Exact controllability for systems describing plate vibrations. A perturbation approach
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 bo...
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.539/ |
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| author | Bournissou, Megane Ervedoza, Sylvain Tucsnak, Marius |
| author_facet | Bournissou, Megane Ervedoza, Sylvain Tucsnak, Marius |
| author_sort | Bournissou, Megane |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-bc08c9f658d04a58ab9010c3ea2d01ad |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-bc08c9f658d04a58ab9010c3ea2d01ad2025-02-07T11:20:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G432735610.5802/crmath.53910.5802/crmath.539Exact controllability for systems describing plate vibrations. A perturbation approachBournissou, Megane0Ervedoza, Sylvain1Tucsnak, Marius2Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence, France.Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence, France.Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence, France.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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.539/ |
| spellingShingle | Bournissou, Megane Ervedoza, Sylvain Tucsnak, Marius Exact controllability for systems describing plate vibrations. A perturbation approach Comptes Rendus. Mathématique |
| title | Exact controllability for systems describing plate vibrations. A perturbation approach |
| title_full | Exact controllability for systems describing plate vibrations. A perturbation approach |
| title_fullStr | Exact controllability for systems describing plate vibrations. A perturbation approach |
| title_full_unstemmed | Exact controllability for systems describing plate vibrations. A perturbation approach |
| title_short | Exact controllability for systems describing plate vibrations. A perturbation approach |
| title_sort | exact controllability for systems describing plate vibrations a perturbation approach |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.539/ |
| work_keys_str_mv | AT bournissoumegane exactcontrollabilityforsystemsdescribingplatevibrationsaperturbationapproach AT ervedozasylvain exactcontrollabilityforsystemsdescribingplatevibrationsaperturbationapproach AT tucsnakmarius exactcontrollabilityforsystemsdescribingplatevibrationsaperturbationapproach |