Stability analysis of Euler-Bernoulli beam with input delay in the boundary control

In this paper, we investigate an Euler-Bernoulli system with input delay in the boundary control. Suppose that there is no delay in observation, y(t), of the system, and a partial input delay in the boundary control. The collocated boundary feedback control law u(t) = alpha y(t) + beta y(t - tau) is applied to obtain the closed loop system. By spectral analysis and Lyapunov method, we show that: when alpha>|beta|, the closed loop system is exponentially stable for any tau>0; when alpha<|beta|, the system is unstable for any tau>0; when a = |beta|, the system is asymptotically stable for almost all tau>0. Finally, we provide numerical simulations to show the spectral distribution for different values of a and beta.