A Recurrence Principle for Stochastic Difference Inclusions

The invariance principle is extended to a recurrence principle and is developed for stochastic difference inclusions. For these systems, random solutions are not unique. Under appropriate Lyapunov-like conditions, it is established for every random solution that almost every complete sample path converges to the largest weakly totally recurrent set contained in a level set of the Lyapunov-like function. Such a set is not larger and is sometimes smaller than the largest weakly invariant set contained in the level set. The principle is useful for establishing robust, uniform asymptotic stability in probability or robust, uniform strong recurrence under weak Lyapunov conditions for stochastic, discrete-time control systems that employ discontinuous feedback laws. Examples demonstrate the achieved results.