Output feedback stabilization of an unstable wave equation with observations subject to time delay

This paper focuses on boundary stabilization of a one-dimensional wave equation with an unstable boundary condition, in which observations are subject to arbitrary fixed time delay. The observability inequality indicates that the open-loop system is observable, based on which the observer and predictor are designed: The state of system is estimated with available observation and then predicted without observation. After that equivalently the authors transform the original system to the well-posed and exponentially stable system by backstepping method. The equivalent system together with the design of observer and predictor give the estimated output feedback. It is shown that the closed-loop system is exponentially stable. Numerical simulations are presented to illustrate the effect of the stabilizing controller.