The characteristics of invariance of attainability set of control systems with random coefficients

This article is continuation of works of L. I. Rodina and E. L. Tonkov in which expansion of the concept of invariance for sets concerning control systems and differential inclusions is entered. This expansion consists in research of the sets which are not invariant in «classical» sense, but possess the property of statistical invariance, and also in studying of statistical characteristics for attainability set of control system.
We consider the characteristics connected with the invariance of the given set $\mathfrak M (\sigma)$ with respect to the control system which display the property of uniformity of stay for the attainability set of the system in $\mathfrak M (\sigma)$ on the finite time interval. We obtain estimates of these characteristics for systems with random coefficients in terms of Lyapunov functions, a derivative owing to differential inclusion and the dynamical system of shifts. In particular, we investigate the estimations with probability one for characteristics of control system which we will name a system with switchings. This system can be identified with a stationary random process whose set of states is finite; for this set there are given the initial probability distribution and the probabilities of finding in each state; the lengths of intervals between the moments of switching system from one state to another are random variables with a given distribution function. The example of estimation of the investigated characteristics for a linear control system with switchings is considered.