Stochastic controllability and stochastic Lyapunov functions

Sufficient conditions are established under which the law of large numbers and related ergodic theorems hold for nonlinear stochastic systems operating under feedback. It is shown that these conditions hold whenever a moment condition is satisfied, which may be interpreted as a generalization of the martingale property. If in addition a stochastic controllability condition holds, then it is shown that the underlying distributions governing the system converge to an invariant probability at a geometric rate. These results are illustrated with an example from nonlinear control theory