Stabilization of Timoshenko Beam by Means of Pointwise Controls

Gen-Qi Xu; Siu Pang Yung

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 579-600
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Xu, Gen-Qi, and Yung, Siu Pang. "Stabilization of Timoshenko Beam by Means of Pointwise Controls." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 579-600. <http://eudml.org/doc/90712>.

@article{Xu2010,
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. },
author = {Xu, Gen-Qi, Yung, Siu Pang},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Timoshenko beam; pointwise feedback control; generalized eigenfunction system; Riesz basis.; generalized eigenfunction system; Riesz basis},
language = {eng},
month = {3},
pages = {579-600},
publisher = {EDP Sciences},
title = {Stabilization of Timoshenko Beam by Means of Pointwise Controls},
url = {http://eudml.org/doc/90712},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Xu, Gen-Qi
AU - Yung, Siu Pang
TI - Stabilization of Timoshenko Beam by Means of Pointwise Controls
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 579
EP - 600
AB - 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.
LA - eng
KW - Timoshenko beam; pointwise feedback control; generalized eigenfunction system; Riesz basis.; generalized eigenfunction system; Riesz basis
UR - http://eudml.org/doc/90712
ER -

References

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