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

TL;DR

This paper develops a new Carleman estimate for ultrahyperbolic operators and leverages it to enhance interior control and observability results for wave equations, allowing for smaller observation regions and interior observation points.

Contribution

The paper introduces a novel Carleman estimate for ultrahyperbolic operators and applies it to improve interior observability and controllability of wave equations with time-dependent terms.

Findings

Smaller observation regions for wave equations.

Observability with interior observation points.

Enhanced interior controllability results.

Abstract

In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in $ \mathbb{R}^m_t \times \mathbb{R}^n_x $. 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.