First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCP6076557, A.A. 2019/20

Information concerning the students who enrolled in A.Y. 2019/20

Information on the course unit
Degree course Second cycle degree in
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2019/20
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 8.0
Type of assessment Mark
Course unit English denomination ADVANCED ANALYSIS
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge GIOVANNI COLOMBO MAT/05
Other lecturers FRANCO RAMPAZZO MAT/05

ECTS: details
Type Scientific-Disciplinary 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 of
Individual study
Practice 4.0 32 68.0 No turn
Lecture 4.0 32 68.0 No turn

Start of activities 30/09/2019
End of activities 18/01/2020
Show course schedule 2019/20 Reg.2011 course timetable

Examination board
Board From To Members of the board
4 Analisi Superiore - a.a. 2019/2020 01/10/2019 30/09/2020 COLOMBO GIOVANNI (Presidente)
RAMPAZZO FRANCO (Membro Effettivo)
ANCONA FABIO (Supplente)

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 Fr├ęchet 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 co-vectors.
Necessary conditions for constrained 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 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).. --: --, --. Cerca nel catalogo
  • Bressan, Piccoli, Introduction to the Mathematical Theory of Control ( AIMS on Applied Mathematics). --: American Institute on Applied Mathematics, --. Cerca nel catalogo

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

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