Wave energy decay under fractional derivative controls

In this article, we investigate the asymptotic behaviour of solutions of the 1D wave equation with a boundary viscoelastic damper of the fractional derivative type. We show that the system is well-posed in the sense of semigroup. We also prove that the associated semigroup is not exponentially stable, but only strongly asymptotically so. Finally, we establish the following result. Provided that the initial states of the system are chosen sufficiently smooth and the relaxation function of the viscoelastic damper is exponentially decreasing, then solutions of the system will decay, as time goes to infinity, as graphic: see PDF A > 0.