Differential Calculus of Set-Valued Maps. An Update

Aubin, J.-P. (1987). Differential Calculus of Set-Valued Maps. An Update. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-87-093

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Abstract

IIASA has played a crucial role in the development of the "graphical approach" to the differential calculus of set-valued maps, around J.-P. Aubin, H. Frankowska, R.T. Rockafellar and allowed to make contacts with Soviet and eastern European mathematicians (C. Olech, B. Pschenichnyiy, E. Polovinkin, V. Tihomirov, etc.) who were following analogous approaches. Since 1981, they and their collaborators developed this calculus and applied it to a variety of problems, in mathematical programming (Kuhn-Tucker rules, sensitivity of solutions and Lagrange multipliers), in nonsmooth analysis (Inverse Functions Theorems, local uniqueness), in control theory (controllability of systems with feedbacks, Pontryagin's Maximum Principle, Hamilton-Jacobi-Bellman equations, observability and other issues), in viability theory (regulation of systems, heavy trajectories), etc.

The first version of this survey appeared at IIASA in 1982, and constituted the seventh chapter of the book "Applied Nonlinear Analysis" published in 1984 by I. Ekeland and the author. Since then, many other results have been motivated by the successful applications of this calculus, and, maybe unfortunately, other concepts (such as the concept of intermediate tangent cone and derivatives introduced and used by H. Frankowska). Infinite-dimensional problems such as control problems or the more classical problems of calculus of variations require the use of adequate adaptations of the same main idea, as well as more technical assumptions.

The time and the place (IIASA) were ripe to update the exposition of this differential calculus. The Russian translation of "Applied Nonlinear Analysis" triggered this revised version.

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