# 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

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topGuo, 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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