Guidance–Evasion Differential Game: Alternative Solvability and Relaxations of the Guidance Problem

A guidance–evasion differential game on a finite time interval is studied. Two sets are assumed to be defined in the position space: a target set (TS) of a player who tries to guarantee the guidance (approach), and a set defining state constraints (SCs). Conditions for alternative solvability are proposed: it is shown that a certain analog of N. N. Krasovskii and A. I. Subbotin's alternative theorem holds provided that the TS is closed in the ordinary sense and the set defining SCs has closed sections. In the case when both these sets are closed, statements with relaxed termination conditions of the guidance game are investigated. In this case, different degrees of relaxation of the requirements concerning the guidance to the TS and the fulfillment of the SCs are considered. A position function is constructed whose values are the minimum sizes of the neighborhoods of the above sets for which the player interested in guidance guarantees its implementation. The construction is based on a variant of the programmed iteration method.