Stabilisation of Timoshenko beam system with delay in the boundary control

Abstract

In this paper, we consider the exponential stabilisation problem of a Timoshenko beam with delay in boundary control. Suppose that the controller outputs of the forms α₁u₁(t) + β₁u₁(t − τ) and α₂u₂(t) + β₂u₂(t − τ) where u₁(t) and u₂(t) are the inputs of boundary controllers. In the past, most of the stabilisation results are required α_j > β_j > 0,  j = 1, 2. In the present paper, we shall give a new dynamic feedback control law that makes the system exponential stabilisation ∀τ > 0 provided that |α_j| ≠ |β_j|(j = 1, 2). The exponential stabilisation result is proved via test of exact observability of the system.

Keywords:

• Timoshenko beam
• input delay
• boundary control; exponential stabilisation
• exact observability

Acknowledgements

This research was supported by the National Science Natural Foundation in China (NSFC–61174080).