Optimal design of boundary observers for the wave equation

¹ Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
(jounieaux@ann.jussieu.fr)
² CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
(yannick.privat@math.cnrs.fr)
³ Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris, France
(emmanuel.trelat@upmc.fr)

Abstract

In this article, we consider the wave equation on a domain of Rⁿ with Lipschitz boundary. For every observable subset Γ of the boundary ∂Ω (endowed with the usual Hausdorff measure H^{n − 1} on ∂Ω), the observability constant provides an account for the quality of the reconstruction in some inverse problem. Our objective is here to determine what is, in some appropriate sense, the best observation domain. After having defined a randomized observability constant, more relevant tan the usual one in applications, we determine the optimal value of this constant over all possible subsets Γ of prescribed area H^{n − 1}(Γ) = LH^{n − 1}(∂Ω), with L ∈ (0,1), under appropriate spectral assumptions on Ω. We compute the maximizers of a relaxed version of the problem, and then study the existence of an optimal set of particular domains Ω. We then define and study an approximation of the problem with a finite number of modes, showing existence and uniqueness of an optimal set, and provide some numerical simulations.