Dynamic Feedback Stabilization for Timoshenko Beam with Locally Input Controls and Input Distributed Delay

WANG Xiaorui(王小瑞),LIU Xilan(劉喜蘭)

( 1.School of Mathematics and Statistics,Qinghai Nationalities University,Xining 810007,China;2.School of Mathematics and Information Science,Baoji University of Arts and Sciences,Baoji 721013,China)

Abstract:In this paper,based on the method of dynamic feedback controller treating the boundary control with delays,we consider the exponential stabilization for a Timoshenko beam with interior locally input delayed controls and input distributed delay control.We design a new dynamic feedback control law to stabilize exponentially this system,and show that the original system is stabilized exponentially.

Key words:Timoshenko beam;Input distributed delay;Dynamic feedback controller;Exponential stabilization

1.Introduction

2.Design of Dynamic Feedback Controller

3.Representation of the System(2.2)

4.The Proof of Theorem 2.1

5.The Proof of Theorem 2.2

6.Conclusion

In this paper,we design a dynamic feedback controller to treat the stabilization of Timoshenko beam with locally input controls and input distributed delay.Under this control strategy,we prove that the closed-loop system(2.8)is exponentially stable for any time delayτ >0 by verifying the exponential stability of the system(2.7)on the some certain conditions.We mainly use the exact observability of the dual system in finite time to verify the exponential stabilization of closed-loop system.The approach proposed in this paper can be applied to more complicated models theoretically.