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 α u ( t ) + β u ( t − τ ) , where u ( t ) is the controller input. It is well known that a system with a collocated feedback control law is exponentially stable if β = 0 , and it is unstable with the same control law when α = 0 . In the present paper, we shall design a dynamic controller that makes the system stabilize exponentially for any | α | ≠ | β | and τ > 0 . Furthermore, we discuss the stability of the system for | α | = | β | .