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.
Highlights
In this article, we establish new Carleman estimates for the wave equation using a geometric approach
The main objective is to apply these estimates in order to derive novel observability inequalities for general linear wave equations, with the following features: (I) The estimates apply to a general class of time-dependent domains, with moving boundaries. (II) The estimates apply to wave equations in any spatial dimension. (III) The estimates apply to general linear waves with time-dependent lower-order coefficients. (IV) The estimates apply for a wide variety of time-dependent observation regions that are smaller than those in standard Carleman-based observability inequalities
As a standard application of these observability estimates, we establish the exact controllability of linear waves on the same general class of time-dependent domains
Summary
We establish new Carleman estimates for the wave equation using a geometric approach. One of the primary goals of the present article is to tackle this problem of Dirichlet boundary controllability of wave equations (with lower-order coefficients varying non-analytically in both space and time) on time-dependent domains in full generality. A larger aim beyond the present paper is to study to similar controllability properties for geometric wave equations, in particular to settings with time-dependent geometry. Carleman estimates have been applied both directly [11, 28] or as part of an intermediate unique continuation argument [10, 34, 32, 33, 35, 36] Another application of the estimates in this article (and their future geometric generalizations) is toward inverse problems for wave equations in settings with time-dependent domains and moving boundaries.
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