### 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

USA

E-mail: sussmann@hamilton.rutgers.edu

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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 C^{0} - 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.
*