Ph. D. Project
Stochastic Mean Field Dynamic Games.
2022/10/01 - 2025/09/30
The aim of the theory of mean field games (MFGs) is the study of deterministic or stochastic differential games as the number of agents tends to infinity. It has been introduced originally and independently in the pioneering works of J-M. Lasry and P-L. Lions [5, 6, 7] and by Caines, Huang, and Malhamé under the name of Nash Certainty Equivalence principle [4]. In MFGs, one supposes that the rational agents are indistinguishable and individually have a negligible influence on the game, and that each individual strategy is influenced by some averages of quantities depending on the states (or the controls) of the other agents. The applications of MFGs are numerous, from economics, large communication networks to the study of crowd motion and so on.

In this thesis we will mainly focus on a particular sub-class of stochastic mean-field systems, namely linear Mckean-Vlasov systems. A remarkable feature of such kind of stochastic differential equations (SDEs) is that it can effectively characterize dynamical systems of large populations subject
to a mean-field interaction with numerous applications in physics, biology, economics and finance, networks and so on. A bibliographical review leads us to the conclusion that there exists a little research on game problems for McKean-Vlasov equations. One of the objectives of this thesis is to enrich the research on game problems for this class of mean-field stochastic systems. As a short term prospect, we would like to extend our recent results ([1]) about LQ zero-sum dynamic games for this class of systems to the the infinite horizon case. By adopting a Ricatti-type approach, our aim is to construct optimal strategies in a feedback-loop form. In this context, we believe that the stabilizing solutions of adequately defined generalized Riccati equations will play a key role in the solution process. One of the main challenges here is the definition of an adequate stability concept. As a mid-term prospect, we would like to address stochastic non-cooperative dynamic games, namely: stochastic nonzero-sum dynamic games. More specifically, we are interested by stochastic Nash and Stackelberg games. We believe that there still substantial open problems in the definition of saddle-point relations for such dynamic games as well as in the characterization of globally defined solutions for the corresponding generalized Riccati equations arising in such games.

The second main research axis that we would like to address in this thesis is related to the impulsive control of mean field stochastic systems. Impulsive control systems are a class of hybrid systems in which the state propagates according to linear continuous-time dynamics except for a countable set of times at which the state can change instantaneously. These systems are useful in representing a number of real world applications, including the problem of drug distribution in the human body, management of renewable resources, sampled-data control systems with consideration of inter-sample behavior, synchronization in heterogeneous networks and so on. Motivated by our recent results about robust control of linear stochastic systems with jumps ([3]), we would like, as a mid-term prospect, to explore how does the Riccati-like theory developed in [3] could be enriched in order to deal with the mean-field case. We believe that the decomposition approach proposed in [9] could be an interesting ingredient in this perspective.

From the application side, we hope that the obtained theoretical results will lead to relevant and innovative control strategies applied to dynamical demand in energy networks, especially in the crucial field of water networks, whether it relates to drinking water distribution systems or sewage
waters drainage systems ([2, 8]). This would constitute a direct follow-up to the recent FUI project SPHEREAU dedicated to efficient water management.
Dynamic games, stochastic systems, mean field.
Control Identification Diagnosis