Wave energy decay under fractional derivative controls

^** Present address: Division of Mathematics and Sciences, Rust College, 150 Rust Avenue, Holly Springs, MS 38635, USA In this article, we investigate the asymptotic behaviour ofsolutions of the 1D wave equation with a boundary viscoelasticdamper of the fractional derivative type. We show that the systemis well-posed in the sense of semigroup. We also prove thatthe associated semigroup is not exponentially stable, but onlystrongly asymptotically so. Finally, we establish the followingresult. Provided that the initial states of the system are chosensufficiently smooth and the relaxation function of the viscoelasticdamper is exponentially decreasing, then solutions of the systemwill decay, as time goes to infinity, as [graphic: see PDF] A > 0.