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 a new theory of generalized differentials—the “generalized differential quotients,” abbr. GDQs— which has good open mapping properties, and to use this theory to state—in Theorem 9.4— 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. The results presented here are much weaker than what can actually be proved by our methods. More general versions, involving systems of differential inclusions, are discussed in a detailed paper currently in preparation. The GDQ concept contains several other notions of generalized differential, but does not include some important theories such as J. Warga’s “derivate containers” (cf. [9]) and the “semidifferentials” and “multidifferentials” proposed by us in previous work (cf. [7]). Recently, we have found another theory—the “pathintegral generalized differentials”, abbr. PIGDs—that contains that of GDQs as well as the other theories mentioned above. We conclude the paper by giving, in §11, a brief sketch of the definition of this concept.