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Stabilization of Timoshenko Beam by Means of Pointwise Controls

Published online by Cambridge University Press:  15 September 2003

Gen-Qi Xu  and
Siu Pang Yung
Gen-Qi Xu
Affiliation:
Department of Mathematics of Shanxi University, TaiYuan 030006, P.R. China.; gqxu@mail.sxu.edu.cn.
Siu Pang Yung
Affiliation:
Department of Mathematics of the University of Hong Kong, Hong Kong, P.R. China; spyung@hku.hk.
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Abstract

We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.

Keywords

Timoshenko beampointwise feedback controlgeneralized eigenfunction systemRiesz basis.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 579 - 600
Copyright
© EDP Sciences, SMAI, 2003

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