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

Abstract

In this note, 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 functional 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 of strings with the Kelvin–Voigt damping are stated.