Barbashin and Krasovskii's asymptotic stability theorem in application to control systems on smooth manifolds

We introduce the notion of so-called standard control system, whose phase space is a finite-dimensional smooth manifold satisfying a series of conditions; in particular, it is supposed to be connected and orientable and have a countable atlas. For a given standard control system, we consider a set of time shifts and construct the closure of this set in the topology of uniform convergence on compact sets. In these terms, we study the conditions of uniform local reachability of a given trajectory. The main result is formulated in terms of a modified Lyapunov function. A simple example is considered.