We will study stability and asymptotic stability for time-varying systems described by ODEs of the form x = f(epsilon (-1)t,x), where f(t,x) is 1-periodic with respect to t and epsilon >0 is a small parameter. Since the discovery of stabilizing effect of vibration in the reverse pendulum example, there 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 algebraic formalism and the formalism of chronological calculus. This latter is a method of asymptotic analysis for flows generated by time-variant ODE. We apply the approach to study stability issues for linear and nonlinear systems. In particular, we derive conditions of stability for the second- and third-order linear differential equations with periodic fast-oscillating coefficients, we study output-feedback stabilization of bilinear systems and consider high-order averaging procedure fur nonlinear systems under homogeneity assumptions. At the end we study the problem of stabilization of nonholonomic (control-linear) systems by means of time-varying feedbacks. (C) 2001 Elsevier Science B.V. All rights reserved. [References: 22]
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