First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SC01111294, 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 6.0
Type of assessment Mark
Course unit English denomination DIFFERENTIAL EQUATIONS
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction Italian
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 MARTINO BARDI MAT/05

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/05 Mathematical Analysis 6.0

Course unit organization
Period Second semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Practice 3.0 24 51.0 No turn
Lecture 3.0 24 51.0 No turn

Start of activities 02/03/2020
End of activities 12/06/2020
Show course schedule 2019/20 Reg.2011 course timetable

Examination board
Examination board not defined

Prerequisites: Differential and integral calculus for functions of several variables; elementary theory of ordinary differential equations; some classical results in Functional Analysis.
Target skills and knowledge: Understanding some methods of analysis for nonlinear partial differential equations of Hamilton-Jacobi type. Applying these equations to problems of optimal control of dynamical systems with one or more agents, via an introduction to the classical theory of games and to differential games, including Mean Field Games.
Examination methods: Oral exam, either on the lectures of the course, including the exercises proposed to the students, or on some additional material related to the topics of the course.
Assessment criteria: The evaluation will be based on the level of understanding of the concepts introduced in the course by the student and on his/her capability to handle them, also in reading and presenting some additional material related to the course.
Course unit contents: Part 1:
- Classical examples of Hamilton-Jacobi equations; the method of characteristics and the onset of singularities.
- The Hopf-Lax formula.
- Viscosity solutions: motivations and basic theory.
- The Comparison Principle and some consequences.
- Introduction to optimal control and the Dynamic Programming method; existence of solutions to H-J equations with convex Hamiltonians; synthesis of optimal feedbacks.
Part 2.
- Zero-sum games, matrix games: the min-max theorem and its consequences.
- Games with N players: Nash equilibria.
- Two-person differential games: verification theorems and feedback Nash equilibria.
- Zero-sum differential games: causal strategies and the definitions of value; Dynamic Programming and the H-J-Isaacs equation; existence of the value.
- Deterministic Mean Field Games: motivations of the theory, derivation of the system of Partial Differential Equations; uniqueness of the solution; some results about existence, with examples.
Planned learning activities and teaching methods: Lectures at the tablet, the text is made available to the student at the end of each week. Exercises are proposed, whose solution can be part of the final exam.
Additional notes about suggested reading: Three reference texts are suggested and some lecture notes of the teacher on the second half of the course are available online.
Textbooks (and optional supplementary readings)
  • L.C. Evans, Partial Differential Equations. Providence: A.M.S., 1998. 2nd edition 2010 Cerca nel catalogo
  • M. Bardi, I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Boston: Birkhauser, 1997. 2nd printing, 2008. Cerca nel catalogo
  • E.N. Barron, Game theory. Hoboken: Wiley, 2008. Cerca nel catalogo
  • M. Bardi, Appunti del corso di Equazioni Differenziali. --: --, 2018. online sul sito del corso
  • P. Cardaliaguet, Notes on Mean Field Games. --: --, 2010. online sul sito del corso

Innovative teaching methods: Teaching and learning strategies
  • Lecturing
  • Problem solving

Innovative teaching methods: Software or applications used
  • One Note (digital ink)

Sustainable Development Goals (SDGs)
Quality Education