Asymptotically stable statistically weakly invariant sets for controlled systems

This note is based on the talk given at the conference “Actual problems of stability and control theory” and dedicated to the extension of the known theorem by E. A. Barbashin and N. N. Krasovskii (1952) on asymptotic stability of an equilibrium state for an autonomous system of differential equations. Our efforts are mainly concentrated on nonautonomous differential inclusions with closed-valued (but not necessarily compact-valued) right-hand sides, where the equilibrium state is a given weakly invariant or statistically weakly invariant (with respect to the solutions of the inclusion) set. The statements are formulated in terms of the Hausdorff–Bebutov metric, the dynamical system of translations corresponding to the right-hand side of the differential inclusion, and the weakly invariant set corresponding to the inclusion.