A differential game of simple approach in manifolds

A game situation is considered in which the players are two points of a manifold with a non-degenerate metric, each with controllable velocity. The payoff in the game is the minimum distance between the players in a semi-infinite interval of the time of motion. The first player minimizes the payoff, the second maximizes it. The phase space of the game is divided into subdomains. In one (the primary domain) the value of the game is the initial distance between the players, in the other (the secondary domain) it is less than the initial distance. It is shown that the boundary of the primary domain consists of singular optimal paths [1], and the regular paths approach it from both sides. Necessary conditions are established for the singular surface to be optimal and the equations of the singular paths are derived. They are of the same form as the analogous relationships in the game of pursuit [2]. A necessary optimality condition, formulated in terms of the geodesic distance between players, is found for the primary domain in the form of an inequality, enabling the boundary of the singular surface to be constructed. The existence of this boundary is a necessary condition for the secondary domain to be non-empty. A generalization of Bellman's equation is obtained; it is shown that the value of the game is constant along secondary optimal paths. On singular paths the distance between the players remains constant. The necessary conditions obtained here provide the basis for an algorithm for constructing optimal paths and the value of the game in the neighbourhood of singularities. The algorithm is then used to work out a complete solution of the problem of approach on a two-dimensional cone, constructing the level curves of the value of the game and an optimal phase portrait. The set of cones for which the secondary domain is empty, i.e. for which the distance between the players is the value throughout the game space, is determined.