Carleman Estimate for Ultrahyperbolic Operators and Improved Interior Control for Wave Equations

Abstract

In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in ${\mathbb{R}}_t^m\times {\mathbb{R}}_x^n$. Then, we use a special case of this estimate to obtain improved observability results for wave equations with time-dependent lower order terms. The key improvements are: (1) we obtain smaller observation regions compared to standard Carleman estimate results, and (2) we also prove observability when the observation point lies inside the domain. Finally, as a corollary of the observability result, we obtain improved interior controllability for the wave equation.

47 pages, moved contents of the appendix into the main section

References in corpus (3)

• Unique continuation from infinity for linear waves
• Global uniqueness theorems for linear and nonlinear waves
• On Carleman and Observability Estimates for Wave Equations on Time-Dependent Domains