Attraction sets in abstract attainability problems and their representations in terms of ultrafilters

Abstract problems about attainability in topological space (TS) under constraints of asymptotic nature (CAN) realized by nonempty family of sets in the space of usual solutions (controls) are considered. As analog of the attainability set defined by image of the target operator (TO) with values in TS, attraction set (AS) in classes of filters or directednesses of usual solutions is considered. Questions connected with the AS dependence under cange in the family of sets of usual solutions generating CAN are investigated. The special attention is paid to the case when this family is a filter (every AS is either generated by a filter used as CAN or is empty set). At the same time, AS under CAN generated by ultrafilters (u/f) that is by maximal filter under unrestrictive conditions on TS and TO there is a singleton, which allows to enter attraction operator which, in the case of regular TS, is continuous under equipment of the set of all ultrafilters on the set of usual solutions with Stone topology. On this basis, it is possible to give a practically exhaustive representation of constructions connected with AS in a regular TS in the class of u/f with their natural factorization based on TO. A whole range of obtained properties extend to the case of TO with values in a Hausdorff TS. Some questions connected with topology weakening of the space in which AS is realized are investigated.