Set-valued differentials and the hybrid maximum principle

We propose an axiomatic definition of the concept of a generalized differentiation theory (GDT) and a precise statement of the directional open mapping property (DOMP), and we outline the definitions of our two most recent GDTs, namely, the generalized differential quotients (GDQs) and path integral generalized differentials (PIGDs). In addition, we give a complete statement of a hybrid maximum principle (MP) for general GDTs, which now amounts to saying that to every GDT that has the DOMP is associated a version of the hybrid MP. Finally, we limit ourselves to theories in which the GDs of a set-valued map F from a C¹ manifold M to a C¹ manifold N at a point (x,y) ∈ M x N are nonempty compact sets of linear maps from T_xM to T_y N-where T_qQ denotes the tangent space of Q at q-thereby excluding theories due to Ioffe, Mordukhovich and others, where the GDs are different kinds of objects. We are fully aware that these choices are somewhat arbitrary, and that, when a truly definitive version is achieved, the details of the definitions might have to be modified