Invariant and statistically weakly invariant sets of control systems

The paper deals with the expansion of the concept of the set invariance with respect to control systems and differential inclusions. Said expansions that statistically invariant and statistically weakly invariant sets are studied. The sufficient conditions for existence of invariant (in the specified sense) sets, which are formulated in terms of the Hausdorff–Bebutov metric, Lyapunov functions and the Clarke derivative, of the given functions are obtained. The work covers both determined systems and the systems with random parameters, for which the concept of statistical invariance with probability one is investigated. The problems about complete controllability of time-varying linear system and about the existence of non-predicting control for linear system with random parameters are considered, too.