Volume 18, Number 1, January-March 2012
|Page(s)||22 - 35|
|Published online||02 December 2010|
Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation∗
Academy of Mathematics and Systems Science, Academia
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
Revised: 15 June 2010
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.
Mathematics Subject Classification: 35B37 / 93B52 / 93C05 / 93C20 / 93D15
Key words: Wave equation / time delay / observer / predictor / feedback control / stability
© EDP Sciences, SMAI, 2010
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