On the question of construction of an attraction set under constraints of asymptotic nature
We study a variant of the reachability problem with constraints of asymptotic character on the choice of controls. More exactly, we consider a control problem in the class of impulses of given intensity and vanishingly small length. The situation is complicated by the presence of discontinuous dependences, which produce effects of the type of multiplying a discontinuous function by a generalized 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 corresponding 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 and discussed with him problems that led to the statement considered in the paper. Krasovskii’s support of this research direction provided possibilities for its fruitful development. His disciples and colleagues will always cherish the memory of Nikolai Nikolaevich in their hearts.