New Theories of Set-valued Differentials and New Versions of the Maximum Principle of Optimal Control Theory⋆

The purpose of this note is to announce two new theories of generalized differentials—the “generalized differential quotients,” abbr. GDQs, and the “path-integral generalized differentials”, abbr. PIGDs—which have good open mapping properties and lead to general versions of the maximum principle. In particular, we use GDQ theory to state—in Theorem 5— a version of the maximum principle for hybrid optimal control problems under weak regularity conditions. For single-valued maps, our GDQ theory essentially coincides with the one proposed by H. Halkin in [4], but GDQ theory applies as well to multivalued maps, thus making it possible to deal with non-Lipschitz vector fields, whose flow maps are in general set-valued.