2026

Multi-valued Analysis and Differential Inclusions

Name: Multi-valued Analysis and Differential Inclusions
Code: MAT11699D
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area: Mathematics

Teaching languages: Portuguese
Languages of tutoring support: Portuguese

Sustainable Development Goals

Learning Goals

Learning of the mathematical objects, which (on the contrary to the classic problems) do not have the
uniqueness property (of solution, of derivative etc.) To develop the basic methods of the study of such
objects and acquire the competences to apply them to Calculus of Variations, to Optimal Control etc.

Contents

Elements of Convex Analysis: convex sets and functions, exposed and extreme faces, Krein-Milman
theorem, duality, subdifferential, normal and tangent cones. Multifunctions in metric spaces. Continuity.
Continuous selections. Multifunctions in measurable spaces. Aumann integral. Elements of Nonsmooth
Analysis: proximal analysis, Clarke’s generalized gradients. Differential Inclusions. Existence theorems.
Topological and other properties of the solution set. Relaxation. Application to Optimal Control.

Teaching Methods

Theoretical lections; distribution of the cards with exercises; personal assistence. Continuous evoluation
(elaboration of the works of the theoretic/practice character) and the oral examination in the end of
semester.