Abstract
Two differential games in feedback strategies [1], [2] are considered, in which players are velocity-controlled points of a Riemannian manifold. The game of pursuit is formulated for the case when the pursuer has advantage in speed. Otherwise the game of approach is considered, i.e. the cost-function is the minimal distance between players during the infinite time-interval of motion. Since the restrictions for the velocities are homogenous, geodesic lines have an important role for optimal paths' construction. The main difference between manifold and Euclidean space is non-uniqueness of the minimal geodesic, connecting two points of the manifold. The analysis of this paper is restricted to the manifolds, which have no more than two minimal geodesics. This gives rise to the singularities of dispersal, equivocal and universal types. Local necessary optimality conditions are found. The players' optimal behaviour in general position is shown to be a (regular) motion along geodesics. The domain, where the latter lie on the geodesic curve, connecting the players, is called the primary domain. A sufficient condition is found for the whole game space to be the primary domain, as is the case in Euclidean space. Necessary conditions are formulated for the existence of singular paths, which are envelopes of the geodesics. The equations of singular motion are obtained and shown to be a generalized Hamiltonian type. An algorithm is suggested for the construction of the optimal paths in the vicinity of a singular surface, its efficiency is demonstrated by complete solutions of both games on a two-dimensional cone. For other approaches to similar game problems see [1]–[5].
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Melikyan, A.A. Games of simple pursuit and approach on the manifolds. Dynamics and Control 4, 395–405 (1994). https://doi.org/10.1007/BF01974143
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DOI: https://doi.org/10.1007/BF01974143