Optimal pursuit-evasion paths in a game on complete cone

Differential games with simple motion in a game space with nonunique shortest geodesic line have complicated structure of singular surfaces. The successful solution of such games on a family of two-dimensional cones was based to the large extent on the parameter analysis of the problem. Simultaneous consideration of the games for all possible parameters was very useful for the determination of the bifurcation points where the structure of optimal phase portraits changes. In the present paper the results of pursuit game on a cone of one nappe are extended to the game on the full cone of two nappes. The latter surface is included in one-parametric family of rotation surfaces (each of which is characterized by two more parameters), and the game on it is considered as the generating problem for the analysis of the games on the perturbed surfaces.