Theoretical and numerical indirect stabilization of coupled wave equations with a single time-delayed damping

TL;DR

This paper investigates how time delay affects the exponential stabilization of coupled wave equations with both delayed and non-delayed damping, providing theoretical conditions and numerical validation in 1D domains.

Contribution

It offers a combined theoretical and numerical analysis of the impact of time delay on stabilization of coupled wave systems, including a new energy-preserving numerical scheme.

Findings

Time delay influences exponential decay rates.

Theoretical conditions guarantee stabilization.

Numerical scheme confirms theoretical results.

Abstract

The focal point of this paper is to theoretically investigate and numerically validate the effect of time delay on the exponential stabilization of a class of coupled hyperbolic systems with delayed and non-delayed dampings. The class in question consists of two strongly coupled wave equations featuring a delayed and non-delayed damping terms on the first wave equation. Through a standard change of variables and semi-group theory, we address the well-posedness of the considered coupled system. Thereon, based on some observability inequalities, we derive sufficient conditions guaranteeing the exponential decay of a suitable energy. On the other hand, from the numerical point of view, we validate the theoretical results in $1D$ domains based on a suitable numerical approximation obtained through the Finite Difference Method. More precisely, we construct a discrete numerical scheme which preserves the energy decay property of its continuous counterpart. Our theoretical analysis and implementation of our developed numerical scheme assert the effect of the time-delayed damping on the exponential stability of strongly coupled wave equations.