In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.
Mots-clés : Timoshenko beam, exponential stability, time delay, Riesz basis, feedback control
@article{COCV_2011__17_2_552_0,
author = {Han, Zhong-Jie and Xu, Gen-Qi},
title = {Exponential stability of {Timoshenko} beam system with delay terms in boundary feedbacks},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {552--574},
publisher = {EDP-Sciences},
volume = {17},
number = {2},
year = {2011},
doi = {10.1051/cocv/2010009},
mrnumber = {2801331},
zbl = {1251.93106},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010009/}
}
TY - JOUR AU - Han, Zhong-Jie AU - Xu, Gen-Qi TI - Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 552 EP - 574 VL - 17 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010009/ DO - 10.1051/cocv/2010009 LA - en ID - COCV_2011__17_2_552_0 ER -
%0 Journal Article %A Han, Zhong-Jie %A Xu, Gen-Qi %T Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 552-574 %V 17 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010009/ %R 10.1051/cocv/2010009 %G en %F COCV_2011__17_2_552_0
Han, Zhong-Jie; Xu, Gen-Qi. Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 552-574. doi : 10.1051/cocv/2010009. http://www.numdam.org/articles/10.1051/cocv/2010009/
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