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

Bao-Zhu Guo; Cheng-Zhong Xu; Hassan Hammouri

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 22-35
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Guo, Bao-Zhu, Xu, Cheng-Zhong, and 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 18.1 (2012): 22-35. <http://eudml.org/doc/277818>.

@article{Guo2012,
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. },
author = {Guo, Bao-Zhu, Xu, Cheng-Zhong, Hammouri, Hassan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Wave equation; time delay; observer; predictor; feedback control; stability},
language = {eng},
month = {2},
number = {1},
pages = {22-35},
publisher = {EDP Sciences},
title = {Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation∗},
url = {http://eudml.org/doc/277818},
volume = {18},
year = {2012},
}

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
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 22
EP - 35
AB - 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.
LA - eng
KW - Wave equation; time delay; observer; predictor; feedback control; stability
UR - http://eudml.org/doc/277818
ER -

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