Long-time behaviour of a translational thermoelastic Timoshenko system with second sound

Abstract

We consider a Timoshenko system coupled with heat equations modelled by Cattaneo's law. The coupling is through the transverse displacement. Both ends of the beam are dynamic. One end of the beam is fixed to a base in a translational motion and a tip mass is attached to the other end. We design a feedback control acting at the base. It is shown that this feedback control is a reasonable one and is capable of stabilizing the system. We prove an exponential and a polynomial stability result using the multiplier technique. To this end, we introduce new functionals to form a suitable Lyapunov functional.

Keywords:

• Vibration control
• exponential decay
• polynomial decay
• translational Timoshenko beam
• second sound

AMS Classifications:

• 35M13
• 35B35
• 35B40

Acknowledgments

This paper was finished during the visit of the first author to King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, KSA, in December 2019–January 2020. The authors would like to express their sincere thanks to KFUPM (Interdisciplinary Research Center for Intelligent Manufacturing and Robotics) for its support through Project No. SB201014. The authors would like to express their gratitude to the anonymous referees for their careful reading and objective suggestions, which allowed to improve the results of this paper and its presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by King Fahd University of Petroleum and Minerals [grant number SB201014].