A note on exponential stability of partially slowly time-varying nonlinear systems

Consider a system (x)over dot = f (x, t, t/beta) with a time-varying vectorfield which contains a regular and a slow time scale (beta large). Assume there exist an alpha(tau) and a K (tau) such that parallel to x(tau) (t, t(0), x(0))parallel to less than or equal to K (tau) parallel to x(0)parallel to e(alpha(tau)(t-t0)) where x(tau) (t, t(0), x(0)) is the solution of the system (x)over dot = f (x, t, tau) with initial state x(0) at t(0). We show that for beta sufficiently large, (x)over dot = f (x, t, t/beta) is exponentially stable when 'on average' alpha(tau) is negative. This result can be used to extend the circle criterion i.e. to obtain a sufficient condition for exponential stability of a feedback interconnection of a slowly time-varying linear system ana a sector nonlinearity. An example is included which shows that the technique can be used to obtain an exponential stability result for a pendulum with a nonlinear partially slowly time-varying friction attaining positive and negative values. Copyright (C) 1998 IFAC.