Delay is Necessary for a Potential to Achieve Exponential Stabilization of the Wave Equation via Internal Control

TL;DR

This paper demonstrates that delay in internal potential control is essential for exponential stabilization of the wave equation, with a novel feedback law ensuring decay, validated through theoretical analysis and a string vibration example.

Contribution

It introduces a delayed potential feedback law that achieves exponential stabilization of the wave equation, highlighting the necessity of delay for stabilization.

Findings

Delay induces stabilization where none is possible without it.

The proposed feedback law guarantees exponential decay.

Application to string vibrations confirms effectiveness.

Abstract

In this work, we study the stabilization of the wave equation using an internal delayed potential. Interestingly, the stabilization mechanism is entirely induced by the delay, since exponential stabilization cannot be achieved in its absence. We first prove the well-posedness of the associated initial--boundary value problem. Then, thanks to the parametric analysis of the corresponding quasipolynomial, we design a delayed po tential feedback law which, together with appropriate initial conditions, ensures the exponential decay rate for the resulting closed-loop system. The control of the transverse vibration of a string illustrates the effectiveness of the result.