Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation^∗

¹ Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, P.R. China
Bao-Zhu.Guo@wits.ac.za
² School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R. China
³ School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
⁴ Université de Lyon, LAGEP, Bâtiment CPE, Université Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

Received: 26 August 2009
Revised: 15 June 2010

Abstract

The stabilization with time delay in observation or control represents difficult mathematical challenges in the control of distributed parameter systems. It is well-known that the stability of closed-loop system achieved by some stabilizing output feedback laws may be destroyed by whatever small time delay there exists in observation. In this paper, we are concerned with a particularly interesting case: Boundary output feedback stabilization of a one-dimensional wave equation system for which the boundary observation suffers from an arbitrary long time delay. We use the observer and predictor to solve the problem: The state is estimated in the time span where the observation is available; and the state is predicted in the time interval where the observation is not available. It is shown that the estimator/predictor based state feedback law stabilizes the delay system asymptotically or exponentially, respectively, relying on the initial data being non-smooth or smooth. Numerical simulations are presented to illustrate the effect of the stabilizing controller.