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Stabilization of wave systems with input delay in the boundary control

Published online by Cambridge University Press:  11 October 2006

Gen Qi Xu ,
Siu Pang Yung  and
Leong Kwan Li
Gen Qi Xu
Affiliation:
Mathematics Department of Tianjin University, Tianjin, 300072, P.R. China; gqxu@tju.edu.cn
Siu Pang Yung
Affiliation:
Mathematics Department of Hong Kong University, Hong Kong, P.R. China; spyung@hku.hk
Leong Kwan Li
Affiliation:
Applied Mathematics Department of the Hong Kong Polytechnic University, Hong Kong, P.R. China; malblkli@polyu.edu.hk
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Abstract

In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu)$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a C0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert space. Furthermore, we show that when the weight $\mu>\frac{1}{2}$, for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When $\mu=\frac{1}{2}$, we show that the system is at most asymptotically stable. When $\mu<\frac{1}{2}$, the system is always unstable.

Keywords

Wave equationtime delaystabilizationRiesz basis.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 4 , October 2006 , pp. 770 - 785
Copyright
© EDP Sciences, SMAI, 2006

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