Existence and stabilization of a kirchhoff moving string with a delay in the boundary or in the internal feedback

In this paper, we study the effect of an internal or boundary time-delay on the stabilization of a moving string. The models adopted here are nonlinear and of “Kirchhoff” type. The well-posedness of the systems is proven by means of the Faedo–Galerkin method. In both cases, we prove that the solution of the system approaches the equilibrium in an exponential manner in the energy norm. To this end we request that the delayed term be dominated by the damping term. This is established through the multiplier technique.