On Carleman and Observability Estimates for Wave Equations on Time-Dependent Domains

TL;DR

This paper develops new Carleman estimates for wave equations on time-dependent domains, leading to improved observability inequalities and exact controllability results in more general and smaller observation regions.

Contribution

It introduces novel Carleman estimates applicable to wave equations on general time-dependent domains with moving boundaries, enabling better control and observation results.

Findings

New Carleman estimates for wave equations on time-dependent domains.

Enhanced observability inequalities with smaller observation regions.

Exact controllability results for linear waves in dynamic domains.

Abstract

We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a fully general class of time-dependent domains, with timelike moving boundaries, (b) they apply to linear wave equations in any spatial dimension and with general time-dependent lower-order coefficients, and (c) they allow for significantly smaller time-dependent regions of observations than allowed from existing Carleman estimate methods. As a standard application, we establish exact controllability for general linear waves, again in the setting of time-dependent domains and regions of control.