Singular surfaces in differential games an introduction

We give a general set up and a version of Isaacs' Verification Theorem that allows us to deal with the various singularities we want to investigate. In particular, we are obliged to allow upper or lower strategies, leading to upper or lower saddle points, that may exists even if the Hamiltonian does not have a saddle point. It is shown that this is needed even for separated games. Then we give a general study of junctions of optimal fields with singular surfaces, which requires a special investigation of the situation where this junction is tangantial extending Caratheodory's General Envelope Theorem. We then proceed to study special singular surfaces, and we end up with an example which shows how a state constraint may appear in the interior of the game space of a separated problem posed with no such constraint to start with.