Pointwise stabilisation of a string with time delay in the observation

Pointwise stabilisation is proposed in this paper for a string equation where the observation signal is subject to a time delay. Different from the boundary control, the feedback stabiliser is acting at the middle joint of the string. Well-posedness of the open-loop system and solvability of the observer are shown first. An observer system is then designed to estimate the state at the time interval when the observation is available, and a predictor system is designed to predict the state at the time interval when the observation is not available. Pointwise output feedback controller is introduced to make the closed-loop system asymptotically stable for the non-smooth initial values and exponentially stable for the smooth initial values, respectively. Simulation results demonstrate that the output feedback based on the observer and predictor effectively stabilises the pointwise control system with time delay.