DOCTORAL COURSE "INTRODUCTIONS TO MEAN FIELD GAMES AND TO STOCHASTIC HYBRID SYSTEMS"AUDIENCE. Doctoral students in mathematics and in industrial engineering, univeristy of Trento. Other interested doctoral students from other doctoral schools. Interested laurea magistrale students are also welcome.

INSTRUCTORS. Fabio Bagagiolo (University of Trento) and Andrew Teel (University of California, Santa Barbara).

PERIOD. February - May 2016 (see below)

DURATION About 30 academic hours

This course is part of the activity of the research project: "OptHySYS: Optimization Techniques for Hybrid Dynamical Systems: from theory to applications", webpage

PROGRAM.

First Part: INTRODUCTION TO MEAN FIELD GAMES.

Duration: about 20 hours.

Period: February 15 - April 15.

Timetable: FIRST AND SECOND LECTURES: Tuesday February 16 and Thursday february 18, 15:00-17:00, Seminar Room of the Department of Mathematics, Povo 0 building.

More than tentative subsequent timetable:

Tue February 23 16:00-18:00 room A108 Povo 1

Wed February 24 14:00-16:00 maths seminar room Povo 0

Tue March 1 16:00-18:00 room A108 Povo 1 CANCELLED

Wed March 2 10:30-12:30 maths seminar room Povo 0

Thu March 3 15:00-17:00 maths seminar room Povo 0

Tue March 8 15:00-17:00 room A108 Povo 1 CANCELLED

Thu March 10 15:00-17:00 maths seminar room Povo 0

Tue March 15 16:00-18:00 maths seminar room Povo 0

Thu March 17 15:00-17:00 maths seminar room Povo 0

Tue March 22 16:00-18:00 PROF. LIONS LECTURE room A101 Povo 1 link

Wed March 23 9:30-11:30 room 7 Povo 0

Thu March 24 15:00-17:00 maths seminar room Povo 0 CANCELLED

Thu March 31 15:00-17:00 maths seminar room Povo 0

Fri April 1 11:00-13:00 maths seminar room Povo 0

Instructor: Fabio Bagagiolo.

The expression "Mean Field Game" stays to represent an optimal control-like situation with a large number of players, possibly infinitely many ones or even a continuum of players. Each single player directly acts on its own evolution in order to optimize a criterium (e. g. to minimize a cost) which depends on its own state and on the states of all other players (e. g. via the mean of all players states: the "mean field", indeed).

This kind of situation is quite common in the applications/motivations: from the pedestrian dynamics in a crowded environment to the opinion dynamics in a social network; from the management of an electrical power grid with many customers to the goods production in a market with many producers.

The mathematical studies of such a problem are quite recent. Usually, in this setting, the dynamics of the players are modelled by a controlled ordinary differential equation with a stochastic perturbation.

The idea is that the optimal behaviour of all players should consist of those trajectories such that, if implemented, the players population moves in a way such that those trajectories themselves become optimal for every single player (recall that the criterium to be optimized also depends on the evolution of the whole population). This can be seen as an adaptation of the concept of Nash equilibrium.

We then get a system of two coupled evolutive second order partial differential equations: namely a Kolmogorov-Fokker-Planck (KFP) equation for the evolution of the probability density of the population and a Hamilton-Jacobi (HJ) type equation for the optimum (the value function). If the dynamics of the players are not stochastically perturbed, then those equations are first order partial differential equations.

We will briefly introduce the optimal control and game theories. We will recall some facts on partial differential equations. For some particular cases, we will show how to obtain the final KFP-HJ systems as description of the limit of Nash equilibria when the number of players goes to infinity, and we will mathematically study such systems. Finally we will give several examples. A possible extension to a "Hybrid Mean Field Games" will be presented: that is a mean field game with two different players populations changing in time (let us think for example to a population of electrical devices connected by a power grid: the two different populations are the devices turned ON and the devices turned OFF, which, of course, are changing in time).

Notes (taken by Marco Andreetto and Stefano Divan)

Second Part: INTRODUCTION TO STOCHASTIC HYBRID SYSTEMS.

Duration: about 15 hours.

Period: May 9 - 20.

Timetable: to be announced

Instructor: Andrew Teel.

We describe a modeling framework that permits the interaction of jumps and flows, as well as stochastic and adversarial effects in dynamical systems. Due to various sources of non-uniqueness, we call these systems "stochastic hybrid inclusions". This feature is in contrast to the uniqueness of solutions that is a part of most existing results for stochastic hybrid systems. We give examples of systems that fit into the inclusion framework, describe what is meant by a solution to these systems, and illustrate Lyapunov-based sufficient conditions for various notions of stochastic stability. To pave the way, we first briefly review existing frameworks for stochastic hybrid systems and recent results for non-stochastic hybrid inclusions.CONTACT Fabio Bagagiolo, fabio.bagagiolo@unitn.it

TEACHING MATERIAL: To be updated. Related to the first part, you may for the moment give a look to the webpage of the 2013 doctoral course Game Theory. In particular to the links: Lecture notes "Noncooperative Differential Games"; Lecture notes "Viscosity solution of Hamilton-Jacobi equations and optimal control problems"; Notes on Mean Field Games. However, more ad hoc material will be posted. Related to the second part, for a short overview on (deterministic) hybrid systems, you may take a look to the first chapter of the book on Hybrid Dynamical Systems by Teel and co-authors.

PARTICIPANTS: