Mathematics

We study the questions of stability and asymptotic stability for time-varying systems described by ODE's with the right-hand sides of the form $f(\omega t, x)$, where $f(t,x)$ is 1-periodic with respect to $t$. The systems of this type draw much attention for a number of reasons. Since the discovery of stabilizing effect of vibration in the reverse pendulum example, there was much study regarding utilization of time varying feedback laws for stabilization of the systems which are not stabilizable by time-invariant feedback. Systems 'with a deficit of control' or control systems for nonholonomic mechanical objects (e.g. mobile robots) are natural examples of this kind. Many questions related to the issue of stability are not yet answered even for the linear time-varying systems. Two main approaches to the stability issue - the one based on Lyapunov functions and another one based on averaging - both encounter difficulties when dealing with the problem of fast oscillations. The Lyapunov functions must be very complicated in order to decrease along the trajectories generated by fast-oscillating controls while the standard averaging procedures result in systems with critical equilibria. Our approach is kind of high-order averaging procedure which is based and makes use of the tools of chronological calculus developed by A.A.Agrachev and R.V.Gamkrelidze in 70's-80's. The method is invariant - the high-order averagings are computed via Lie algebraic brackets of vector fields (right-hand sides). We apply the high-order averaging to study stability issues for both 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, study output-feedback stabilization of bilinear systems, consider averaging procedures for nonlinear systems under homogeneity assumptions, study the problem of stabilization of nonholonomic (control-linear) systems by means of time-varying feedbacks.