[Contrôle exact pour systèmes décrivant les vibrations des plaques. Une approche perturbative]
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.
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.
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Marius Tucsnak 1 ; Megane Bournissou 1 ; Sylvain Ervedoza 1
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@article{CRMATH_2024__362_G4_327_0,
author = {Marius Tucsnak and Megane Bournissou and Sylvain Ervedoza},
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},
volume = {362},
year = {2024},
doi = {10.5802/crmath.539},
language = {en},
}
TY - JOUR AU - Marius Tucsnak AU - Megane Bournissou AU - Sylvain Ervedoza TI - Exact controllability for systems describing plate vibrations. A perturbation approach JO - Comptes Rendus. Mathématique PY - 2024 SP - 327 EP - 356 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.539 LA - en ID - CRMATH_2024__362_G4_327_0 ER -
%0 Journal Article %A Marius Tucsnak %A Megane Bournissou %A Sylvain Ervedoza %T Exact controllability for systems describing plate vibrations. A perturbation approach %J Comptes Rendus. Mathématique %D 2024 %P 327-356 %V 362 %I Académie des sciences, Paris %R 10.5802/crmath.539 %G en %F CRMATH_2024__362_G4_327_0
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/
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