Stability of a Nonlinear Axially Moving String With the Kelvin-Voigt Damping

TL;DR

This paper analyzes the stability of a nonlinear axially moving string with Kelvin-Voigt damping, demonstrating exponential decay of displacement below a critical speed and bounded response under external forces.

Contribution

It establishes the stability conditions and exponential decay of the string's displacement using Lyapunov functions, advancing understanding of damping effects in moving strings.

Findings

String displacement converges to zero below critical speed

Displacement remains bounded under bounded external forces

Lyapunov function decays exponentially

Abstract

In this paper, a nonlinear axially moving string with the Kelvin-Voigt damping is considered. It is proved that the string is stable, i.e., its transversal displacement converges to zero when the axial speed of the string is less than a certain critical value. The proof is established by showing that a Lyapunov function corresponding to the string decays to zero exponentially. It is also shown that the string displacement is bounded when a bounded distributed force is applied to it transversally. Furthermore, a few open problems regarding the stability and stabilization of strings with the Kelvin-Voigt damping are stated.