In this paper, we consider a system modeling an axially moving viscoelastic string under a spatiotemporally varying tension. A mechanism consisted of a hydraulic touch-roll actuator pointed at the right boundary to suppress the transverse vibrations. We adopt the multiplier method to design a boundary control law and to prove an exponential stability result. However, this result is obtained provided that the lower bound of the tension in the string is larger than its time derivative. The effectiveness of the proposed control law is demonstrated via simulations. 

This paper deals with the stabilization problem of an axially moving string with a tip mass attached at the free end and subject to an external disturbance. The disturbance here is not uniformly bounded, and it is assumed to be exponentially increasing. First, the tip mass equation is designed under a boundary controller. By using this equation, the active disturbance rejection control (ADRC) technique is applied to design a disturbance observer, and it is shown that the observer can be estimated exponentially. Then, the closed-loop system is formulated and the well-posedness of the model is proved in the framework of the semigroup theory. The stability of the closed-loop system is then proved by means of the multiplier technique, where the energy system converges to equilibrium with an exponential manner. The efficiency of the obtained results is verified through numerical simulations.