Boundary feedback stabilization by time delay for one-dimensional wave equations

Abstract

In the application of feedback controls, a delay may appear as a perturbation caused by the computation of the controls. For vibrating systems, this delay can destroy the stabilizing effect of the control. To avoid this problem, we consider feedback laws where a certain delay is included as a part of the control law and not as a perturbation. We consider systems that are governed by the wave equation. As a first system, we consider a string that is fixed at one end and stabilized with a boundary feedback with constant delay at the other end. As a second example, we consider a circular string where both ends of the string are coupled by a feedback law. For both systems, we show the exponential stability of the proposed feedback with retarded input. Moreover, for the first system, we show robustness with respect to variations in time of the feedback parameter.