The program iteration method and the relaxation problem

The issues related to an approach–evasion differential game are considered: alternative solvability, construction of relaxations of an approach game problem, and construction of a solution based on the program iteration method. The case is considered when the set defining the phase constraints in a differential game with a closed target set may be nonclosed in the position space but has closed sections. For this situation, an alternative is established that is ideologically similar to the Krasovskii–Subbotin alternative under a certain correction of the classes of strategies. The question of constructing relaxations of the problem of approaching the target set in the presence of phase constraints is considered; it is assumed that the weakening of the conditions in terms of bringing the system to the target set and in terms of observing the phase constraints may be different, which is achieved by introducing a special priority coefficient. When a position of the game is fixed, the smallest size of a neighborhood of the target set is determined for which, with a proportional (in the sense of the mentioned coefficient) weakening of the phase constraints, the player interested in the approach can still guarantee it in an appropriate class of strategies (here, nonanticipation strategies or quasi-strategies). For the resulting main function of the position, a sequence of functions (positions) converging to this function is introduced based on a variant of the program iteration method operating in the space of sets with elements in the form of game positions. After that, a special operator on the function space (a program operator) is constructed, which implements this sequence by means of a “direct ” iterative procedure and for which the main function itself is a fixed point. Thus, a new version of the program iteration method is implemented. A type of the quality functional with the following property is proposed: when a position is fixed, the value of the main function is the value of a game for the minimax–maximin of this functional.