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Archive for Rational Mechanics and Analysis

, Volume 211, Issue 1, pp 113–187 | Cite as

Controllability of Two Coupled Wave Equations on a Compact Manifold

  • Belhassen Dehman
  • Jérôme Le RousseauEmail author
  • Matthieu Léautaud
Article

Abstract

We consider the exact controllability problem on a compact manifold Ω for two coupled wave equations, with a control function acting on one of them only. Action on the second wave equation is obtained through a coupling term. First, when the two waves propagate with the same speed, we introduce the time \({T_{\omega \rightarrow \mathcal{O} \rightarrow \omega}}\) for which all geodesics traveling in Ω go through the control region ω, then through the coupling region \({\mathcal{O}}\), and finally come back in ω. We prove that the system is controllable if and only if both ω and \({\mathcal{O}}\) satisfy the Geometric Control Condition and the control time is larger than \({T_{\omega \rightarrow \mathcal{O} \rightarrow \omega}}\). Second, we prove that the associated HUM control operator is a pseudodifferential operator and we exhibit its principal symbol. Finally, if the two waves propagate with different speeds, we give sharp sufficient controllability conditions on the functional spaces, the geometry of the sets ω and \({\mathcal{O}}\), and the minimal time.

Keywords

Compact Manifold Pseudodifferential Operator Principal Symbol Unique Continuation Fourier Integral Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Belhassen Dehman
    • 1
  • Jérôme Le Rousseau
    • 2
    Email author
  • Matthieu Léautaud
    • 3
  1. 1.Département de Mathématiques, Faculté des sciences de TunisUniversité de Tunis El ManarEl ManarTunisia
  2. 2.Laboratoire de Mathématiques, Analyse, Probabilités, ModélisationOrléans CNRS UMR 6628, Fédération Denis-Poisson, FR CNRS 2964, Université d’OrléansOrléans Cedex 2France
  3. 3.Université Paris-Sud 11, MathématiquesOrsay CedexFrance

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