Numerical construction of Nash solutions in a two-player linear positional differential game in which the phase space has more than two dimensions

The problem of constructing Nash solutions in a two-player non-zero-sum positional differential game with terminal payoffs, linear dynamics, and constraints on the players' controls in the form of convex polyhedra is considered. The formalization of the players' strategies and of the motions generated by them is based on the formalization and results of the theory of zero-sum positional differential games developed by N. N. Krasovskii and his scientific school. The problem of finding game solutions is reduced to solving nonstandard control problems. We propose algorithms for the construction of the algebraic sum and geometric difference of convex polyhedra. The algorithms extend the applicability domain of an earlier developed algorithm, which constructed Nash solutions, to problems with dynamics in phase spaces with more than two dimensions.