High-gain adaptive boundary stabilization for an axially moving string subject to unbounded boundary disturbance

In this paper, we are interested to stabilize an axially moving string subject external disturbances. We assume that the disturbance may increases exponentially. employ active rejection control (ADRC) approach estimate disturbance. design a observer has time-varying gain so can be estimated with exponential way. order closed loop system, use constructed through high-gain adaptive velocity feedback. The existence and uniqueness of solution system is dealt in framework nonlinear semigroup theory by using theorem due Crandall-Liggett. It shown formulated capable stabilizing exponentially system. obtained results also valid for immobile case ($v=0$) present work improves certain previous results.