The space $\mathrm{clcv}(\mathbb R^n)$ with the Hausdorff–Bebutov metric and statistically invariant sets of control systems

We obtain conditions that allow one to evaluate the relative frequency of occurrence of the reachable set of a control system in a given set. If the relative frequency of occurrence in this set is $1$, then the set is said to be statistically invariant. It is assumed that the images of the right-hand side of the differential inclusion corresponding to the given control system are convex, closed, but not necessarily compact. We also study the basic properties of the space $\mathrm{clcv}(\mathbb R^n)$ of nonempty closed convex subsets of $\mathbb R^n$ with the Hausdorff–Bebutov metric.