The stabilization of one-dimensional wave equation by delayed output feedback

In this paper, we study the stability of a string equation with time delays in the output feedback loop. When the delay is equal to the even multiples of the wave propagation time, we develop the necessary and sufficient conditions for the feedback gain and time delay which guarantee the exponential stability of the closed-loop system. We also show that when the delay is an odd multiple of the wave propagation time, the closed-loop system is unstable. In the particular case of delay equal to two, the lack of robustness to a small perturbation in time delay is discussed. The semigroup theory and Riesz basis approach are adopted in investigation. Finally, some numerical simulation for the case of the delay equal to two is presented to illustrate the convergence.