Lie- and chronologico-algebraic tools for studying stability of time-varying systems

We will study stability and asymptotic stability for time-varying systems d escribed by ODEs of the form x = f(epsilon (-1)t,x), where f(t,x) is 1-peri odic with respect to t and epsilon >0 is a small parameter. Since the disco very of stabilizing effect of vibration in the reverse pendulum example, th ere have been a lot of study regarding stability of such systems and design of fast-oscillating stabilizing feedback laws. In this paper we suggest an approach which is kind of high-order averaging procedure based on Lie alge braic formalism and the formalism of chronological calculus. This latter is a method of asymptotic analysis for flows generated by time-variant ODE. W e apply the approach to study stability issues for linear and nonlinear sys tems. In particular, we derive conditions of stability for the second- and third-order linear differential equations with periodic fast-oscillating co efficients, we study output-feedback stabilization of bilinear systems and consider high-order averaging procedure fur nonlinear systems under homogen eity assumptions. At the end we study the problem of stabilization of nonho lonomic (control-linear) systems by means of time-varying feedbacks. (C) 20 01 Elsevier Science B.V. All rights reserved.