
Course unit
DIFFERENTIAL EQUATIONS
SC01111294, 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 
6.0 
Course unit organization
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Lecture 
3.0 
24 
51.0 
No turn 
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 HamiltonJacobi 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 HamiltonJacobi equations; the method of characteristics and the onset of singularities.
 The HopfLax 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 HJ equations with convex Hamiltonians; synthesis of optimal feedbacks.
Part 2.
 Zerosum games, matrix games: the minmax theorem and its consequences.
 Games with N players: Nash equilibria.
 Twoperson differential games: verification theorems and feedback Nash equilibria.
 Zerosum differential games: causal strategies and the definitions of value; Dynamic Programming and the HJIsaacs 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

M. Bardi, I. CapuzzoDolcetta, Optimal control and viscosity solutions of HamiltonJacobiBellman equations. Boston: Birkhauser, 1997. 2nd printing, 2008.

E.N. Barron, Game theory. Hoboken: Wiley, 2008.

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
Sustainable Development Goals (SDGs)

