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.
Keywords: wave equation, time delay, observer, predictor, feedback control, stability
@article{COCV_2012__18_1_22_0,
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},
year = {2012},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
doi = {10.1051/cocv/2010044},
mrnumber = {2887926},
zbl = {1246.35120},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010044/}
}
TY - JOUR AU - Guo, Bao-Zhu AU - Xu, Cheng-Zhong AU - Hammouri, Hassan TI - Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 22 EP - 35 VL - 18 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010044/ DO - 10.1051/cocv/2010044 LA - en ID - COCV_2012__18_1_22_0 ER -
%0 Journal Article %A Guo, Bao-Zhu %A Xu, Cheng-Zhong %A Hammouri, Hassan %T Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 22-35 %V 18 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010044/ %R 10.1051/cocv/2010044 %G en %F COCV_2012__18_1_22_0
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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