Uniform Stabilization of aNonlinear Axially Moving String by a Boundary Control of Memory Type

A nonlinear problem modeling a moving string with memory condition at the boundary is considered. This work aims at establishing reasonable sufficient conditions capable of suppressing the transversal vibrations that occur during the movement of the string. Unlike the fixed strings case here we need to employ the Reynold Transport Theorem to compute the ”total” derivative. This results in the appearance of several boundary terms which necessitates appropriate handling. We shall prove that the motion of the string plays an important role in determining the rate of the decay of solutions.