Stabilization of an Euler–Bernoulli beam with input delay in the boundary control

In this paper, we consider a cantilever Euler–Bernoulli beam with controller delay. Suppose that the output of the boundary controller is of the form αu(t)+βu(t−τ)αu(t)+βu(t−τ), where u(t)u(t) is the controller input. It is well known that a system with a collocated feedback control law is exponentially stable if β=0β=0, and it is unstable with the same control law when α=0α=0. In the present paper, we shall design a dynamic controller that makes the system stabilize exponentially for any |α|≠|β||α|≠|β| and τ>0τ>0. Furthermore, we discuss the stability of the system for |α|=|β||α|=|β|.