A simple pursuit-and-evasion game on a two-dimensional cone

A differential pursuit-and-evasion game is considered in which the players-velocity-controlled points in three-dimensional Euclidean space-move on a two-dimensional conical surface, i.e. at each instant of time the players may choose their velocity vectors in an arbitrary direction along the tangent of the cone (the magnitude of the velocity vectors is bounded by a constant). The pursuer has a strict velocity advantage. It is shown that self-similar variables reduce the original game with dynamic equations of fourth order to a two-dimensional game. The necessary conditions of optimality are applied to construct a complete solution of the positional pursuit-and-evasion game. It is shown that in the main part of the phase space the optimal motion of the players is along the connecting geodesies. In the other part of the space, each player moves along his own geodesic; the envelopes of these geodesies are singular equivocal trajectories. The equivocal surface is a basic element of synthesis, enabling a complete optimal phase portrait of the game to be constructed. A third kind of motion is obtained for certain parameter values. In the corresponding subregion of the phase space, the pursuit time is independent of the evader position; starting from any point, the players meet at the vertex of the cone. Sufficiency of the optimality conditions is not considered. The present paper uses the methods described in [1] and develops its results.