The statistically invariant sets of controllable systems with random parameters

We investigate the properties of statistical invariance and statistically weak invariance with probability one for control systems with random parameters. We obtain the sufficient conditions for the invariance of the given set with respect to the control system formulated in terms of Lyapunov functions and the dynamical system of shifts. We prove the extension for the theorem of S. A. Chaplygin about differential inequalities and obtain the conditions of existence for the upper solution of Cauchy problem with piecewise continuous on $t$ right-hand part without assumption of uniqueness of solution.