Existence and Boundary Control of a Nonlinear Axially Moving String Subject to Disturbances

This paper addresses the stabilization question for a nonlinear model of an axially moving string. The string is tensioned and is subjectto spatiotemporary varying disturbances. The Hamilton principle ofchanging mass is employed to formulate mathematically the problem. By means of the Faedo–Galerkin method, we establish the wellposedness. A boundary control with a time-varying delay is designedto stabilize uniformly the string. Then, we derive a decay rate of thesolution under the condition that the retarded term be dominated bythe damping one. Some examples are given to clarify when the rateis exponential or polynomial.