Work

Boundary sidewise observability of the wave equation

Year: 2025

Type: article

Abstract: The wave equation on a bounded domain of \mathbb{R}^{n} with nonhomogeneous boundary Dirichlet data or sources supported on a subset of the boundary is considered. We analyze the problem of observing the source out of boundary measurements done away from its support. We first notice that due to the existence of solutions that are arbitrarily concentrated near the source, for any given integer N , these observability inequalities may not hold even if we allow a loss of N derivatives. We then establish observability inequalities in Sobolev norms, under a suitable microlocal geometric condition on the support of the source and the measurement set, for sources fulfilling pseudo-differential conditions that exclude these concentration phenomena. The proof relies on microlocal arguments and is essentially based on the use of microlocal defect measures.

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/1688

Authors: Belhassen Dehman, Enrique Zuazua

Institutions: Tunis El Manar University, Universidad de Deusto, Friedrich-Alexander-Universität Erlangen-Nürnberg, Universidad Autónoma de Madrid

Language: English

Topic: Stability and Controllability of Differential Equations

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: diamond

OpenAlex ID: W7081996960

FWCI:

Cites:

Cited by:

Related to: