Stabilization of wave systems with input delay in the boundary control

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: 1 July 2004
Revised: 10 May 2005
Revised: 13 October 2005

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 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 , for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When , we show that the system is at most asymptotically stable. When , the system is always unstable.