Stabilization of wave systems with input delay in the boundary control

Issue		ESAIM: COCV Volume 12, Number 4, October 2006


Page(s)		770 - 785
DOI		10.1051/cocv:2006021
Published online		11 October 2006

ESAIM: COCV, October 2006, Vol. 12, pp. 770-785
DOI: 10.1051/cocv:2006021

Stabilization of wave systems with input delay in the boundary control

Gen Qi Xu¹, Siu Pang Yung² and Leong Kwan Li³

¹ Mathematics Department of Tianjin University, Tianjin, 300072, P.R. China; gqxu@tju.edu.cn
² Mathematics Department of Hong Kong University, Hong Kong, P.R. China; spyung@hku.hk
³ Applied Mathematics Department of the Hong Kong Polytechnic University, Hong Kong, P.R. China; malblkli@polyu.edu.hk

(Received July 1st, 2004. Revised May 10 and October 13, 2005. Published online 11 October 2006.)

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.

Mathematics Subject Classification. 34H05, 49J25, 49K25, 93D15

Key words: Wave equation, time delay, stabilization, Riesz basis.

© EDP Sciences, SMAI 2006

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