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Geometric control
condition for the wave equation with a time-dependent
observation domain
Jérôme Le Rousseau, Gilles Lebeau, Peppino Terpolilli and Emmanuel Trélat |
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Vol. 10 (2017), No. 4, 983–1015
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Abstract |
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We characterize the observability property (and, by duality, the controllability and the stabilization) of the wave equation on a Riemannian manifold , with or without boundary, where the observation (or control) domain is time-varying. We provide a condition ensuring observability, in terms of propagating bicharacteristics. This condition extends the well-known geometric control condition established for fixed observation domains. As one of the consequences, we prove that it is always possible to find a time-dependent observation domain of arbitrarily small measure for which the observability property holds. From a practical point of view, this means that it is possible to reconstruct the solutions of the wave equation with only few sensors (in the Lebesgue measure sense), at the price of moving the sensors in the domain in an adequate way. We provide several illustrating examples, in which the observation domain is the rigid displacement in of a fixed domain, with speed , showing that the observability property depends both on and on the wave speed. Despite the apparent simplicity of some of our examples, the observability property can depend on nontrivial arithmetic considerations. |
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Keywords
wave equation, geometric control condition, time-dependent
observation domain
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Mathematical Subject Classification 2010
Primary: 35L05, 93B07, 93C20
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Milestones
Received: 13 July 2016
Revised: 25 January 2017
Accepted: 7 March 2017
Published: 9 May 2017
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Authors |
| Jérôme Le Rousseau | |
| Université Paris-Nord, CNRS UMR
7339 Laboratoire d’analyse géométrie et applications (LAGA) Institut universitaire de France 99 Avenue Jean Baptiste Clément 93430 Villetaneuse France |
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| Gilles Lebeau | |
| Université de Nice Sophia-Antipolis,
CNRS UMR 7351 Laboratoire J.-A. Dieudonné Parc Valrose 06108 Nice France |
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| Peppino Terpolilli | |
| Total Centre scientifique et technique Jean-Feger Avenue Larribau 64000 Pau France |
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| Emmanuel Trélat | |
| Sorbonne Universités, UPMC Univ
Paris 06, CNRS UMR 7598 Laboratoire Jacques-Louis Lions Institut universitaire de France 75005 Paris France |
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