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IMA Journal of Mathematical Control and Information Advance Access originally published online on July 27, 2005
IMA Journal of Mathematical Control and Information 2006 23(1):85-96; doi:10.1093/imamci/dni045
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Institute of Mathematics and its Applications 2005; all rights reserved for permissions please email: journals.permissions@oupjournals.org

The group and Riesz basis properties of string systems with time delay and exact controllability with boundary control

Xu Gen Qi1,** and Jia Jun Guo2,***

1 Mathematics Department of Tianjin University, Tianjin 300072, P. R. China, 2 Mathematics Department of Xian Jiaotong University, Xian 710049, P. R. China

** Email: gqxu{at}tju.edu.cn

*** Email: Jiajg{at}mailst.xjtu.edu.cn

The group property of a string system with time delay in boundary and the Riesz basis property of eigenvectors of the system are discussed in the present paper. It is proved that, when the feedback with delay {tau} > 0, the system also associates a C0 group, and its eigenvectors (generalized eigenvectors) form a Riesz basis in Hilbert space H. This result shows that time delay may destroy the stability of the system, but the group and Riesz basis properties are kept. As a consequence, the exact controllability of the system with boundary control is given.

Keywords: string equation; time delay; C0 group; Riesz basis property; exact controllability.

Received on 12 April 2004. accepted on 8 October 2004.


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