Geometric control condition for the wave equation with a time-dependent observation domain

By Jérôme Le Rousseau, Gilles Lebeau, P. Terpolilli, Emmanuel Trélat. 2016

74 citations recorded by the source corpus.

We characterize the observability property (and, by duality, the controllability and the stabilization) of the wave equation on a Riemannian manifold $\Omega,$ 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 observationdomain is the rigid displacement in $\Omega$ of a fixed domain, withspeed $v,$ showing that the observability property depends both on $v$and on the wave speed. Despite the apparent simplicity of some of ourexamples, the observability property can depend on nontrivial arithmeticconsiderations.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12e49841ada31cc8ca693161e577a54059ba489d

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In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Characterization by observability inequalities of controllability and stabilization properties (2018, background): “Let ω ⊂ Ω be an open subset satisfying the Geometric Control Condition in some time T (see [3, 19]).”
Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn (2017, background): “Thus, we can say that the GCC condition is a characteristic of observable open sets on ∂Ω, though this condition is not strictly necessary (see [29]).”
Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs (2020, background): “Geometric Control Condition(GCC) (see [2, 57]): The domain D ′ T ⊂ ?̄?T , is said to satisfy the geometric control condition (gcc) in DT if every compressed generalized bicharacteristic (x(s), t (s), τ(s), ξ (s) intersects the set D ′ T for some s ∈ R We refer the reader to [2, 57] for the rather te”