Stabilization of wave systems with input delay in the boundary control

Abstract

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 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.Mathematics Subject Classification. 34H05, 49J25, 49K25, 93D15.

Global existence and energy decay of solutions to a Bresse system with delay terms

“…The same results were showed if both the damping and the delay act in the boundary domain. We also recall the result by Xu, Yung and Li [13], where the authors proved the same result as in [12] for the one space dimension by adopting the spectral analysis approach.…”

“…For example, it was proved in [11] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay. To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary (see [12] and [13]). For instance, in [12] the authors studied the wave equation with a linear internal damping term with constant delay and determined suitable relations between µ 1 and µ 2 , for which the stability or alternatively instability takes place.…”

Stabilization of Second Order Evolution Equations with Unbounded Feedback with Time-Dependent Delay

“…If (3) is not satisfied, there exist cases where instabilities may appear, see [16,17,23] for the wave equation. Hence this condition appears to be quite realistic.…”

Well-Posedness and Asymptotic Stability of Solutions to the Bresse System under Cattaneo's Law with Infinite Memories and Time-Varying Delays

“…In [8], the authors showed that a small delay in a boundary control can destabilize a system which is uniformly asymptotically stable in the absence delays. To stabilize a hyperbolic system involving input delay terms, additional control will be necessary [22,29]. Kirane and Said-Houari [20] considered a viscoelastic wave equation with a linear damping and a delay of the form…”