Multidifferential Calculus: Chain Rule, Open Mapping and Transversal Intersection Theorems
The purpose of this note is to develop a calculus of “multidifferentials” of setvalued maps between finite-dimensional real linear spaces. This will be done by generalizing the concept of the classical differential (CD), following the ideas of F. Clarke’s theory of generalized Jacobians (CGJ’s) and J. Warga’s theory of derivate containers (WDC’s), cf. Clarke  and Warga [17, 18, 19, 20].
KeywordsCompact Subset Open Mapping Convex Cone Closed Convex Cone Pontryagin Maximum Principle
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- Browder, F.E. (1960), “On the fixed point index for continuous mappings of locally connected spaces,” Summa Brazil. Math., Vol. j, 253293.Google Scholar
- Lojasiewicz Jr., S., “Local controllability of parametrized differential equations,” in preparation.Google Scholar
- Sternberg, S. (1964), Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar
- Sussmann, H.J. (1994), “A strong version of the Lojasiewicz Maximum Principle,” in Optimal Control of Differential Equations, Nicolai H. Pavel, ed., Marcel Dekker, New York.Google Scholar
- Sussmann, H.J. (1994), “A strong version of the Maximum Principle under weak hypotheses,” in Proc. 33rd IEEE Conference on Decision and Control, Orlando, Florida, 1950–1956.Google Scholar
- Sussmann, H.J. (1997), “Some recent results on the maximum principle of optimal control theory,” in Systems and Control in the Twenty-First Google Scholar
- Century,C. I. Byrnes, B. N. Datta, D. S. Gilliam and C. F. Martin, eds., Birkhäuser, Boston, 351–372.Google Scholar
- Sussmann, H.J. (1996), “A strong maximum principle for systems of differential inclusions,” in Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, Dec. 1996, IEEE Publications.Google Scholar
- Wazewski, T. (1947), “Sur l’évaluation du domaine d’existence des fonctions implicites réelles ou complexes,” Ann. Soc. Polon. Math., Vol. 20.Google Scholar