Programmed iteration method and sets of positional absorption

Abstract

A pursuit–evasion differential game is considered, and the programmed iteration method is used to construct a set of positional absorption corresponding to the Krasovskii–Subbotin alternative theorem. The case is considered where the set of positions determining the state constraints may not be closed (in the position space), but has closed sections corresponding to fixed times. Properties are established that are interpreted as the (one-sided) continuity of the positional absorption set from above, and the relation to the solution of the game in the class of set-valued quasi-strategies is shown.