The strong stabilization of a 1-D conservative wave equation by boundary displacement only

Abstract

In this paper, we consider the stabilization of a 1-D conservative wave equation by collocated boundary displacement signal only. We first design an observer for the original system, in which we introduce an ordinary differential equation so that the unavailable velocity can be recovered. Then we use the C0-semigroup theory to verify the well-posedness and stability of the error system. Next we propose a feedback law in terms of the observer. Consequently, we obtain the asymptotic stability of the closed-loop system. Finally, using the finite element method, the numerical simulations show that the open-loop system is conservative and the closed-loop system is asymptotically stable.