Second Level Degree in Mathematics
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DIFFERENTIAL EQUATIONS - 6 CFU
Teacher
Martino Bardi
Scheduled Period
I Year - 2 Semester | 02/03/2020 - 12/06/2020
Hours: 48 (24 esercitazione, 24 lezione)
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 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)
- P. Cardaliaguet, Notes on Mean Field Games, 2010 online sul sito del corso
- L.C. Evans, Partial Differential Equations, Providence, A.M.S., 1998. 2nd edition 2010
- E.N. Barron, Game theory, Hoboken, Wiley, 2008.
- M. Bardi, Appunti del corso di Equazioni Differenziali, 2018 online sul sito del corso
- M. Bardi, I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Boston, Birkhauser, 1997. 2nd printing, 2008.