Extension of E. A. Barbashin's and N. N. Krasovskii's stability theorems to controlled dynamical systems

The known theorems by E. A. Barbashin and N. N. Krasovskii (1952) about the asymptotic and global stability of an equilibrium state for an autonomous system of differential equations are extended to nonautonomous differential inclusions with closed-valued (but not necessarily compact-valued) right-hand sides, where the equilibrium state is a 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.