A Result on Tracking Equilibria for Slowly Varying Nonlinear Systems

This paper considers the problem of tracking a time varying equilibrium point for a nonlinear system. The fundamental issue of concern is described roughly as follows: Given a nonlinear system x¿(t) = f(x(t), u(t)) with compact restraint set for u(t), assume that asymptotic tracking to equilibrium is guaranteed whenever u(t) is constant. Then, does it follow that tracking will be preserved when u(t) and the associated equilibrium point are "slowly" varying? In this paper, we answer this question in the affirmative subject to regularity assumptions on f(.,.,) and a sufficiently small bound on |¿(t)|.