Stability analysis for 1-D wave equation with delayed feedback control

In this paper, we investigate the stability problem of 1-D wave equations with delayed feedback control on the boundary. By a delicate spectral analysis, the sufficient and necessary conditions for the feedback gain and the time delay are derived to guarantee the exponential stability of the closed-loop system. We discuss about all the situations for the time delay $\tau >0$ including the case that τ is irrational. The stability region of the feedback gain exists if and only if the time delay τ is an even number. In this case, an explicit formula of the stability region is obtained accordingly and it characterizes the shrink of the stability region as τ tends to infinity. In addition, we find that the small perturbation of magnitude in the time delay can only trigger the excitation of high frequency modes. That completely proves the judgement in [3, Page 5, Remark] and gives a mathematical explanation why numerical experiments usually do not demonstrate the non-robustness when a small perturbation is added to the time delay.