Comptes Rendus
Research article - Partial differential equations, Control theory
Exact controllability for systems describing plate vibrations. A perturbation approach
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 327-356.

The aim of this paper is to prove new exact controllability properties of systems described by perturbations of the classical Kirchhoff plate equation. We first consider systems described by an abstract plate equation with a bounded control operator. The generator of these systems is perturbed by bounded operators which are not necessarily compact, thus not falling in the range of application of compactness-uniqueness arguments. Our first main result is abstract and can be informally stated as follows: if the system described by the corresponding unperturbed abstract wave equation, with the same control operator, is exactly controllable (in some time), then the considered perturbed plate system is exactly controllable in arbitrarily small time. The employed methodology is based, in particular, on frequency-dependent Hautus type tests for systems with skew-adjoint operators. When applied to systems described by the classical Kirchhoff equations, our abstract results, combined with some elliptic Carleman-type estimates, yield exact controllability in arbitrarily small time, provided that the system described by the wave equation in the same spatial domain and with the same control operator is exactly controllable. The same abstract results can be used to prove the exact controllability of the system obtained by linearizing the von Kármán plate equation around a real analytic stationary state. This leads, via a fixed-point method, to our second main result: the nonlinear system described by the von Kármán plate equations is locally exactly controllable around any stationary state defined by a real analytic function. We also discuss the possible application of the methods in this paper to systems described by Schrödinger type equations on manifolds or by the related Berger’s nonlinear plate equation.

L’objectif de ce travail est l’obtention des nouvelles propiétés de contrôlabilité exacte de systèmes décrits par des perturbations de l’équation de plaque de Kirchhoff classique. Nous considérons d’abord les systèmes décrits par une équation des plaques abstraite avec un opérateur de contrôle borné. Le générateur de ces systèmes est perturbé par des opérateurs bornés qui ne sont pas nécessairement compacts, donc hors du domaine d’application des arguments compacité-unicité. Notre premier résultat principal est abstrait et peut être énoncé de manière informelle comme suit : si le système décrit par l’équation d’onde abstraite non perturbée correspondante, avec le même opérateur de contrôle, est exactement contrôlable (dans un certain temps), alors le système de plaques perturbé considéré est exactement contrôlable en un temps arbitrairement petit. La méthodologie employée s’appuie notamment sur des tests de type Hautus dépendant de la fréquence pour des systèmes à opérateurs anti-adjoints. Lorsqu’ils sont appliqués à des systèmes décrits par les équations classiques de Kirchhoff, nos résultats abstraits, combinés à des estimations elliptiques de type Carleman, donnent une contrôlabilité exacte en un temps arbitrairement petit, à condition que le système décrit par l’équation d’onde dans le même domaine spatial et avec le même l’opérateur de contrôle est exactement contrôlable. Les mêmes résultats abstraits peuvent être utilisés pour prouver la contrôlabilité exacte du système obtenu en linéarisant l’équation de la plaque de von Karman autour d’un état stationnaire analytique réel. Ceci conduit, via une méthode de point fixe, à notre deuxième résultat principal : le système non linéaire décrit par les équations de von Kárman est localement exactement contrôlable autour de tout état stationnaire défini par une fonction analytique réelle. Nous discutons également de l’application possible des méthodes de cet article à des systèmes décrits par des équations de type Schrodinger sur des variétés ou par l’équation de plaque non linéaire de Berger associée.

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DOI: 10.5802/crmath.539

Marius Tucsnak 1; Megane Bournissou 1; Sylvain Ervedoza 1

1 Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence, France.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Exact controllability for systems describing plate vibrations. {A} perturbation approach},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {327--356},
     publisher = {Acad\'emie des sciences, Paris},
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Marius Tucsnak; Megane Bournissou; Sylvain Ervedoza. Exact controllability for systems describing plate vibrations. A perturbation approach. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 327-356. doi : 10.5802/crmath.539. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.539/

[1] N. Anantharaman; M. Léautaud; F. Macià Wigner measures and observability for the Schrödinger equation on the disk, Invent. Math., Volume 206 (2016) no. 2, pp. 485-599 | DOI | Zbl

[2] B. Barnes Majorization, range inclusion, and factorization for bounded linear operators, Proc. Am. Math. Soc., Volume 133 (2005) no. 1, pp. 155-162 | DOI | MR | Zbl

[3] J. Bourgain; N. Burq; M. Zworski Control for Schrödinger operators on 2-tori: rough potentials, J. Eur. Math. Soc., Volume 15 (2013) no. 5, pp. 1597-1628 | DOI | MR

[4] M. S. Berger; P. C. Fife On von Kármán’s equations and the buckling of a thin elastic plate, Bull. Am. Math. Soc., Volume 72 (1966) no. 6, pp. 1006-1011 | DOI | Zbl

[5] C. Bardos; G. Lebeau; J. Rauch Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Rend. Semin. Mat., Torino Fasc. Spec. (1988), pp. 11-31 | MR | Zbl

[6] C. Bardos; G. Lebeau; J. Rauch Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992), pp. 1024-1065 | DOI | MR

[7] L. Baudouin; J.-P. Puel Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Probl., Volume 18 (2002) no. 6, pp. 1537-1554 | DOI

[8] N. Burq Contrôle de l’équation des ondes dans des ouverts peu réguliers, École polytechnique, 1995

[9] N. Burq; M. Zworski Control for Schrödinger operators on tori, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 309-324 | DOI | MR | Zbl

[10] O. Calin; D.-C. Chang Geometric mechanics on Riemannian manifolds. Applications to partial differential equations, Applied and Numerical Harmonic Analysis, Birkhäuser, 2005 | Zbl

[11] G. Chen; S. A. Fulling; F. J. Narcowich; S. Sun Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math., Volume 51 (1991) no. 1, pp. 266-301 | DOI | MR

[12] I. Chueshov; I. Lasiecka Von Karman evolution equations. Well-posedness and long-time dynamics, Springer Monographs in Mathematics, Springer, 2010 | DOI | MR | Zbl

[13] J.-M. Coron; P. Lissy Local null controllability of the three-dimensional Navier–Stokes system with a distributed control having two vanishing components, Invent. Math., Volume 198 (2014) no. 3, pp. 833-880 | DOI | MR | Zbl

[14] J.-M. Coron Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007 | DOI | MR | Zbl

[15] Ph. G. Ciarlet; P. Rabier Les équations de von Kármán, Lecture Notes in Mathematics, 826, Springer, 1980 | DOI | MR | Zbl

[16] N. Cîndea; M. Tucsnak Local exact controllability for Berger plate equation, Math. Control Signals Syst., Volume 21 (2009) no. 2, pp. 93-110 | DOI | MR

[17] N. Cîndea; M. Tucsnak Internal exact observability of a perturbed Euler–Bernoulli equation, Ann. Acad. Rom. Sci., Math. Appl., Volume 2 (2010) no. 2, pp. 205-221 | MR | Zbl

[18] S. Dyatlov; L. Jin; S. Nonnenmacher Control of eigenfunctions on surfaces of variable curvature, J. Am. Math. Soc., Volume 35 (2022) no. 2, pp. 361-465 | DOI | MR | Zbl

[19] M. L. Duprez; P. Lissy Indirect controllability of some linear parabolic systems of m equations with m-1 controls involving coupling terms of zero or first order, J. Math. Pures Appl., Volume 106 (2016) no. 5, pp. 905-934 | DOI | MR

[20] M. L. Duprez; P. Lissy Positive and negative results on the internal controllability of parabolic equations coupled by zero- and first-order terms, J. Evol. Equ., Volume 18 (2018) no. 2, pp. 659-680 | DOI | MR

[21] M. L. Duprez; G. Olive Compact perturbations of controlled systems, Math. Control Relat. Fields, Volume 8 (2018) no. 2, pp. 397-410 | DOI | MR | Zbl

[22] M. Eller; D. Toundykov Semiglobal exact controllability of nonlinear plates, SIAM J. Control Optim., Volume 53 (2015) no. 4, pp. 2480-2513 | DOI | MR

[23] A. Favini; M. A. Horn; I. Lasiecka; D. Tataru Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Differ. Integral Equ., Volume 9 (1996) no. 2, pp. 267-294 | MR | Zbl

[24] A. Favini; M. A. Horn; I. Lasiecka; D. Tataru Addendum to the paper: “Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation” [Differential Integral Equations 9 (1996), no. 2, 267–294; MR1364048 (97a:35065)], Differ. Integral Equ., Volume 10 (1997) no. 1, pp. 197-200 | MR

[25] A. V. Fursikov; O. Y. Imanuvilov Controllability of Evolution Equations, Lecture Notes Series, Seoul, 34, Seoul National University Research Institute of Mathematics, Global Analysis Research Center, 1996 | MR | Zbl

[26] A. Haraux Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl., Volume 68 (1989) no. 4, pp. 457-465 | MR

[27] O. Y. Imanuvilov On exact controllability for the Navier-Stokes equations, ESAIM, Control Optim. Calc. Var., Volume 3 (1998), pp. 97-131 | DOI | Numdam | MR | Zbl

[28] S. Jaffard Contrôle interne exact des vibrations d’une plaque rectangulaire, Port. Math., Volume 47 (1990) no. 4, pp. 423-429 | MR | Zbl

[29] L. Jin Control for Schrödinger equation on hyperbolic surfaces, Math. Res. Lett., Volume 25 (2018) no. 6, pp. 1865-1877 | DOI | MR

[30] F. John Plane waves and spherical means applied to partial differential equations, Interscience Tracts in Pure and Applied Mathematics, 2, Interscience Publishers, 1955 | MR | Zbl

[31] V. Komornik On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl., Volume 71 (1992) no. 4, pp. 331-342 | MR

[32] J. E. Lagnese Local controllability of dynamic von Kármán plates, Control Cybern., Volume 19 (1990) no. 3-4, pp. 155-168 | MR

[33] J. Le Rousseau On Carleman estimates with two large parameters, Indiana Univ. Math. J., Volume 64 (2015), pp. 55-113 | DOI | MR

[34] G. Lebeau Contrôle de l’équation de Schrödinger, J. Math. Pures Appl., Volume 71 (1992) no. 3, pp. 267-291

[35] J.-L. Lions Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1: Contrôlabilité exacte. (Exact controllability, perturbations and stabilization of distributed systems. Vol. 1: Exact controllability), Recherches en Mathématiques Appliquées, 8, Masson, 1988 (with appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch) | MR | Zbl

[36] P. Lissy Sur la contrôlabilité et son coût pour quelques équations aux dérivées partielles, Ph. D. Thesis, Université Pierre et Marie Curie - Paris VI (2013) (https://theses.hal.science/tel-00918763)

[37] K. Liu Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., Volume 35 (1997) no. 5, pp. 1574-1590 | DOI | MR

[38] K. Liu; Z. Liu; B. Rao Exponential stability of an abstract nondissipative linear system, SIAM J. Control Optim., Volume 40 (2001) no. 1, pp. 149-165 | DOI | MR

[39] J. Le Rousseau; G. Lebeau; L. Robbiano Elliptic Carleman estimates and applications to stabilization and controllability. Vol. II. General boundary conditions on Riemannian manifolds, Progress in Nonlinear Differential Equations and their Applications, 98, Birkhäuser/Springer, 2022 (PNLDE Subseries in Control) | DOI | MR | Zbl

[40] L. Miller Controllability cost of conservative systems: resolvent condition and transmutation, J. Funct. Anal., Volume 218 (2005) no. 2, pp. 425-444 | DOI | MR | Zbl

[41] L. Miller Resolvent conditions for the control of unitary groups and their approximations, J. Spectr. Theory, Volume 2 (2012) no. 1, pp. 1-55 | DOI | MR | Zbl

[42] G. P. Menzala; E. Zuazua Timoshenko’s plate equation as a singular limit of the dynamical von Kármán system, J. Math. Pures Appl., Volume 79 (2000) no. 1, pp. 73-94 | DOI | Zbl

[43] A. H. Nayfeh; D. T. Mook Nonlinear oscillations, John Wiley & Sons, 2008 | Zbl

[44] J. Rauch; M. Taylor Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., Volume 24 (1974), pp. 79-86 | DOI | MR | Zbl

[45] K. Ramdani; T. Takahashi; G. Tenenbaum; M. Tucsnak A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator, J. Funct. Anal., Volume 226 (2005) no. 1, pp. 193-229 | DOI | MR

[46] F. Treves Analytic partial differential equations, Grundlehren der Mathematischen Wissenschaften, 359, Springer, 2022 | DOI | MR | Zbl

[47] M. Tucsnak; G. Weiss Observation and control for operator semigroups, Birkhäuser Advanced Texts. Basler Lehrbücher, Springer, 2009 | DOI | Zbl

[48] G. Yuan; M. Yamamoto Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality, Chin. Ann. Math., Ser. B, Volume 31 (2010) no. 4, pp. 555-578 | DOI | Zbl

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