Boundary Output Feedback Stabilization of a One-Dimensional Wave Equation System with Time Delay

The stabilization with time delay in observation and control represents extremely difficult mathematical challenges in distributed parameter systems control. It is well-known that the closed-loop stability achieved by some stabilizing output feedback laws is not robust for any small time delay. When this happens, it necessary to reconsider the stabilizing feedback control laws. We are concerned with a particularly interesting case: boundary output feedback stabilization of one-dimensional wave equation system for which the boundary observation suffers a time delay. This stabilization problem has been unsolved for over two decades. We construct an infinite-dimensional observer such that the estimation error converges exponentially to zero as time goes to infinity and we design a stabilizing state feedback law. Using the separation principle as in the finite dimensional cases, we show that the delay system is exponentially stabilized by the feedback law based on the estimated state.