Stabilization of wave systems with input delay in the boundary control

Abstract

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 C₀ 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 Hubert 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.