Abstract

We investigate the statistical characteristics of attainability set $A(t,\sigma,X)$ of control system $$ \dot x=f(h^t\sigma,x,u),\quad (t,\sigma,x,u)\in\mathbb R\times\Sigma\times\mathbb R^n\times\mathbb R^m, \qquad (1) $$ 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.

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