On the result equivalence of constraints of asymptotic nature

An abstract control problem with constraints of integral nature is considered. Conditions are established for the equivalence of different variants of asymptotic constraints corresponding to weakening the traditional constraints on the choice of the controls. The proposed approach is based on an extension construction in the class of finitely additive measures. For a long time, the author has had the wonderful opportunity to discuss his results concerning the mentioned (rather nontraditional) approach with N. N. Krasovskii; these discussions and the support of this approach by N. N. Krasovskii played an important role in its formation and development.