Stability results for second-order evolution equations with memory and switching time-delay

TL;DR

This paper investigates the stability of second-order evolution equations with memory and intermittent delay feedback, demonstrating conditions under which asymptotic or exponential stability is maintained despite the presence of delays.

Contribution

It provides new stability results for equations with memory and switching delay, extending known stability conditions to more complex, realistic scenarios.

Findings

Asymptotic stability holds if delay feedback coefficient is in L^1 and off intervals are large.

Exponential stability is preserved under certain assumptions on memory kernel and delay feedback.

Stability is maintained despite intermittent delay feedback effects.

Abstract

It is well-known that wave-type equations with memory, under appropriate assumptions on the memory kernel, are uniformly exponentially stable. On the other hand, time delay effects may destroy this behavior. Here, we consider the stabilization problem for second-order evolution equations with memory and intermittent delay feedback. We show that, under suitable assumptions involving the delay feedback coefficient and the memory kernel, asymptotic or exponential stability are still preserved. In particular, asymptotic stability is guaranteed if the delay feedback coefficient belongs to $L^1(0;+\infty)$ and the time intervals where the delay feedback is off are sufficiently large.