Exponential stability of abstract evolution equations with time delay

TL;DR

This paper proves that semilinear evolution equations with time delay maintain exponential stability if the linear part is stable and the delay feedback is sufficiently small, supported by illustrative examples.

Contribution

It establishes conditions under which exponential stability persists in delayed evolution equations, extending stability results to systems with small delay feedback.

Findings

Exponential stability of the linear part implies stability of the delayed system under small feedback.

A smallness condition on the delay feedback ensures stability retention.

Examples demonstrate the applicability of the abstract results.

Abstract

We consider abstract semilinear evolution equations with a time delay feedback. We show that, if the $C_0$-semigroup describing the linear part of the model is exponentially stable, then the whole system retains this good property when a suitable smallness condition on the time delay feedback is satisfied. Some examples illustrating our abstract approach are also given.