Statistical characteristics of attainability set and periodic processes of control systems

We investigate the statistical characteristics of attainability set $A(t,\sigma,X)$ of control system
\begin{equation} \dot x=f(h^t\sigma,x,u),\quad(t,\sigma,x,u)\in\mathbb R\times\Sigma\times\mathbb R^n\times\mathbb R^m, \tag{1} \end{equation}
which is parametrized by means of topological dynamic system $(\Sigma,h^t)$. We obtained the lower estimations for such characteristics as the relative frequency of containing, the upper and lower relative frequencies of containing of attainability set of the system (1) in the given set $M$ as well as new sufficient conditions of statistical invariance of the set $M$ with respect to control system. We received the conditions for system (1) and set $X$ at which for given $\sigma\in\Sigma$ and $\varkappa_0\in (0,1]$ the relative frequency of containing of attainability set $A(t,\sigma,X)$ of systems (1) in the set $M$ not less $\varkappa_0$. Results of the work are illustrated by the example of control system which describes periodic processes in a chemical reactor.