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