Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center

In the terms of Lyapunov functions we obtain the conditions that allow to estimate the relative frequency of occurrence of the attainable set of a controllable system in a given set $\mathfrak M$. The set $\mathfrak M$ is called statistically invariant if the relative frequency of occurrence in $\mathfrak M$ is equal to one. We also derive the conditions of the statistically weak invariance of $\mathfrak M$ with respect to controllable system, that is, for every initial point from $\mathfrak M$, at least one solution of the controllable system is statistically invariant. We obtain the conditions for the attainable set to be non-wandering as well as the conditions of existence of the minimal attraction center.