Some questions of the structure of the programmed iterations method

Two abstract versions of the programmed iterations method are considered. This method was used in the theory of differential games for constructing the value function and the stable bridges in the sense of N. N. Krasovskii. The connection of the direct version of the programmed iterations method and the known iterated procedure operating in the space of sets is investigated. The direct iterated procedure is realized in the space of set-valued mappings. For the indirect iterated procedure the natural process of the compression of phase constraints is realized; the corresponding goal set is fixed. The duality of two above-mentioned versions of the programmed iterations method is established.