Exponential Stability and exact controllability of a system of coupled wave equations by second order terms (via Laplacian) with only one non-smooth local damping

TL;DR

This paper proves exponential stability and exact controllability of coupled wave equations with local damping using advanced mathematical techniques, without geometric restrictions or assumptions on wave speeds.

Contribution

It introduces new methods combining Carleman estimates and frequency domain techniques to establish stability and controllability without geometric or wave speed restrictions.

Findings

System is strongly stable without geometric conditions.

System is exponentially stable under (PMGC) condition.

No restriction on wave propagation speeds.

Abstract

The purpose of this work is to investigate the exponential stability of a second order coupled wave equations by laplacian with one locally internal viscous damping. Firstly, using a unique continuation theorem combined with a Carleman estimate, we prove that our system is strongly stable without any geometric condition. Secondly, using a combination of the multiplier techniques and the frequency domain approach, we show that our system is exponentially stable under \textbf{(PMGC)} condition on the damping region without any restriction on wave propagation speed (i.e whether the two wave equations propagate at the same speed or not)