Stabilization of Timoshenko beam by means of pointwise controls

Xu, Gen-Qi; Yung, Siu Pang

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 579-600.

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.

Classification : 34K35, 47A65, 93B52, 93C20
Mots clés : Timoshenko beam, pointwise feedback control, generalized eigenfunction system, Riesz basis

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