| New Theories of Set-Valued Dierentials and New Versions of the Maximum Principle of Optimal Control Theory (2000) | |||||||||||||||
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| Introduction The purpose of this note is to announce a new theory of generalized di#erentials---the "generalized di#erential 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 di#erential inclusions, are discussed in a detailed paper currently in preparation. The GDQ concept contains several other notions of generalized di#erential, but does not include some important theories such as J. Warga's "derivate container | |||||||||||||||
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