Differential Approach–Evasion Game: Alternative Solvability and the Construction of Relaxations

Abstract

Abstract: A nonlinear differential approach–evasion game is studied. For this game, Krasovskii andSubbotin established a fundamental theorem on the alternative. The game parameters are thetarget set and the set determining state constraints; in the above-mentioned theorem, both ofthese sets are assumed to be closed in the position space. In the present study, we do not assumethe closedness of the set defining the state constraints, and only the closedness of all of its sectionscorresponding to fixed moments of time is postulated; the target set is assumed to be closed.Under these conditions, a version of the assertion about the alternative solvability is established,and relaxations of the original differential game are constructed that are determined by theweakening of the conditions for the termination of the pursuit game. This construction uses thewell-known method of programmed iterations, implemented on the space of sets whose points arepositions in the game. As a result, a special sequence of functions is formed that convergespointwise to a certain limit function of position. The values of the latter have the meaning of thesmallest size of neighborhoods of the parameter sets at which a player interested in approachingthe target set is guaranteed to solve his problem under conditions weakened in the above way; inthis case, however, some degree of priority is allowed concerning the approach to the target setand compliance with the state constraints. To construct the above-mentioned limit function, a“direct” iterative procedure of the method of programmed iterations in a function space is alsoproposed, and the desired limit function turns out to be a fixed point of the operator generatingthis procedure. In addition, it is shown that each value of this function is the cost of some targetset with a special merit functional.