Some Fundamental Qualifications of Optimal Strategies and Transition Surfaces in Differential Games

Abstract

At each point of a regular region of a differential game, there are two tangent cones which are complementary to each other. The two vectograms defined by the pair of pure optimal strategies must be contained separately in the two tangent cones. The velocity vector resulting from the selection of the optimal strategies consequently must represent a semipermeable direction. These conditions, which reveal a fundamental separating property of the optimal velocity vector, are weaker than that of Isaac’s main equation. Moreover they hold even on singular surfaces in the regular regions. The conditions also reveal a necessary condition for a differential game to have a regular region. Each isovalued surface in the regular region is essentially a semipermeable surface. A transition surface arises only when the resulting directed isovalued surface of the previous optimal strategies cannot have a smooth semipermeable extension at the singular surface. This observation yields a necessary condition for a transition surface to occur. The localization of transition surfaces is then possible. Finally, a jump and smooth condition of the isovalued surfaces is given.