Exponential stability estimates for an axially travelling string damped at one end

Abstract

We study the small vibrations of an axially travelling string with a dashpot damping at one end. The string is modelled by a wave equation in a time-dependent interval with two endpoints moving at a constant speed $\mathbf{v}$. For the undamped case, we obtain a conserved functional equivalent to the energy of the solution. We derive precise upper and lower exponentially decaying estimates for the energy with explicit constants. These estimates do not seem to be reported in the literature even for the non-travelling case $\mathbf{v}=0$.

Keywords:

• Axially travelling strings
• Fourier series
• energy estimates
• boundary stabilisation

AMS CLASSIFICATIONS:

• 35L05
• 35C10
• 93D15

Acknowledgments

The authors also thank the reviewers of this paper for their valuable suggestions on earlier version of this manuscript.

Disclosure statement

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

Additional information

Funding

The authors have been supported by the General Direction of Scientific Research and Technological Development (Algerian Ministry of Higher Education and Scientific Research) PRFU # C00L03UN280120220010.