Vol. 10, No. 4, 2017

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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

Vol. 10 (2017), No. 4, 983–1015

We characterize the observability property (and, by duality, the controllability and the stabilization) of the wave equation on a Riemannian manifold Ω, 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 Ω of a fixed domain, with speed v, showing that the observability property depends both on v and on the wave speed. Despite the apparent simplicity of some of our examples, the observability property can depend on nontrivial arithmetic considerations.

wave equation, geometric control condition, time-dependent observation domain
Mathematical Subject Classification 2010
Primary: 35L05, 93B07, 93C20
Received: 13 July 2016
Revised: 25 January 2017
Accepted: 7 March 2017
Published: 9 May 2017
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
Gilles Lebeau
Université de Nice Sophia-Antipolis, CNRS UMR 7351
Laboratoire J.-A. Dieudonné
Parc Valrose
06108 Nice France
Peppino Terpolilli
Centre scientifique et technique Jean-Feger
Avenue Larribau
64000 Pau
Emmanuel Trélat
Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598
Laboratoire Jacques-Louis Lions
Institut universitaire de France
75005 Paris