We study a variant of the reachability problem with constraints of asymptotic character on the choiceof controls. More exactly, we consider a control problem in the class of impulses of given intensity and vanishingly small length. The situaton is complicated by the presence of discontinuou dependences, which produce effects of the type of multiplying a discontinuous function by a generalizd function. The constructed extensions in the special class of finitely additive measures make it possible to present the required solution,defined as an asymptotic analog of a reachable  set, in terms of a continuous image of a compact, which is described with the use of the Stone space orresponding to the natural algebra of sets of the control interval. 



One of the authors had the honor of communicating with Nikolai Nikolaevich Krasovskii for many years nd discussed with him problems that led to the statement considered in the paper. Krasovskii.s support of this research direction provided possbilities for its fruitful development. His