Stabilization of wave systems with input delay in the boundary control

Xu, Gen Qi; Yung, Siu Pang; Li, Leong Kwan

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 $(1-\mu )$ 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.

Classification : 34H05, 49J25, 49K25, 93D15
Mots clés : wave equation, time delay, stabilization, Riesz basis

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