Scientific Disciplinary Sector (SSD)
MAT/05 - MATHEMATICAL ANALYSIS
I sem. dal Oct 1, 2014 al Jan 30, 2015.
In this course we will provide an introduction to Convex Analysis in finite and infinite-dimensional spaces. We will show also some applications to problems of nonlinear optimizations and control theory arising from physics and economics.
Review of weak topology on Banach spaces: convex sets, Minkowski functional, linear continuous operators, weak topology, separation of convex sets.
Convex functions: general properties, lower semicontinuous convex functions, convex conjugate, subdifferential in the sense of Convex Analysis. Introduction to Calculus of Variations.
Generalizations of convexity: differential calculus in Hilbert and Banach spaces, proximal and limiting subdifferential, the density theorem, sum rule, chain rule, generalized gradient in Banach space.
Introduction to control theory: multifunctions and trajectories of differential inclusions, viability,
equilibria, invariance, stabilization, reachability, Pontryagin Maximum Principle, necessary conditions
Application to optimization problems arising from physical or economic models.
|Ivar Ekeland and Roger Témam
||Convex Analysis and Variational Problems
|F.H. Clarke, Y.S. Ledyaev, Ronald J. Stern, P.R. Wolenski
||Nonsmooth Analysis and Control Theory
||Springer-Verlag New York Inc.
|Frank H. Clarke
||Optimization and Nonsmooth Analysis
Written and oral examination. There will be also two partial tests during the course.