Contrôlabilité asymptotique de systèmes hyperboliques linéaires

Tatsien Li; Bopeng Rao

doi:10.1016/j.crma.2011.03.006

Équations aux dérivées partielles/Contrôle optimal

Contrôlabilité asymptotique de systèmes hyperboliques linéaires
[Asymptotic controllability for linear hyperbolic systems]

Presented by: Philippe G. Ciarlet

Tatsien Li ^1,² ; Bopeng Rao ³

¹ School of Mathematical Sciences, Fudan University, Shanghai 200433, China
² Shanghai Key Laboratory for Contemporary Applied Mathematics; Nonlinear Mathematical Modeling and Methods Laboratory, China
³ Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 663-668.

Abstract
Résumé

In this Note we introduce the asymptotic controllability and the asymptotic null controllability for 1-D linear hyperbolic systems under the lack of boundary controls. We claim that they are equivalent, respectively, to the strong observability and the weak observability for the dual system. An example of $4 \times 4$ hyperbolic system with only one boundary control is shown to be asymptotically controllable but not exactly controllable.

Dans cette Note, nous considérons la contrôlabilité asymptotique et la contrôlabilité nulle asymptotique pour des systèmes hyperboliques linéaires en dimension dʼespace un. Nous établissons quʼelles sont équivalentes, respectivement, à lʼobservabilité forte et lʼobservabilité faible du système dual. Nous donnons un exemple dʼun système hyperbolique $4 \times 4$ sousmis à un seul contrôle frontière, qui est asymptotiquement contrôlable mais non exactement contrôlable.

Received: 2010-12-21
Accepted: 2011-02-24
Published online: 2011-05-25

DOI: 10.1016/j.crma.2011.03.006

Author's affiliations:

Tatsien Li ^{1, 2}; Bopeng Rao ³

¹ School of Mathematical Sciences, Fudan University, Shanghai 200433, China
² Shanghai Key Laboratory for Contemporary Applied Mathematics; Nonlinear Mathematical Modeling and Methods Laboratory, China
³ Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg, France

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     author = {Tatsien Li and Bopeng Rao},
     title = {Contr\^olabilit\'e asymptotique de syst\`emes hyperboliques lin\'eaires},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {663--668},
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     year = {2011},
     doi = {10.1016/j.crma.2011.03.006},
     language = {fr},
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Tatsien Li; Bopeng Rao. Contrôlabilité asymptotique de systèmes hyperboliques linéaires. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 663-668. doi : 10.1016/j.crma.2011.03.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.03.006/

References
Cited by

[1] Vilmos Komornik; Paola Loreti Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, 2005

[2] Tatsien Li Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, vol. 3, American Institute of Mathematical Sciences & Higher Education Press, 2010

[3] Tatsien Li; Bopeng Rao Local exact boundary controllability for a class of quasilinear hyperbolic systems, Chin. Ann. Math. B, Volume 23 (2002), pp. 209-218

[4] Tatsien Li; Bopeng Rao Exact boundary controllability for quasilinear hyperbolic systems, SIAM J. Control Optim., Volume 41 (2003), pp. 1748-1755

[5] Tatsien Li; Bopeng Rao Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems, Chin. Ann. Math. B, Volume 31 (2010), pp. 723-742

[6] Tatsien Li; Bopeng Rao Exact controllability and exact observability for quasilinear hyperbolic systems: Known results and open problems (Tatsien Li; Yuejun Peng; Bopeng Rao, eds.), Series in Contemporary Applied Mathematics, vol. 15, Higher Education Press, World Scientific, 2010, pp. 374-385

[7] Jacques-Lious Lions; Contrôlabilié Exacte Perturbations et Stabilisation de Systèmes Distribués, tome 1, Masson, 1988

[8] David L. Russell Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., Volume 20 (1978), pp. 639-739

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