Stabilizing a vibrating string by time delay

In the application of feedback controls, the computation of the controls may cause a delay. For vibrating systems, a constant delay can destroy the stabilizing effect of the control. To avoid this problem we consider a feedback where a certain delay is a part of the control law and not a perturbation. We consider a string that is fixed at one end and controlled with a boundary feedback with constant delay at the other end. We show the exponential stability of this system that is governed by the wave equation. Moreover, we show the robustness of the stability with respect to variations in time of the feedback parameter that appears as a factor in the control law.