Uniform observability estimates for linear waves

TL;DR

This paper presents a constructive proof of uniform observability estimates for the wave equation on compact manifolds, improving understanding of control properties with explicit constants and handling low- and high-frequency components.

Contribution

It introduces a fully constructive proof of observability for wave equations under optimal geometric conditions, contrasting with previous non-constructive methods.

Findings

Explicit estimates of the blowup of the observability constant near the geometric control time

Dependence of the observability constant on added bounded potentials

Application to wave equations with lower order terms

Abstract

In this article, we give a completely constructive proof of the observability/controllability of the wave equation on a compact manifold under optimal geometric conditions. This contrasts with the original proof of Bardos-Lebeau-Rauch, which contains two non-constructive arguments. Our method is based on the Dehman-Lebeau Egorov approach to treat the high-frequencies, and the optimal unique continuation stability result of the authors for the low-frequencies. As an application, we first give estimates of the blowup of the observability constant when the time tends to the limit geometric control time (for wave equations with possibly lower order terms). Second, we provide (on manifolds with or without boundary) with an explicit dependence of the observability constant with respect to the addition of a bounded potential to the equation.