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Second Level Degree in Mathematics

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ADVANCED ANALYSIS - 8 CFU

Teacher

Giovanni Colombo

Scheduled Period

I Year - 2 Semester | 28/02/2022 - 11/06/2022

Hours: 64 (32 esercitazione, 32 lezione)

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Prerequisites

Basic real and functional analysis (some results will be recalled during the first lecture)

Target skills and knowledge

Students will be gradually introduced to some of the most important methods and ideas of modern nonlinear analysis. At the end this should provide the students with the ability of approaching a broad spectrum of topics, both applied and theoretical.

Examination methods

An oral exam on the topics covered by the course, that will include some exercises, among those that were assigned during the course.

Assessment criteria

The understanding of topics, results, and main ideas presented in the course will be evaluated. Possibly, the student's focusing on a particular subject or application will be also taken into consideration.

Course contents

Fixed point theorems by Brouwer and Schauder, with applications; the hairy ball theorem.
Gateaux and Fréchet differentiability. The differential of the norm in L^p spaces.

Ekeland variational principle with some applications (Banach fixed point theorem; local inveribility of smooth functions in infinite dimensional spaces). Further applications to PDE and control theory.

An introduction to Convex analysis: regularity of convex functions; subdifferential and normal vectors to convex sets; the convex conjugate; convex minimization problems and variational inequalities. Variational methods: applications to the p-Laplacian and to elliptic problems with an obstacle. Applications of convex analysis to quasi stati evolutions.

An introduction to the mathematical Control Theory. Closedness of the set of trajectories under convexity assumptions; existence of optimal controls for minimum problems.
Optimal control and necessary conditions for minima. Pontryagin Maximum principle.
Families of vector fields and controllability of control systems. An introduction to Rashewskii-Chow Theorem.

Planned learning activities and teaching methods

Lectures and exercises during the classes, with the possibility of personal focusing on particular subjects.

Additional notes about suggested reading

All lectures will be made on a tablet projected on a screen, and will be put on the Moodle platform in pdf format, together with a video, during the same day.

Textbooks (and optional supplementary readings)

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