The program iterations method in game problem of guidance and set-valued quasistrategies

We consider a differential game of guidance-evasion, which is solved with the program iterations method. The iterated procedure is realized in the space of sets, the elements of which are the game's positions. The objective of this procedure is to construct the alternative partition of the space of positions as established by N.N. Krasovskii and A.I. Subbotin. In addition, more general assumptions on topological properties of the set defining the phase constraints are considered. The connection with the game's solution in the class of quasistrategies is investigated. These quasistrategies are defined as set-valued mappings on spaces of strategic (countably additive) Borel measures.