Optimal location of controllers for the one-dimensional wave equation
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, pp. 1097-1126.

In this paper, we consider the homogeneous one-dimensional wave equation defined on (0,π). For every subset ω[0,π] of positive measure, every T2π, and all initial data, there exists a unique control of minimal norm in L 2 (0,T;L 2 (ω)) steering the system exactly to zero. In this article we consider two optimal design problems. Let L(0,1). The first problem is to determine the optimal shape and position of ω in order to minimize the norm of the control for given initial data, over all possible measurable subsets ω of [0,π] of Lebesgue measure . The second problem is to minimize the norm of the control operator, over all such subsets. Considering a relaxed version of these optimal design problems, we show and characterize the emergence of different phenomena for the first problem depending on the choice of the initial data: existence of optimal sets having a finite or an infinite number of connected components, or nonexistence of an optimal set (relaxation phenomenon). The second problem does not admit any optimal solution except for L=1/2. Moreover, we provide an interpretation of these problems in terms of a classical optimal control problem for an infinite number of controlled ordinary differential equations. This new interpretation permits in turn to study modal approximations of the two problems and leads to new numerical algorithms. Their efficiency will be exhibited by several experiments and simulations.

DOI: 10.1016/j.anihpc.2012.11.005
Classification: 49J20, 35L05, 49J15, 49Q10, 49K35
Keywords: Wave equation, Exact controllability, HUM method, Shape optimization, Relaxation, Optimal control, Pontryagin Maximum Principle
@article{AIHPC_2013__30_6_1097_0,
     author = {Privat, Yannick and Tr\'elat, Emmanuel and Zuazua, Enrique},
     title = {Optimal location of controllers for the one-dimensional wave equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1097--1126},
     publisher = {Elsevier},
     volume = {30},
     number = {6},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.11.005},
     mrnumber = {3132418},
     zbl = {1296.49004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.11.005/}
}
TY  - JOUR
AU  - Privat, Yannick
AU  - Trélat, Emmanuel
AU  - Zuazua, Enrique
TI  - Optimal location of controllers for the one-dimensional wave equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 1097
EP  - 1126
VL  - 30
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.11.005/
DO  - 10.1016/j.anihpc.2012.11.005
LA  - en
ID  - AIHPC_2013__30_6_1097_0
ER  - 
%0 Journal Article
%A Privat, Yannick
%A Trélat, Emmanuel
%A Zuazua, Enrique
%T Optimal location of controllers for the one-dimensional wave equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 1097-1126
%V 30
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.11.005/
%R 10.1016/j.anihpc.2012.11.005
%G en
%F AIHPC_2013__30_6_1097_0
Privat, Yannick; Trélat, Emmanuel; Zuazua, Enrique. Optimal location of controllers for the one-dimensional wave equation. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, pp. 1097-1126. doi : 10.1016/j.anihpc.2012.11.005. http://www.numdam.org/articles/10.1016/j.anihpc.2012.11.005/

[1] N.I. Akhiezer, Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs vol. 79, American Mathematical Society (1990) | MR | Zbl

[2] B. Bonnard, J.-B. Caillau, E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. Calc. Var. 13 no. 2 (2007), 207-236 | EuDML | Numdam | MR | Zbl

[3] D. Bucur, G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations vol. 65, Birkhäuser Verlag, Basel (2005) | MR | Zbl

[4] M. Cerf, T. Haberkorn, E. Trélat, Continuation from a flat to a round Earth model in the coplanar orbit transfer problem, Optimal Control Appl. Methods 33 no. 6 (2012), 654-675, http://dx.doi.org/10.1002/oca.1016 | MR | Zbl

[5] S. Cox, E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial Differential Equations 19 (1994), 213-243 | MR | Zbl

[6] R. Fourer, D.M. Gay, B.W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press (2002)

[7] P. Hébrard, A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Systems Control Lett. 48 (2003), 199-209 | MR | Zbl

[8] P. Hébrard, A. Henrot, A spillover phenomenon in the optimal location of actuators, SIAM J. Control Optim. 44 (2005), 349-366 | MR | Zbl

[9] A. Henrot, M. Pierre, Variation et optimisation de formes, Mathématiques Et Applications, Springer (2005) | MR

[10] A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), 367-379 | EuDML | MR | Zbl

[11] X.J. Li, J.M. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim. 29 no. 4 (1991), 895-908 | MR | Zbl

[12] J.-L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev. 30 (1988), 1-68 | MR | Zbl

[13] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Recherches en Mathématiques Appliquées, Masson (1988) | MR | Zbl

[14] S. Mandelbrojt, Quasi-analycité des séries de Fourier, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (2) 4 no. 3 (1935), 225-229 | EuDML | JFM | Numdam | MR

[15] K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control 56 no. 1 (2011), 113-124 | MR

[16] A. Münch, Optimal location of the support of the control for the 1-D wave equation: numerical investigations, Comput. Optim. Appl. 42 (2009), 443-470 | MR | Zbl

[17] A. Münch, F. Periago, Optimal distribution of the internal null control for the 1D heat equation, J. Differential Equations 250 (2011), 95-111 | MR | Zbl

[18] S.L. Padula, R.K. Kincaid, Optimization strategies for sensor and actuator placement, Report NASA TM-1999-209126, 1999.

[19] F. Periago, Optimal shape and position of the support for the internal exact control of a string, Systems Control Lett. 58 no. 2 (2009), 136-140 | MR | Zbl

[20] Y. Privat, E. Trélat, E. Zuazua, Optimal observation of the one-dimensional wave equation, Preprint Hal, 2012. | MR

[21] Y. Privat, E. Trélat, E. Zuazua, Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains, Preprint Hal, 2012. | MR

[22] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York, London (1962) | MR | Zbl

[23] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 no. 4 (1978), 639-739 | MR | Zbl

[24] O. Sigmund, J.S. Jensen, Systematic design of phononic band-gap materials and structures by topology optimization, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 361 no. 1806 (2003), 1001-1019 | Zbl

[25] J. Simon, Compact sets in the space L p (0,T;B), Ann. Mat. Pura Appl. 146 no. 4 (1987), 65-96 | Zbl

[26] E. Trélat, Contrôle optimal, Théorie Et Applications, Vuibert, Paris (2005) | MR

[27] E. Trélat, Optimal control and applications to aerospace: some results and challenges, J. Optim. Theory Appl. 154 no. 3 (2012), 713-758 | MR | Zbl

[28] M. Van De Wal, B. Jager, A review of methods for input/output selection, Automatica 37 no. 4 (2001), 487-510 | MR | Zbl

[29] A. Wächter, L.T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. 106 (2006), 25-57 | MR | Zbl

Cited by Sources: