The programmed iterations method in a game problem of guidance

The solution of a differential game of guidance–evasion on the basis of the programmed iterations method is considered. The basic goal consists in the construction of a set of positional absorption corresponding to alternative partition following from the fundamental alternative theorem of N. N. Krasovskii and A. I. Subbotin. For construction, an operator of programmed absorption defined by the target set in a guidance problem is used. The set defining phase constraints is gradually transformed by the above-mentioned operator; therefore, the sequence for which the corresponding limit coincides with the set of positional absorption is realized. It is assumed that the target set is closed and the set defining phase constraints of initial problem has closed sections corresponding to fixation of time. Properties having the sense of one-sided continuity of the positional absorption set under variation of sets defining initial differential game are established. It is shown that the limit of iterated procedure coincides with the set of successful solvability in a class of set-valued generalized quasistrategies.