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EXPONENTIAL STABILISATION OF A TREE-SHAPED NETWORK OF STRINGS WITH VARIABLE COEFFICIENTS*

Published online by Cambridge University Press:  10 March 2011

YAN NI GUO  and
GEN QI XU
YAN NI GUO
Affiliation:
Institute of Applied Mathematics, College of Science, Civil Aviation University of China, Tianjin 300300, P.R. China e-mail: gyann@126.com
GEN QI XU
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China e-mail: gqxu@tju.edu.cn
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Abstract

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In this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.

Keywords

93C2093D1535B3535P1047A70
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 481 - 499
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

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