On existence and stabilization of an axially moving string subject to time varying delay

Abstract

This paper aims to study the impact of a time-varying delay on the behavior of a internal or boundary stabilized moving system. Our study is based on a nonlinear and of "Kirchhoff" type. We establish the well-posedness of the systems by using of the Faedo-Galerkin method. For this system, we prove that the solution attains the equilibrium with the energy norm in an exponential manner. To this aim we claim that the retarded term be dominated by the damping one. The presence of a strong damping with some restrictive conditions makes the system exponentially stable even in the presence of the delay term. This is established through the multiplier technique.