Qualification of optimal strategies and singular surfaces of a class of differential games

Abstract

At each point of a regular region, there exist two tangent cones which are complementary to each other. The two vectograms defined by the pair of 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 Isaacs' main equation. However, they hold even on singular surfaces. A singular surface arises only when the resulting velocity vector of the previous optimal strategies cannot continuously represent a semipermeable direction after the optimal path crosses over the surface. This observation yields a necessary condition for a singular surface to occur. The localization of singular surfaces is then possible. Finally, a smooth condition with respect to optimal paths is given.