On certain problems of the structure of ultrafilters related to extensions of abstract control problems

We study the procedures for constructing ultrafilters in measurable spaces that are in a general sense based on Cartesian products. Our constructions intend to use ultrafilters as generalized elements in extension constructions for abstract reachability problems with asymptotic constraints. Constrains of this kind may arise, for instance, with sequential relaxation of boundary and intermediate conditions in control problems. In this case, families of sets of admissible regular controls in practically interesting cases represent filter bases, which makes it natural to use ultrafilters (maximal filters) that are admissible, in a certain sense, with respect to these families. We assume that counterparts of reachability sets considered in this work are sets in a topological space; the latter may be non-metrizable which occurs, for instance, in typical applications of the pointwise convergence topology. Such a topology may also be useful in impulse control problems, where one studies sheafs of motions as counterparts of reachability sets for exact and approximate satisfaction of trajectory constraints.