Issue |
ESAIM: COCV
Volume 18, Number 1, January-March 2012
|
|
---|---|---|
Page(s) | 22 - 35 | |
DOI | https://doi.org/10.1051/cocv/2010044 | |
Published online | 02 December 2010 |
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
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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