Energy decay for evolution equations with delay feedbacks

TL;DR

This paper investigates the stability and energy decay in linear and nonlinear evolution equations with delay feedbacks, establishing conditions for exponential decay using semigroup theory and Gronwall's inequality.

Contribution

It introduces a systematic approach to prove existence, uniqueness, and exponential decay for systems with delay feedbacks, extending previous results to more general settings.

Findings

Existence and uniqueness of solutions established.

Exponential decay estimates proven under certain conditions.

Applicable to both linear and nonlinear systems with multiple delays.

Abstract

We study abstract linear and nonlinear evolutionary systems with single or multiple delay feedbacks, illustrated by several concrete examples. In particular, we assume that the operator associated with the undelayed part of the system generates an exponentially stable semigroup and that the delay damping coefficients are locally integrable in time. A step by step procedure combined with Gronwall's inequality allows us to prove the existence and uniqueness of solutions. Furthermore, under appropriate conditions we obtain exponential decay estimates.