The statistically weak invariant sets of control systems

We obtain the conditions that allow to estimate the relative frequency of occurrence of the attainable set of a control system in some given set. The set is called statistically invariant if the relative frequency of occurrence in this set is equal to one. We also derive the conditions of the statistically weak invariance of the given set with respect to controllable system, that is, for every initial point from this set, at least one solution of the control system is statistically invariant. We suggest that the images of the right hand part of the differential inclusions corresponding for the given control system are closed but may be not compact. The main results are formulated in the terms of Lyapunov functions, metric of Hausdorff–Bebutov and the dynamical system of shifts that attended in the right hand part of the differential inclusion.