Exponential stabilisation of a simple tree-shaped network of Timoshenko beams system

Abstract

The stabilisation problem of a forked triple system of Timoshenko beams, which is a simple tree-shaped network, is considered in this article. Suppose that the root of the fork is clamped. At the interior node of the fork, the displacement and rotational angle of the beams are continuous; the shearing force and bending moments of the beams satisfy the transmission conditions. The feedback controllers are designed at the exterior tips to stabilise this system. Then the closed-loop system is proved to be well-posed. By the eigenfrequency analysis together with the asymptotic technique, the system is shown to satisfy the spectrum-determined-growth condition. Finally, this closed-loop system is proved to be exponentially stable. Furthermore, a simulation is given to support these results.

Keywords:

• Timoshenko beam
• stabilisation
• network
• feedback control
• spectrum-determined-growth condition
• exponential stability

Acknowledgements

The authors would like to thank the referees for their useful and helpful comments and suggestions. This research is supported by the Natural Science Foundation of China Grant NSFC-60874034.