Existence Result and Exponential Stabilization of a Moving String with Time Dependent Delay in the Boundary Feedback

This paper aims to study the impact of a time-varying delay on the behavior of boundary stabilzed moving system. Our study is based on a Kirchhoff model. We establish the an existence and uniqueness result of the system by using of the Galerkin approximations. We show an unifom exponential stabilization result using the modified multiplier technique. Our result is obtained under the fact that the damping term dominates the delayed term and the time delay function and its derivatives should be bounded and of finite norm. Moreover, if the strong damping exists in the equation with certain conditions, the system attains its equilibrium exponentially in despite of the existence of the retarded term. This result ameliorates and generalizes the earliest obtained results where the delay is considered constant.