On uniform observability of gradient flows in the vanishing viscosity limit

Laurent, Camille; Léautaud, Matthieu

Journal de l’École polytechnique - Mathématiques, Tome 8 (2021), pp. 439-506.

Nous considérons l’équation de transport par un champ de gradient avec une petite perturbation visqueuse $-\epsilon {\Delta }_g$. Nous étudions la propriété d’observabilité (resp. de contrôlabilité) uniforme dans la limite (singulière) de viscosité évanescente $\epsilon \to 0^{+}$, c’est-à-dire la possibilité d’avoir une constante d’observabilité (resp. un coût du contrôle) uniforme. Nous prouvons avec une série d’exemples que le temps minimal pour l’observabilité uniforme peut être bien plus grand que le temps minimal pour l’équation limite $\epsilon =0$. Nous montrons aussi que les deux temps minimaux coïncident pour les solutions positives. Les preuves reposent sur une reformulation semiclassique du problème ainsi que (a) des estimées d’Agmon de décroissance des fonctions propres dans la zone classiquement interdite [HS84] (b) des estimées fines du noyaux de la chaleur semiclassique [LY84].

We consider a transport equation by a gradient vector field with a small viscous perturbation $-\epsilon {\Delta }_g$. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit $\epsilon \to 0^{+}$, that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation $\epsilon =0$. We also prove that the two minimal times coincide for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning the decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY84].

Classification : 93B07, 93B05, 35B25, 35F05, 35K05, 93C73
Keywords: Transport equation, gradient flow, vanishing viscosity limit, parabolic equation, minimal control time, semiclassical Schrödinger operator
Mot clés : Équation de transport, flot gradient, limite de viscosité évanescente, équation parabolique, temps minimal de contrôle, opérateur de Schrödinger semiclassique

[ABB20] Agrachev, Andrei; Barilari, Davide; Boscain, Ugo A comprehensive introduction to sub-Riemannian geometry, Cambridge Studies in Advanced Math., 181, Cambridge University Press, Cambridge, 2020 | MR | Zbl

[All98] Allibert, Brice Contrôle analytique de l’équation des ondes et de l’équation de Schrödinger sur des surfaces de révolution, Comm. Partial Differential Equations, Volume 23 (1998) no. 9-10, pp. 1493-1556 | DOI | MR | Zbl

[AM19a] Amirat, Youcef; Münch, Arnaud Asymptotic analysis of an advection-diffusion equation and application to boundary controllability, Asymptot. Anal., Volume 112 (2019) no. 1-2, pp. 59-106 | DOI | MR | Zbl

[AM19b] Amirat, Youcef; Münch, Arnaud On the controllability of an advection-diffusion equation with respect to the diffusion parameter: asymptotic analysis and numerical simulations, Acta Math. Appl. Sinica (English Ser.), Volume 35 (2019) no. 1, pp. 54-110 | DOI | MR | Zbl

[Bes78] Besse, Arthur L. Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb. (3), 93, Springer-Verlag, Berlin-New York, 1978 | MR | Zbl

[BP20a] Bárcena-Petisco, Jon Asier Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit, SIAM J. Control Optim., Volume 58 (2020) no. 3, pp. 1597-1625 | DOI | MR | Zbl

[BP20b] Bárcena-Petisco, Jon Asier Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term, 2020 | HAL

[CF96] Coron, Jean-Michel; Fursikov, Andrei V. Global exact controllability of the $2$-D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys., Volume 4 (1996), pp. 429-448 | MR | Zbl

[CFKS87] Cycon, Hans L.; Froese, Richard G.; Kirsch, Werner; Simon, Barry Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987 | Zbl

[CG05] Coron, Jean-Michel; Guerrero, Sergio Singular optimal control: A linear $1$-D parabolic-hyperbolic example, Asymptot. Anal., Volume 44 (2005), pp. 237-257 | MR

[Cha43] Chandresekhar, Subrahmanyan Stochastic problems in physics and astronomy, Rev. Modern Phys., Volume 15 (1943), pp. 1-89 | DOI | MR

[Cha09] Chapouly, Marianne On the global null controllability of a Navier-Stokes system with Navier slip boundary conditions, J. Differential Equations, Volume 247 (2009), pp. 2094-2123 | DOI | MR | Zbl

[CMS20] Coron, Jean-Michel; Marbach, Frédéric; Sueur, Franck Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc. (JEMS), Volume 22 (2020) no. 5, pp. 1625-1673 | DOI | MR | Zbl

[Cor96] Coron, Jean-Michel On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var., Volume 1 (1996), pp. 35-75 | DOI | Numdam | Zbl

[Cor07] Coron, Jean-Michel Control and nonlinearity, Math. Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007 | MR | Zbl

[Daf00] Dafermos, Constantine M. Hyperbolic conservation laws in continuum physics, Springer-Verlag, Berlin, 2000 | Zbl

[DGLLPN20] Di Gesù, Giacomo; Lelièvre, Tony; Le Peutrec, Dorian; Nectoux, Boris The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1, J. Math. Pures Appl. (9), Volume 138 (2020), pp. 242-306 | DOI | MR | Zbl

[DL09] Dehman, Belhassen; Lebeau, Gilles Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time, SIAM J. Control Optim., Volume 48 (2009) no. 2, pp. 521-550 | DOI | MR | Zbl

[DR77] Dolecki, Szymon; Russell, David L. A general theory of observation and control, SIAM J. Control Optim., Volume 15 (1977) no. 2, pp. 185-220 | DOI | MR | Zbl

[DR20] Dang, Nguyen Viet; Rivière, Gabriel Pollicott-Ruelle spectrum and Witten Laplacians, J. Eur. Math. Soc. (JEMS) (2020) (to appear)

[DS99] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Math. Society Lect. Note Series, 268, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl

[ET74] Ekeland, Ivar; Temam, Roger Analyse convexe et problèmes variationnels, Dunod–Gauthier-Villars, Paris, 1974 | Zbl

[Eva98] Evans, Lawrence C. Partial differential equations, Graduate Studies in Math., American Mathematical Society, Providence, RI, 1998 | Zbl

[FI96] Fursikov, Andrei V.; Imanuvilov, Oleg Yu. Controllability of evolution equations, Lect. Notes Series, 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996 | MR | Zbl

[GG07] Glass, Olivier; Guerrero, Sergio On the uniform controllability of the Burgers equation, SIAM J. Control Optim., Volume 46 (2007), pp. 1211-1238 | DOI | MR

[GG08] Glass, Olivier; Guerrero, Sergio Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., Volume 60 (2008), pp. 61-100 | DOI | MR | Zbl

[GG09] Glass, Olivier; Guerrero, Sergio Uniform controllability of a transport equation in zero diffusion-dispersion limit, Math. Models Methods Appl. Sci., Volume 19 (2009), pp. 1567-1601 | DOI | MR | Zbl

[GL07] Guerrero, Sergio; Lebeau, Gilles Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations, Volume 32 (2007), pp. 1813-1836 | DOI | MR | Zbl

[Gla10] Glass, Olivier A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit, J. Funct. Anal., Volume 258 (2010), pp. 852-868 | DOI | MR | Zbl

[Hel88] Helffer, Bernard Semi-classical analysis for the Schrödinger operator and applications, Lect. Notes in Math., 1336, Springer-Verlag, Berlin, 1988 | DOI | Zbl

[HKL15] Han-Kwan, Daniel; Léautaud, Matthieu Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium, Ann. PDE, Volume 1 (2015) no. 1, 3, 84 pages | DOI | MR | Zbl

[HKN04] Helffer, Bernard; Klein, Markus; Nier, Francis Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp., Volume 26 (2004), pp. 41-85 | MR | Zbl

[HN06] Helffer, Bernard; Nier, Francis Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary, Mém. Soc. Math. France (N.S.), 105, Société Mathématique de France, Paris, 2006 | DOI | Numdam | Zbl

[HS84] Helffer, Bernard; Sjöstrand, Johannes Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations, Volume 9 (1984) no. 4, pp. 337-408 | DOI | MR

[HS85] Helffer, Bernard; Sjöstrand, Johannes Puits multiples en mécanique semi-classique IV. Étude du complexe de Witten, Comm. Partial Differential Equations, Volume 10 (1985) no. 3, pp. 245-340 | DOI | Zbl

[Kru70] Kružkov, Stanislav N. First order quasilinear equations with several independent variables. (Russian), Mat. Sb. (N.S.), Volume 81 (1970), pp. 228-255

[LB20] Le Balc’h, Kévin Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl. (9), Volume 135 (2020), pp. 103-139 | DOI | MR | Zbl

[Lee13] Lee, John M. Introduction to smooth manifolds, Graduate Texts in Math., 218, Springer, New York, 2013 | MR | Zbl

[Lio88] Lions, Jacques-Louis Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués 2. Perturbations, Recherches en Math. Appliquées, 9, Masson, Paris, 1988 | Zbl

[Lis12] Lissy, Pierre A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation, Comptes Rendus Mathématique, Volume 350 (2012) no. 11-12, pp. 591-595 | DOI | MR | Zbl

[Lis14] Lissy, Pierre An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit, Systems Control Lett., Volume 69 (2014), pp. 98-102 | DOI | MR

[Lis15] Lissy, Pierre Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, Volume 259 (2015) no. 10, pp. 5331-5352 | DOI | MR | Zbl

[LL16] Laurent, Camille; Léautaud, Matthieu Uniform observability estimates for linear waves, ESAIM Control Optim. Calc. Var., Volume 22 (2016) no. 4, pp. 1097-1136 | DOI | MR | Zbl

[LL21a] Laurent, Camille; Léautaud, Matthieu Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller, Anal. PDE (2021) (to appear) | DOI

[LL21b] Laurent, Camille; Léautaud, Matthieu On uniform controllability of 1D transport equations in the vanishing viscosity limit (2021) (work in progress)

[LL21c] Laurent, Camille; Léautaud, Matthieu The cost function for the approximate control of waves (2021) (work in progress)

[LP10] Le Peutrec, Dorian Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian, Ann. Fac. Sci. Toulouse Math. (6), Volume 19 (2010) no. 3-4, pp. 735-809 http://afst.cedram.org/item?id=AFST_2010__19_3-4_735_0 | DOI | Numdam | MR | Zbl

[LR95] Lebeau, Gilles; Robbiano, Luc Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995), pp. 335-356 | DOI

[LSU68] Ladyženskaja, Ol’ga A.; Solonnikov, Vsevolod A.; Ural’ceva, Nina N. Linear and quasilinear equations of parabolic type, Translations of Math. Monographs, 23, American Mathematical Society, Providence, R.I., 1968 | MR

[LY86] Li, Peter; Yau, Shing-Tung On the parabolic kernel of the Schrödinger operator, Acta Math., Volume 156 (1986) no. 3-4, pp. 153-201 | Zbl

[Léa10] Léautaud, Matthieu Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., Volume 258 (2010), pp. 2739-2778 | DOI | MR | Zbl

[Léa12] Léautaud, Matthieu Uniform controllability of scalar conservation laws in the vanishing viscosity limit, SIAM J. Control Optim., Volume 50 (2012) no. 3, pp. 1661-1699 | DOI | MR | Zbl

[Léa18] Léautaud, Matthieu Sur quelques questions de prolongement unique, de propagation et de contrôle, Habilitation à diriger des recherches, Université Paris Diderot (2018) (http://leautaud.perso.math.cnrs.fr/files/HdR.pdf)

[Mar14] Marbach, Frédéric Small time global null controllability for a viscous Burgers’ equation despite the presence of a boundary layer, J. Math. Pures Appl. (9), Volume 102 (2014) no. 2, pp. 364-384 | DOI | MR | Zbl

[Mic19] Michel, Laurent About small eigenvalues of the Witten Laplacian, Pure Appl. Anal., Volume 1 (2019) no. 2, pp. 149-206 | DOI | MR | Zbl

[Mil04] Miller, Luc Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, Volume 204 (2004) no. 1, pp. 202-226 | DOI | MR | Zbl

[Mün18] Münch, Arnaud Numerical estimations of the cost of boundary controls for the equation $y_t-\epsilon y_{xx}+M{y}_x=0$ with respect to $\epsilon$, Recent advances in PDEs: analysis, numerics and control (SEMA SIMAI Springer Ser.), Volume 17, Springer, Cham, 2018, pp. 159-191

[RS80] Reed, Michael; Simon, Barry Methods of modern mathematical physics I. Functional analysis, Academic Press, Inc., New York, 1980 | Zbl

[Sim82] Simon, Barry Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), Volume 7 (1982) no. 3, pp. 447-526 | DOI | Zbl

[Sim83] Simon, Barry Instantons, double wells and large deviations, Bull. Amer. Math. Soc. (N.S.), Volume 8 (1983) no. 2, pp. 323-326 | DOI | MR | Zbl

[SM79] Schuss, Zeev; Matkowsky, Bernard J. The exit problem: a new approach to diffusion across potential barriers, SIAM J. Appl. Math., Volume 36 (1979) no. 3, pp. 604-623 | DOI | MR | Zbl

[Var67] Varadhan, Srinivasa R. S. Diffusion processes in a small time interval, Comm. Pure Appl. Math., Volume 20 (1967), pp. 659-685 | DOI | MR | Zbl

[Wit82] Witten, Edward Supersymmetry and Morse theory, J. Differential Geometry, Volume 17 (1982) no. 4, p. 661-692 (1983) http://projecteuclid.org/euclid.jdg/1214437492 | DOI | MR | Zbl