Control of an axially moving string

This project has a twofold aim. We intend to study the dynamic of an axially moving string (beam) under two types of control at the boundary. In particular, we would like to stabilize these structures, which are often subject to undesirable transversal vibrations. We shall seek appropriate internal and boundary controllers capable of driving the structures to a stable position in a reasonable amount of time (preferably exponentially). To this end, we shall explore the possibility of using viscoelastic material, which have already proved very efficient for structures, which are not moving longitudinally. We expect to be able to prove exponential stabilization results although passing from fixed strings (beams) at both ends or at least one of them to moving ones is more challenging. Indeed, while studying axially moving strings (beams), we will be faced with new boundary terms coming from the differentiation. Due to the longitudinal motion, there will be a rate of flow across the boundary to take into account. These terms are usually not easy to deal with. We will use mainly the Leibniz-Reynolds transport theorem, some suitably chosen controls, the multiplier technique, eventually some ad-hoc methods and some ideas from the viscoelasticity theory.