On stabilization of an elastic system by fast-oscillating time-variant feedback

Abstract

We study a class of elastic systems described by a (hyperbolic) partial differential equation. Our working example is the equation of a vibrating string subject to a destabilizing linear disturbance. Our main goal is to establish conditions for stabilization and asymptotic stabilization of the equilibrium configuration of the string by applying to it fast oscillating controlled force. In first situation studied we assume that the string is subject to damping; after it we consider the same system without damping. We extend the tools of high-order averaging and of chronological calculus for studying stability of this distributed parameter system.