Geodesic
Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 22-35

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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.

DOI : 10.1051/cocv/2010044
Classification : 35B37, 93B52, 93C05, 93C20, 93D15
Keywords: wave equation, time delay, observer, predictor, feedback control, stability 
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     author = {Guo, Bao-Zhu and Xu, Cheng-Zhong and Hammouri, Hassan},
     title = {Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {22--35},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {1},
     year = {2012},
     doi = {10.1051/cocv/2010044},
     mrnumber = {2887926},
     zbl = {1246.35120},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010044/}
}
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Guo, Bao-Zhu; Xu, Cheng-Zhong; Hammouri, Hassan. Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 22-35. doi: 10.1051/cocv/2010044

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