Stabilization of Timoshenko beam by means of pointwise controls
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.

DOI : 10.1051/cocv:2003028
Classification : 34K35, 47A65, 93B52, 93C20
Keywords: Timoshenko beam, pointwise feedback control, generalized eigenfunction system, Riesz basis 
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     author = {Xu, Gen-Qi and Yung, Siu Pang},
     title = {Stabilization of {Timoshenko} beam by means of pointwise controls},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {579--600},
     year = {2003},
     publisher = {EDP Sciences},
     volume = {9},
     doi = {10.1051/cocv:2003028},
     mrnumber = {1998716},
     zbl = {1068.93024},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2003028/}
}
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Xu, Gen-Qi; Yung, Siu Pang. Stabilization of Timoshenko beam by means of pointwise controls. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 579-600. doi: 10.1051/cocv:2003028

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