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IMA Journal of Mathematical Control and Information Advance Access published online on July 27, 2005
IMA Journal of Mathematical Control and Information, 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
Received April 12, 2004
Accepted October 8, 2004

Article

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

Xu Gen Qi 1* and Jia Jun Guo 2

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

* To whom correspondence should be addressed.
Xu Gen Qi, E-mail: gqxu{at}tju.edu.cn


  Abstract

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.
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