One representation of the results of action of approximate solutions in a problem with constraints of asymptotic nature

We consider an abstract attainability problem with constraints of asymptotic nature defined in the form of a nonempty family of subsets in the space of usual solutions. Various variants of implementing asymptotic effects are considered (convergence in a topological space, cycles, and so on). A rather general method is suggested for presenting the results of action of approximate solutions; this method generalizes constructions based on sequences in the space of usual solutions.