + µ 2 < -µ 2 and (4.10a). With(4.10a), (4.10b), (4.36), (4.37), and (4.39), and by an argument similar to that presented in Case 1, we have proved kUnkH = 0, which is a contradiction to (4.6). Case 4. Let (, , ) R4, where R4 = (, , ) | 1 2 , 2 - 2 - > 0, -2 + + 1 > 0, 0 < 1, and is shown in Figure 7 where is 1 4 , 1 2 , and 3 4 , respectively. Take µ = 2 2- , then µ (0, 1) because 2 - 2 - > 0 in R4. Note that - + 2 < 0 in R4. Thus, kA 2 vn + A-+ 2 nk = O(1). (4.40) REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 17 On the other hand, k-1 n A- 2 (A 2 vn + A-+ 2 )nk = O(1) (4.41) by (4.8c). Acting A-+ 2 to (4.8c) yields, 1-µ n A-+ 2 n + -µ n A (A 2 vn + A-+ 2 n) = o(1). (4.42) Since - 2 > 0 in R4, by interpolation, we obtain, -µ n kA (A 2 vn + A-+ 2 n)k k-1 n A- 2 (A 2 vn + A-+ 2 n)kµ kA 2 vn + A-+ 2 nk1-µ . (4.43) By (4.40), (4.41), and (4.43), -µ n kA (A 2 vn + A-+ 2 n)k = O(1). (4.44) It follows from (4.42) and (4.44) that k1-µ n A-+ 2 nk = O(1). (4.45) Let b = 2-2- 2-- (0, 1] in R4. It is straightforward to verify the following equality - µ 2 b + (1 - µ)(1 - b) = 0. (4.46) Then, with 0 < - - 2 < - 2 - 2 and by interpolation, knk =kA-- 2 (A-+ 2 n)k kA- 2 - 2 (A-+ 2 n)kb kA-+ 2 nk1-b =kA 2 nkb kA-+ 2 nk1-b = k - µ 2 n A 2 nkb k1-µ n A-+ 2 nk1-b = o(1), (4.47) where the penultimate equality is due to (4.46). Next, note that 1 - < 1 2 in R4, which implies that kA1- unk = O(1). Here, we have used (4.8a). Taking inner product of (4.8c) with µ-1 n A1- un yields, o(1) = ihA1- un, ni + h-1 n A 1 2 (A 1 2 vn + A-+ 1 2 n), uni = h-1 n A 1 2 vn, A 1 2 uni + h-1 n A-+ 1 2 n, A 1 2 uni = ikA 1 2 unk2 + o(1), (4.48) where we have used (4.6), (4.8a), (4.47), - + 1 2 < 2 , and -1 < -µ 2 to obtain the last equality. Finally, we take inner product of (4.8c) with µ-1 n n to get o(1) = iknk2 + -1 n hA 1 2 vn, A- 1 2 ni + -1 n kA 2 nk2 = h-1 n A 1 2 vn, A- 1 2 ni, (4.49) where the last equality is obtained similarly to that in (4.48). With (4.10b), (4.47), (4.48), and (4.49), and by an argument similar to that presented in Case 1, we have proved kUnkH = 0, which is a contradiction to (4.6). 18 Z. KUANG ET AL. 5. Asymptotic behavior of eigenvalues In this section, we are going to study the asymptotic behavior of some eigenvalue sequences for the operator A,,. Recall that we assume there exists a sequence µn of eigenvalues of A such that 0 < µ1 µ2 · · · , lim n µn = . The following lemma is a direct consequence. Lemma 5.1. Let f(, µ) = (1 + µ )3 + (µ+ + µ )2 + (µ + µ2 ) + µ+1 , (, µ) C × R+, (5.1) with = 1. If the following holds: f(n, µn) = 0, (5.2) then n is an eigenvalue of A,,. Now, for any n 1, we consider the following equation: f(, µn) (1 + µ n)3 + (µ+ n + µ n)2 + (µ2 n + µn) + µ+1 n = 0. (5.3) Let us denote an = 1 + µ n, bn = µ+ n + µ , cn = µ2 n + µn, dn = µ+1 n . (5.4) Then (5.3) takes the following form: an3 + bn2 + cn + dn = 0, (5.5) with an, bn, cn, dn R+. Let hn = 3ancn - b2 n, gn = 2b3 n - 9anbncn + 27a2 ndn, pn = hn 3a2 n , qn = gn 27a3 n . (5.6) Define n = qn 2 2 + pn 3 3 = Wn 108a4 n , where Wn = 27a2 nd2 n - 18anbncndn + 4anc3 n + 4b3 ndn - b2 nc2 n. (5.7) We also define n,± = - qn 2 ± p n - qn 2 ± rqn 2 2 + pn 3 3 . (5.8) In our paper, for any = ||ei , we define 1 2 = || 1 2 ei 2 , and 1 3 = || 1 3 ei 3 . Note that since an, bn, cn, dn R+, n is either real or purely imaginary. Also notice that n,+n,- = - pn 3 3 . Since pn is real, we have that REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 19 1 3 n,+ 1 3 n,- = -pn 3 . This statement is obviously true when n is real. On the other hand, when n is imaginary, 1 3 n,+ and 1 3 n,- are complex conjugate so the statement still holds. Notice that n,+ + n,- = -qn, n,+ - n,- = 2 p n. We can define An and Bn as: An = 1 3 n,+ + 1 3 n,- = -qn 2 3 n,+ - 1 3 n,+ 1 3 n,- + 2 3 n,- , Bn = 1 3 n,+ - 1 3 n,- = 2 n 2 3 n,+ + 1 3 n,+ 1 3 n,- + 2 3 n,- . (5.9) With the above notations, we have the following result ([15]) regarding the roots of (5.5). Lemma 5.2 (Cardano's formula). Equation (5.5) admits three roots which are given by the following: ( n,0 = An - bn 3an , n,± = -1 2 An - bn 3an ± i 3 2 Bn. (5.10) As mentioned in the beginning of this section, we are interested in the asymptotic behavior of some eigenvalue sequences for the operator A,,. Specifically, with the Cardano's formula, we are interested in characterizing the leading terms of n,0 and n,± as n . This will be the focus of the rest of the section. For the ease of presentation, we first partition G = {(, , ) | 0 1, 0 1, 0 < 1} (5.11) into various regions, and report the leading terms of n,0 and n,± in corresponding regions in Section 5.1. It should be noticed that E G. Therefore, by identifying the leading terms of n,0 and n,± as in G, we will have complete knowledge of these leading terms in E. The leading terms of n,0 and n,± in E are also reported in Section 5.1. As of how we can determine the leading terms of n,0 and n,±, based on the Cardano's Formula (5.10), we start from determining the leading terms of An and Bn as n . This is the focus in Section 5.2. In our derivation, we will show that just using the leading term of An does not suffice to estimate the leading terms of n,0 and n,± as desired. This motivates us to further characterize the lower order terms of An, which will be the focus of Section 5.3. Upon acquiring these lower order terms, we can compute the leading terms of n,0 and n,±, which is reported in Section 5.4. 5.1. Partition of G and leading terms of n,0 and n,± The partition of G is given by Definition 5.3. 20 Z. KUANG ET AL. Figure 8. Partition of G according to Definition 5.3, choosing = 1 8 . The dotted lines represent the contours of various regions in E. Definition 5.3. G = {(, , ) | 0 1, 0 1, 0 < 1} is partitioned into various regions given in (5.12). The parameter space E considered in this paper is a subset of G. V1 = (, , ) | 0 < 1 2 , 0 < 1 2 (1 - ), 0 < < 1 , V2 = (, , ) | 1 2 < 1, 0 < 1 2 (2 - ), 0 < 1 , V3 = (, , ) | 1 4 (1 + 2 + ) < < 1, 1 2 (2 - ) < 1, 0 < < 1 , F3 = (, , ) | 1 4 (2 + 2) < < 1, 1 2 (2 - 1) < < 1, = 1 , V4 = (, , ) | 0 < 1 4 (1 + 2 + ), 1 2 (1 - ) < 1, 0 < 1 , F12 = (, , ) | = 1 2 , 0 < 1 2 (1 - ), 0 < < 1 , F14 = (, , ) | 0 < 1 2 , = 1 2 (1 - ), 0 < 1 , F23 = (, , ) | 1 2 < 1, = 1 2 (2 - ), 0 < 1 , F34 = (, , ) | = 1 4 (1 + 2 + ), 1 2 (1 - ) < 1, 0 < 1 , L1234 = (, , ) | = 1 2 , = 1 2 (1 - ), 0 < 1 . (5.12) Note that all the regions in (5.12) forms a partition of G. The notation used in Definition 5.3 is suggestive. We use Vi to represent a volume (polytope), where i {1, 2, 3, 4}. We use S and the corresponding subscripts to represent a surface (plane) that is the common boundary between two volumes. For example, F12 represents the surface that is the common boundary between V1 and V2. Finally we use L and the corresponding subscripts to represent a line segment generated by the intersection of multiple volumes. Figure 8 visualizes the IDs, location, and the geometries of various regions by partitioning G. Figure 9 shows how the partition changes at various levels. The leading terms of n,0 and n,± in these various regions of G are given by Theorem 5.4. Theorem 5.4. n,0 and n,± are given respectively in the regions of G defined in Definition 5.3 as: REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 21 Figure 9. Partitions of G at various 's. The dotted lines represent the contours of various regions in E. When = 1, the contours overlapped with F14 and F34. ­ When (, , ) 6 L1234, n,0 = -µ n(1 + o(1)), n,± = -1 2 µ2+-1 n (1 + o(1)) ± iµ 1 2 - 2 n (1 + o(1)), (, , ) V1; n,0 = -µ-2++1 n (1 + o(1)), n,± = - µ n 2 (1 + o(1)) ± iµ - 2 n (1 + o(1)), (, , ) V2; n,0 = -µ n(1 + o(1)), n,+ = -µ2-- n (1 + o(1)), n,- = -µ-2++1 n (1 + o(1)), (, , ) V3 F3; n,0 = -µ n(1 + o(1)), n,± = -1 2 µ2-- n (1 + o(1)) ± iµ 1 2 - 2 n (1 + o(1)), (, , ) V4; n,0 = - µ n 2 (1 + o(1)), n,± = - µ n 4 (1 + o(1)) ± i 2µ 1 2 - 2 n (1 + o(1)), (, , ) F12; n,0 = -µ 1 2 - 2 n (1 + o(1)), n,± = -1 4 µ 2- 2 - 1 2 n (1 + o(1)) ± iµ 1 2 - 2 n (1 + o(1)), (, , ) F14; n,0 = -µ -- 2 +1 n (1 + o(1)), n,± = (-1 2 ± i 3 2 )µ - 2 n (1 + o(1)), (, , ) F23; n,0 = -µ n(1 + o(1)), n,± = (-1 2 ± i 3 2 )µ 1 2 - 2 n (1 + o(1)), (, , ) F34. ­ When (, , ) L1234, n,0 = 1 6 2 2 3 (3 69 - 11) 1 3 - 2 2 3 (3 69 + 11) 1 3 - 2 µ 1 2 - 2 n (1 + o(1)), n,+ = 1 12 2 2 3 -1 + i 3 (3 69 - 11) 1 3 + 2 2 3 1 + i 3 (3 69 + 11) 1 3 - 4 µ 1 2 - 2 n (1 + o(1)), n,- = 1 12 2 2 3 -1 - i 3 (3 69 - 11) 1 3 + 2 2 3 1 - i 3 (3 69 + 11) 1 3 - 4 µ 1 2 - 2 n (1 + o(1)). Since E G is the parameter space of interest in this paper, we also summarize the leading terms of n,0 and n,± in various regions of E in Corollary 5.5. Corollary 5.5. Respectively in R1, R2, R3, R4, S2, and S3, the leading terms of n,0 and n,± are n,0 = -µ n, n,+ = -µ2-- n , n,- = -µ-2++1 n , (, , ) R1\(F23 F34 L1234); n,0 = -µ n, n,± = -1 2 µ2-- n ± iµ 1 2 - 2 n , (, , ) R2\F14; n,0 = -µ n, n,± = -1 2 µ2+-1 n ± iµ 1 2 - 2 n , (, , ) R3; n,0 = -µ-2++1 n , n,± = - µ n 2 ± iµ - 2 n , (, , ) R4\F12; n,0 = -µ n, n,± = -1 2 µ2-- n ± iµ 1 2 - 2 n , (, , ) S2\F14; n,0 = -µ n, n,± = -1 2 µ2+-1 n ± iµ 1 2 - 2 n , (, , ) S3. 22 Z. KUANG ET AL. Corollary 5.5 plays a crucial role: the decomposition of the parameter space E into regions of polynomial stability, exponential stability, and regularity, as well as the order and the corresponding optimality of polynomial stability and Gevrey class, is all conjectured and derived based on the behavior of the leading terms of the roots of the characteristic equation reported in this corollary. Subsequent sections will provide technical results leading to the proof of Theorem 5.4. 5.2. Leading Terms of An and Bn This section determines the leading terms of An and Bn as n . Lemma 5.6 presents a method to compute these leading terms. Lemma 5.6. If -qn 2 = o( n) as n , then lim n An = lim n - qn pn = lim n - 1 9an · gn hn , lim n Bn = lim n 6 n pn = lim n 3 · Wn hn . (5.13) If n = o(-qn 2 ) as n , then lim n An = lim n 3qn pn = lim n 1 3an · gn hn , lim n Bn = lim n - 2 n pn = lim n - 3 3 · Wn hn . (5.14) If n = O(-qn 2 ) as n , then lim n An, Bn = (1 + 2) 1 3 ± (1 - 2) 1 3 µx n(1 + o(1)), (5.15) where 1, 2, and x are quantities related to -qn 2 and via - qn 2 = 1µ3x n (1 + o(1)), p n = 2µ3x n (1 + o(1)). (5.16) In (5.16), 3x is the power of the leading term shared by -qn 2 and n because we consider n = O(-qn 2 ). We further assume that 1 ± 2 6= 0. We omit the discussion when 1 ± 2 = 0 as it is not involved in our problem. Proof. ­ If -qn 2 = o( n), then by (5.8), n,± = O( n) and limn n,± = limn ± n limn n,+ = - limn n,- limn 1 3 n,+ = - limn 1 3 n,-. Recall that 1 3 n,+ 1 3 n,- = -pn 3 . We therefore have that limn 2 3 n,± = limn pn 3 . Therefore, by (5.9), we can show that (5.13) holds. ­ If n = o -qn 2 , then by (5.8), n,± = O(-qn 2 ) and limn n,± = limn -qn 2 limn 1 3 n,+ = limn 1 3 n,-. By 1 3 n,+ 1 3 n,- = -pn 3 , we have that limn 2 3 n,± = - limn pn 3 . Therefore, by (5.9), we can show that (5.14) holds. ­ Finally, when n = O(-qn 2 ), without loss of generality, we can write: - qn 2 = 1µ3x n (1 + o(1)), p n = 2µ3x n (1 + o(1)), where we further assume that 1 ± 2 6= 0. Then n,± = (1 ± 2)µ3x n (1 + o(1)) 1 3 n,± = (1 ± 2) 1 3 µx n(1 + o(1)). Therefore, by (5.9), (5.15) also holds. REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 23 Table 1. Expression of an, hn, gn, and Wn in various regions. Region an hn gn Wn V1 µ n(1 + o(1)) 3µ+1 n (1 + o(1)) 18µ+2+1 n (1 + o(1)) 4µ+3 n (1 + o(1)) V2 µ n(1 + o(1)) 3µ2+ n (1 + o(1)) -9µ2++2 n (1 + o(1)) 4µ6+ n (1 + o(1)) V3 F3 µ n(1 + o(1)) -µ2+2 n (1 + o(1)) 2µ3+3 n (1 + o(1)) -µ4+2+2 n (1 + o(1)) V4 µ n(1 + o(1)) -µ2+2 n (1 + o(1)) 2µ3+3 n (1 + o(1)) 4µ4+3+1 n (1 + o(1)) F12 µ n(1 + o(1)) 6µ+1 n (1 + o(1)) 9µ+2+1 n (1 + o(1)) 32µ+3 n (1 + o(1)) F14 µ n(1 + o(1)) 2µ+1 n (1 + o(1)) 20µ 3 2 + 3 2 n (1 + o(1)) 16µ+3 n (1 + o(1)) F23 µ n(1 + o(1)) 2µ2+ n (1 + o(1)) -7µ 3+ 3 2 n (1 + o(1)) 3µ6+ n (1 + o(1)) F34 µ n(1 + o(1)) -µ2+2 n (1 + o(1)) 2µ3+3 n (1 + o(1)) 3µ4+3+1 n (1 + o(1)) L1234 µ n(1 + o(1)) 5µ+1 n (1 + o(1)) 11µ 3 2 + 3 2 n (1 + o(1)) 23µ+3 n (1 + o(1)) Lemma 5.6 shows that the leading terms of An and Bn are given by various analytic expressions involving hn, gn, an, and n, depending on the relative order between -qn 2 and n. Furthermore, from (5.6), (5.7), and (5.8), -qn 2 and n are also dependent on gn, Wn, and an. Therefore, to determine the leading terms of An and Bn, it suffices to determine the leading terms of an, hn, gn and Wn. Without loss of generality, we show in detail how we can determine the leading term of hn in region V1. Note that from (5.4) and (5.6), hn = 3µn + 3µ2 n + 3µ2+ n - 2µ2+ n - µ2+2 n - µ2 n + 3µ+1 n . (5.17) The leading term of hn in region V1 is the term in (5.17) with the largest exponent when (, , ) V1. For each of the seven terms in (5.17), we assume in turn that the term in question is the leading term, and verify the assumption by solving a feasibility problem. For example, if we assume that 3µn is the leading term, define V1 = {(, , ) V1 | 1 > 2, 1 > 2 + , 1 > 2 + , 1 > 2 + 2, 1 > 2, 1 > + 1} . (5.18) We need to verify the assumption by showing that in (5.18), V1 6= . Furthermore, if V1 = V1, then 3µn is the unique leading term of hn in region V1. In this way, the leading terms of an, hn, gn, and Wn in various regions can be determined by solving a series of feasibility problems. In practice, such feasibility problems can be solved by the Reduce function in Mathematica. The leading terms of an, hn, gn, and Wn are summarized in Table 1. With the quantities in Table 1, we are able to determine the leading terms of -qn 2 and n using (5.6), (5.7), and (5.8). Based on the leading terms of -qn 2 and n, we can further determine their relative order in these regions as n . The results are summarized in Table 2. As an illustrative example, we show how to compute n in region V1. From (5.7) and Table 1: p n = Wn 6 3a2 n = q 4µ+3 n (1 + o(1)) 6 3(µ n(1 + o(1)))2 = 2µ +3 2 n (1 + o(1)) 6 3µ2 n (1 + o(1)) = 3 9 µ 3 2 - 3 2 n (1 + o(1)). With Tables 1 and 2, the leading terms of An and Bn can be calculated according to Lemma 5.6. These leading terms are summarized in Table 3. 24 Z. KUANG ET AL. Table 2. Expression of -qn 2 , n and their relative order in various regions. Region -qn 2 n Relative order V1 -1 3 µ-+1 n (1 + o(1)) 3 9 µ 3 2 - 3 2 n (1 + o(1)) -qn 2 = o( n) V2 1 6 µ2+- n (1 + o(1)) 3 9 µ 3- 3 2 n (1 + o(1)) -qn 2 = o( n) V3 F3 - 1 27 µ3 n (1 + o(1)) i 3 18 µ2+- n (1 + o(1)) n = o -qn 2 V4 - 1 27 µ3 n (1 + o(1)) 3 9 µ 2- 2 + 1 2 n (1 + o(1)) n = o -qn 2 F12 -1 6 µ-+1 n (1 + o(1)) 2 6 9 µ 3 2 - 3 2 n (1 + o(1)) -qn 2 = o( n) F14 -10 27 µ 3 2 - 3 2 n (1 + o(1)) 2 3 9 µ 3 2 - 3 2 n (1 + o(1)) -qn 2 = o( n) F23 7 54 µ 3- 3 2 n (1 + o(1)) 1 6 µ 3- 3 2 n (1 + o(1)) -qn 2 = o( n) F34 - 1 27 µ3 n (1 + o(1)) 1 6 µ 2- 2 + 1 2 n (1 + o(1)) n = o -qn 2 L1234 -11 54 µ 3 2 - 3 2 n (1 + o(1)) 69 18 µ 3 2 - 3 2 n (1 + o(1)) -qn 2 = o( n) Table 3. Expression of An and Bn in various regions. Region An Bn V1 -2 3 µ n(1 + o(1)) 2 3 3 µ 1 2 - 2 n (1 + o(1)) V2 1 3 µ n(1 + o(1)) 2 3 3 µ - 2 n (1 + o(1)) V3 F3 -2 3 µ n(1 + o(1)) i 3 3 µ2-- n (1 + o(1)) V4 -2 3 µ n(1 + o(1)) 2 3 3 µ 1 2 - 2 n (1 + o(1)) F12 -1 6 µ n(1 + o(1)) 2 6 3 µ 1 2 - 2 n (1 + o(1)) F14 -2 3 µ 1 2 - 2 n (1 + o(1)) 2 3 3 µ 1 2 - 2 n (1 + o(1)) F23 1 3 µ - 2 n (1 + o(1)) µ - 2 n (1 + o(1)) F34 -2 3 µ n(1 + o(1)) µ 1 2 - 2 n (1 + o(1)) L1234 3 3 69-11- 3 11+3 69 3 3 2 µ 1 2 - 2 n (1 + o(1)) 3 3 69-11+ 3 11+3 69 3 3 2 µ 1 2 - 2 n (1 + o(1)) 5.3. Lower order terms of An While Section 5.2 provides a characterization of the leading term of An, in subsequent derivation, we will show that just using the leading term of An does not suffice to estimate the leading terms of n,0 and n,± as desired. This motivates us to further characterize the lower order terms of An, yielding Lemma 5.7. Lemma 5.7. Let An = Zn + wn, where Zn is the leading term of An as n , and wn is the lower order term. i.e. wn = o(Zn). Then lim n wn = lim n - pnZn + qn + Z3 n pn + 3Z2 n = lim n - 27a3 nZ3 n + 9anhnZn + gn 81a3 nZ2 n + 9anhn . (5.19) Proof. We first define xn,0 = An, xn,± = - 1 2 An ± i 3 2 Bn. REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 25 By Cadano's formula, xn,0, xn,± are solutions to x3 n + pnxn + qn = 0. We further define yn,0 = wn, yn,± = - wn 2 - 3 2 Zn ± i 3 2 Bn, (5.20) where An = Zn + wn, Zn is the leading term of An, and wn = o(Zn). Apparently, xn,0 = Zn + yn,0, xn,± = Zn + yn,±. Therefore, yn,0 and yn,± are the three solutions for: (yn + Zn)3 + pn(y + Zn) + qn = 0 y3 n + 3Zny2 n + (pn + 3Z2 n)yn + pnZn + qn + Z3 n = 0. On the other hand, using (5.20), by Vieta's formula with respect to yn,0, and yn,±, yn,0yn,+ + yn,0yn,- + yn,+yn,- = 3 4 B2 n + 9 4 Z2 n - 3 2 Znwn - 3 4 w2 n = pn + 3Z2 n, (5.21a) yn,0 + yn,+ + yn,- = 3 4 B2 nwn + 9 4 Z2 nwn + 3 2 Znw2 n + 1 4 w3 n = - pnZn + qn + Z3 n . (5.21b) If -qn 2 = o( n), by (5.13), Zn = o(Bn). From (5.21a) and (5.21b), as n , lim n 3 4 B2 n = lim n pn + 3Z2 n, lim n 3 4 B2 nwn = lim n - pnZn + qn + Z3 n , which indicates that (5.19) is true. If n = o -qn 2 , by (5.14), Bn = o(Zn). From (5.21a) and (5.21b), as n , lim n 9 4 Z2 n = lim n pn + 3Z2 n, lim n 9 4 Z2 nwn = lim n - pnZn + qn + Z3 n , which indicates that (5.19) is also true in this case. Finally, if n = O -qn 2 , by (5.15), Zn = O(Bn). From (5.21a) and (5.21b), as n , lim n 3 4 B2 n + 9 4 Z2 n = lim n pn + 3Z2 n, lim n 3 4 B2 nwn + 9 4 Z2 nwn = lim n - pnZn + qn + Z3 n , which validates (5.19) in this case. Given the leading term of An, denoted as Zn, Lemma 5.7 provides an analytic solution to the lower order term of An to Zn, denoted as wn, via (5.19) as a function of Zn, an, hn, and gn. For all the regions defined in Definition 5.3, Zn is given by Table 3, and the leading terms of an, hn, gn can be looked up via Table 1. We summarize the expression of wn in Table 4. 5.4. Determine the leading terms of n,0 and n,± With the results from Sections 5.2 and 5.3, we are now able to apply Cardano's Formula presented in Lemma 5.10 to compute the leading terms of n,0 and n,± in various regions. To proceed, we also need to 26 Z. KUANG ET AL. Table 4. Expression of wn in various regions. Region wn V1 µ2+-1 n (1 + o(1)) V2 -µ-2++1 n (1 + o(1)) V3 F3 µ2-- n (1 + o(1)) V4 µ2-- n (1 + o(1)) F12 - 1 16 µ3+-1 n (1 + o(1)) F14 1 2 µ 2- 2 - 1 2 n (1 + o(1)) F23 -µ -- 2 +1 n (1 + o(1)) F34 µ 1 2 - 2 n (1 + o(1)) L1234 Not needed Table 5. Intermediate quantities in (5.10). Region An -1 2 An V1 -2 3 µ n + µ2+-1 n (1 + o(1)) 1 3 µ n - 1 2 µ2+-1 n (1 + o(1)) V2 1 3 µ n - µ-2++1 n (1 + o(1)) -1 6 µ n + 1 2 µ-2++1 n (1 + o(1)) V3 F3 -2 3 µ n + µ2-- n (1 + o(1)) 1 3 µ n - 1 2 µ2-- n (1 + o(1)) V4 -2 3 µ n + µ2-- n (1 + o(1)) 1 3 µ n - 1 2 µ2-- n (1 + o(1)) F12 -1 6 µ n - 1 16 µ3+-1 n (1 + o(1)) 1 12 µ n + 1 32 µ3+-1 n (1 + o(1)) F14 -2 3 µ 1 2 - 2 n + 1 2 µ 2- 2 - 1 2 n (1 + o(1)) 1 3 µ 1 2 - 2 n - 1 4 µ 2- 2 - 1 2 n (1 + o(1)) F23 1 3 µ - 2 n - µ -- 2 +1 n (1 + o(1)) -1 6 µ - 2 n + 1 2 µ -- 2 +1 n (1 + o(1)) F34 -2 3 µ n + µ 1 2 - 2 n (1 + o(1)) µ n 3 - 1 2 µ 1 2 - 2 n (1 + o(1)) L1234 3 3 69-11- 3 11+3 69 3 3 2 µ 1 2 - 2 n (1 + o(1)) 3 3 69-11- 3 11+3 69 6 3 2 µ 1 2 - 2 n (1 + o(1)) Region ±i 3 2 Bn - bn 3an V1 ±iµ 1 2 - 2 n (1 + o(1)) -1 3 µ n V2 ±iµ - 2 n (1 + o(1)) -1 3 µ n V3 F3 1 2 µ2-- n (1 + o(1)) -1 3 µ n V4 ±iµ 1 2 - 2 n (1 + o(1)) -1 3 µ n F12 ±i 2µ 1 2 - 2 n (1 + o(1)) -1 3 µ n F14 ±iµ 1 2 - 2 n (1 + o(1)) -1 3 µ 1 2 - 2 n F23 ±i 3 2 µ - 2 n (1 + o(1)) -1 3 µ - 2 n F34 ±i 3 2 µ 1 2 - 2 n (1 + o(1)) -1 3 µ n L1234 ± i 3 3 69-11+ 3 11+3 69 2 3 2 3 µ 1 2 - 2 n (1 + o(1)) -1 3 µ 1 2 - 2 n notice that bn 3an = 1 3 µ n. (5.22) REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 27 Applying the results from Table 3, Table 4 and (5.22) to (5.10), we can compute the leading terms of n,0 and n,± given in Theorem 5.4. The intermediate quantities are summarized in Table 5. Using Table 5, we can verify most of the results in Theorem 5.4. Furthermore, the necessity of computing the lower order terms of An is evident as - bn 3an could cancel out the leading terms in An or 1 2 An. However, notice that in V3 F3, n,0 = -µ n(1 + o(1)), n,+ = -µ2-- n (1 + o(1)), and n,- = o(µ2-- n ), where we are not sure of the leading term of n,- due to cancellation. To identify the leading term of n,-, consider the Vieta's formula lim n n,0n,+n,- = lim n - dn an lim n µ2- n n,- = -µ-+1 n n,- = -µ-2++1 n (1 + o(1)). We therefore have concluded the determination of the leading terms of n,0 and n,± in various regions of the parameter space. 6. Sharpness of the order of Gevrey class and the order of polynomial stability We now present the proof of Theorem 2.2, which shows that the orders of Gevrey class corresponding to (2.2) of Theorem 2.1 and the orders of polynomial stability corresponding to (1.6) are sharp. Proof. From (3.8) and (3.14), we have that lim R, || ||µ(,,) k(i - A)-1 k < , (, , ) R2 R3 R4 S2 S3. (6.1) where µ(, , ) takes value from (2.2) corresponding to the order of Gevrey class in R2, R3, and R4 respec- tively; and µ(, , ) takes value from (1.6) corresponding to the order of polynomial stability in S2 and S3. Subsequently, it suffices to show that lim R, || ||µ(,,) k(i - A)-1 k 2, (, , ) R2 R3 R4 S2 S3. (6.2) This is because through an argument of contradiction, if the orders corresponding to (2.2) and (1.6) are not sharp, then there exists > 0 such that lim R, || ||µ(,,)+ k(i - A)-1 k < , (, , ) R2 R3 R4 S2 S3. (6.3) However, from (6.2), for > 0, lim R, || ||µ(,,) k(i - A)-1 k 2 lim R, || ||µ(,,)+ k(i - A)-1 k , which contradicts against (6.3). Therefore, in what follows, we show why (6.2) is true. To this end, notice that if C, with Re 6= 0, is an eigenvalue of A which is a densely defined closed operator on some Hilbert space H such that (iIm - A)-1 exists, then |Re | k(iIm - A)-1 k 1. (6.4) 28 Z. KUANG ET AL. Indeed, there exists an x D(A) with kxk = 1 such that Ax = (µ + i)x µ(i - A)-1 x = -x. Thus (6.4) follows. Now, from Theorem 5.4, we know that A,, has a sequence of complex conjugate eigenvalues of the form: n,± = -aµ n 1 + o(1) ± ibµ n 1 + o(1) , where a > 0, b > 0, > 0, and R. Let = bµ n and with (6.4), we have 1 aµ n 1 + o(1) ibµ n - A,, -1 = a b || i - A,,)-1 1 + o(1) || (i - A,,)-1 b/ a 1 + o(1) . (6.5) Now, we show that satisfying (6.5) in R2, R3, R4, S2, and S3, respectively implies that (6.2) is also satisfied. In region R2 and S2, n,± = ( -1 2 µ2-- n 1 + o(1) ± iµ 1 2 - 2 n 1 + o(1) , > 1- 2 ; -1 4 µ 2- 2 - 1 2 n 1 + o(1) ± µ 1 2 - 2 n 1 + o(1) , = 1- 2 . Thus, = 2-- (1-)/2 , b/ a = 2, > 1- 2 ; = 2- 2 - 1 2 1 2 - 2 , b/ a = 4, = 1- 2 ; which leads to || 2-- (1-)/2 (i - A,,)-1 2 1 + o(1) . In region R3 and S3, n,± = - 1 2 µ2+-1 n 1 + o(1) ± iµ 1 2 - 2 n 1 + o(1) . Hence, = 2 + - 1 (1 - )/2 , b/ a = 2. (6.6) Consequently, || 2+-1 (1-)/2 (i - A,,)-1 2. Finally, in R4, n,± = ( -1 2 µ n 1 + o(1) ± iµ - 2 n 1 + o(1) , > 1 2 ; -1 4 µ n 1 + o(1) ± i 2µ 1 2 - 2 n 1 + o(1) , = 1 2 . (6.7) REGULARITY ANALYSIS FOR AN ABSTRACT THERMOELASTIC SYSTEM WITH INERTIAL TERM 29 Thus, ( = - 2 , b/ a = 2, > 1 2 , = (1-)/2 , b/ a = 2+2 , = 1 2 . (6.8) and we again have || - 2 (i - A ,)-1 2 1 + o(1) . Combining the above, we see that (6.2) is satisfied, which concludes our proof. References [1] F. Ammar-Khodja, A. Bader and A. Benabdallah, Dynamic stabilization of systems via decoupling techniques. 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