Optimization and Systems Theory Seminar
Monday, August 30, 1999, 14.00-15.00, Room 3721, Lindstedtsv. 25

Professor Hector Sussmann
Department of Mathematics
Rutgers University
Piscataway, New Jersey
E-mail: sussmann@hamilton.rutgers.edu

Generalized differentials, open mapping theorems, and transversality

Motivated by problems in optimization theory (Lagrange multipliers, nonlinear programming) and optimal control (the Pontryagin maximum principle), several authors (e.g., F. Clark, J. Warga, H. Halkin) have proposed concepts of "generalized derivative" for maps which are not differentiable in the ordinary sense. All these concepts deceptively yield the same results for some simple scalar-valued functions of one variable (for example, if f(x)=|x|, then they all give the interval [-1,1] as the "derivative of f at 0"), but lead to different results for general vector-valued maps. We will present a new concept - the "generalized differential quotient" - which is both simpler and more powerful than those of the previous theories, and show how it leads to strong versions of the usual open mapping and transversal intersection theorems. The basic mathematical tool is the systematic use of a class of set-valued maps (the "regular," or C0 - approximable maps) which extends the class of maps with convex values to which the Kakutani fixed point theorem applies but, unlike this class, has good invariance properties under nonlinear transformations. The crucial technical result is a generalization to this class of maps of a theorem of Leray and Schauder on connected sets of zeros of a homotopy.
Calendar of seminars
Last update: August 30, 1999 by Anders Forsgren, anders.forsgren@math.kth.se.