Dynamic Boundary Control of Vibration Systems Based on Passivity

Abstract

In this chapter, we consider feedback stabilization of a class of passive infinite dimensional systems by means of dynamic boundary control. The notion of passivity was developed in connection with circuit theory in the late ′50s where the basic motivation was to investigate the behavior of circuits composed of passive circuit elements such as resistors, capacitors and inductors, see [61]. This concept was then introduced into control systems, see [2], [50], [156], [164]. To motivate the concept of passivity, let us consider the following situation: Let S be a dynamical system with an input vector u= (u₁,…,u_m)^T∈R^mand an output vector y = (y₁,…,y_m)^T∈R^m. Let H be the Hilbert space in which the solutions of S evolves, and let E(t): H → R be a positive time function which depends on the solutions of S. Assume that the time derivative of E(t) along the solutions of S satisfies

$$ \dot E(t) = {u^T}y = \sum\limits_{i = 1}^m {{u_i}{y_i}.} $$

((5.1))