no. 4
Stabilization of wave systems with input delay in the boundary control
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 770-785

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-μ) 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 μ>1 2, for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When μ=1 2, we show that the system is at most asymptotically stable. When μ<1 2, the system is always unstable.

DOI : 10.1051/cocv:2006021
Classification : 34H05, 49J25, 49K25, 93D15
Keywords: wave equation, time delay, stabilization, Riesz basis 
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     title = {Stabilization of wave systems with input delay in the boundary control},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {770--785},
     year = {2006},
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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, Tome 12 (2006) no. 4, pp. 770-785. doi: 10.1051/cocv:2006021

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