
Course unit
ADVANCED ANALYSIS
SCP6076557, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/05 
Mathematical Analysis 
8.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
4.0 
32 
68.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
Prerequisites:

Basic real and functional analysis 
Target skills and knowledge:

Students will be gradually introduced to some of the main 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 may include doing some simple exercises. 
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 unit 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.
An introduction to the mathematical Control Theory. Closedness of the set of trajectories under convexity assumptions; existence of optimal controls for minimum problems. Set separation and cone (non)transversality as basic tools for abstract constrained minimization.
Optimal Control.
Nonlinear ordinary differential equations and transport of vectors and covectors.
Necessary conditions for constrained minima. Pontryagin Maximum principle.
Families of vector fields and controllability of control systems. An introduction to RashewskiiChow 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 during the same day. Moreover, in the second part of the course printed lecture notes will be available. 
Textbooks (and optional supplementary readings) 

Ekeland,Temam, Convex analysis and variational problems (Classics in Applied Mathematics).. : , .

Bressan, Piccoli, Introduction to the Mathematical Theory of Control ( AIMS on Applied Mathematics). : American Institute on Applied Mathematics, .

Innovative teaching methods: Teaching and learning strategies
 Loading of files and pages (web pages, Moodle, ...)
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)
 One Note (digital ink)
 Latex
 Mathematica
Sustainable Development Goals (SDGs)

