A note on exponential stability of partially slowly time - varyingnonlinear

Consider a system _ x = f(x; t; t) with a time-varying vectorreld which contains a regular and a slow time scale (large). Assume there exists () such that kx (t; t 0 ; x 0)k K()kx 0 ke ()(t?t0) where x (t; t 0 ; x 0) is the solution of the system _ x = f(x; t;) with initial state x 0 at t 0. We show that for suuciently large, _ x = f(x; t; t) is exponentially stable when the average of () is negative. This result can be used to extend the circle criterion i.e. to obtain a suucient condition for exponential stability of a feedback interconnection of a slowly time-varying linear system and a sector non-linearity. 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.