In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu )$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a ${C}_{0}$ group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert space. Furthermore, we show that when the weight $\mu >\frac{1}{2}$, for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When $\mu =\frac{1}{2}$, we show that the system is at most asymptotically stable. When $\mu <\frac{1}{2}$, the system is always unstable.

Keywords: wave equation, time delay, stabilization, Riesz basis

@article{COCV_2006__12_4_770_0, author = {Xu, Gen Qi and Yung, Siu Pang and Li, Leong Kwan}, title = {Stabilization of wave systems with input delay in the boundary control}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {770--785}, publisher = {EDP-Sciences}, volume = {12}, number = {4}, year = {2006}, doi = {10.1051/cocv:2006021}, mrnumber = {2266817}, zbl = {1105.35016}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2006021/} }

TY - JOUR AU - Xu, Gen Qi AU - Yung, Siu Pang AU - Li, Leong Kwan TI - Stabilization of wave systems with input delay in the boundary control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 770 EP - 785 VL - 12 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2006021/ DO - 10.1051/cocv:2006021 LA - en ID - COCV_2006__12_4_770_0 ER -

%0 Journal Article %A Xu, Gen Qi %A Yung, Siu Pang %A Li, Leong Kwan %T Stabilization of wave systems with input delay in the boundary control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 770-785 %V 12 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2006021/ %R 10.1051/cocv:2006021 %G en %F COCV_2006__12_4_770_0

Xu, Gen Qi; Yung, Siu Pang; Li, Leong Kwan. Stabilization of wave systems with input delay in the boundary control. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, pp. 770-785. doi : 10.1051/cocv:2006021. http://archive.numdam.org/articles/10.1051/cocv:2006021/

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