Stabilization of the uniform Timoshenko beam by one locally distributed feedback

Abstract

In this article, we study the energy decay rate for an elastic Timoshenko system. This system consists of two coupled wave equations. Only the equation about the rotation angle is damped by one locally distributed feedback at the neighbourhood of the boundary. The equation for the transverse displacement of the beam is only indirectly damped through the coupling. First, we establish an exponential energy decay rate in the case of the same speed of propagation. Next, when the wave speeds are different, a polynomial-type decay rate is obtained. These results are proved by verifying the frequency domain conditions.

Keywords:

• Timoshenko system
• locally distributed feedback
• exponential and polynomial stability
• frequency domain conditions

Acknowledgement

The authors would like to thank professor Fatiha Alabau-Boussouira for very helpful discussions and suggestions.