Stabilization of wave systems with input delay in the boundary control. (English) Zbl 1105.35016

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

ESAIM, Control Optim. Calc. Var. 12, 770-785 (2006).

Summary: 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.

MSC:

35B35	Stability in context of PDEs
49J25	Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
49K25	Optimal control problems with equations with ret.arguments (nec.) (MSC2000)
93D15	Stabilization of systems by feedback
35B37	PDE in connection with control problems (MSC2000)
47D06	One-parameter semigroups and linear evolution equations
35L20	Initial-boundary value problems for second-order hyperbolic equations

Keywords:

Riesz basis; boundary velocity feedback law; closed loop system

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