Exponential Stability of a Transmission Problem with History and Delay

Abstract

In this paper, we consider a transmission problem in the presence of history and delay terms.Under appropriate assumptions, we prove well-posedness by using the semigroup theory. Our stability estimate proves that the unique dissipation given by the history term is strong enough to stabilize exponentially the system in presence of delay by introducing a suitable Lyaponov functional.

Decay Properties for Transmission System with Infinite Memory and Distributed Delay

“…Under suitable assumptions on the delay term and the function g, an exponential stability result is proved for these two cases: µ 2 < µ 1 and µ 2 = µ 1 . Also, system (1)- (5) was studied by Bahri et al [6]; the distributed delay is replaced with the fixed delay and µ 1 = 0. Under certain appropriate hypotheses on the relaxation function and the weight of the delay, the authors proved the well-posedness by using the semigroup theory technique.…”