Stability of a nonlinear axially moving string

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.