Output feedback stabilization of cascaded ODE‐Wave equations with time delay in observation

In this paper, we are concerned with the stabilization of a cascaded ODE‐wave system subject to boundary observation with time delay. Well‐posedness of the open‐loop system is observable to show first. Both observer and predictor systems are designed to estimate the state variable on the time interval [0,t−τ] when the observation is available, and to predict the state variable on the time interval [t−τ,t] when the observation is not available respectively. Next, we transform the original system to the well‐posed and exponentially stable system by the backstepping method. Based on the estimated state variable and the output feedback stabilizing controller using the backstepping method. It is shown that the closed‐loop system is exponentially stable. The numerical simulation is presented to illustrate the effect of the stabilizing controller.