PDEs and optimal control methods in mean field games, population dynamics and multi-agent models (n. 2022W58BJ5)
Type
PRIN 2022, D.D. n. 104 del 2 febbraio 2022
CUP
C53D23002590006
Principal Investigator
Alessio Porretta (Università degli Studi di ROMA "Tor Vergata")
Other research units
UO1 - Alessio PORRETTA - Università degli Studi di ROMA "Tor Vergata"
UO2 - Marco Alessandro CIRANT - Università degli Studi di Padova
UO3 - Luca ROSSI - Università degli Studi di ROMA "La Sapienza"
Duration
28/09/2023 - 28/02/2026
Description
This project is devoted to investigate many directions of research arising from the theory of mean field games, the control of multi-agent systems and population dynamics. Our focus is on models and methods related to partial differential equations (PDEs) and dynamic optimization. Our motivation is twofold: to analyse models which are significant in applied mathematics (specifically, in social sciences and economics) and, to this purpose, to develop advanced theoretical tools of broad interest in the field of PDEs and
control theory.
The heart of the project lies in mean field game (MFG) theory, which describes (Nash) equilibria in the strategic interactions of a large population of rational agents. In the PDE description, those equilibria appear as solutions of forward-backward systems
coupling the Hamilton-Jacobi equation for the value function of the agents and the Kolmogorov equation for the distribution law of the population. This theory is having a huge impact in applied mathematics; among others, we will develop the analysis of MFG
models in crowd dynamics, data analysis, economics and finance. Accordingly, this will lead us to a big enlargement of the existing theory of MFG systems: we will address issues related to boundary conditions, state constraints, stability/instability and pattern
formation in long time. We will investigate those systems under different kind of structure conditions, coming either from the individual dynamics of the agents or from specific interactions between the individual strategies and the collective state. This involves nonlocal diffusions and sub-diffusive dynamics (including time or space nonlocal operators in Hamilton-Jacobi and Fokker-Planck equations), kinetic models or PDE systems coupling Hamilton-Jacobi with reaction-diffusion equations.
The interaction between Hamilton-Jacobi and transport-diffusion equations will be central in our project and will lead to significant advances in at least three directions: (i) the study of mean field planning problems and optimal transport under congestion effects, (ii) the development of adjoint methods for regularity, ergodicity and long time estimates of nonlocal diffusive (or sub-diffusive) Hamilton-Jacobi and Fokker-Planck equations, (iii) the extension of Aubry-Mather and weak-Kam theory to sub-Riemannian control problems.
Other approaches to multi-agent control and population dynamics will be considered in our project, both independently and in relation with MFG theory. We will address sparse control methods (or control via leaders) in the context of differential games, microscopic or multiscale approaches to opinion-formation models and mean-field control problems on the Wasserstein space of measures.
We will also analyze models in population dynamics described by reaction-diffusion equations, including reaction-diffusion systems appearing in SIR-type epidemiological models, analyzing front propagation, existence and stability of traveling waves.
Activities
Conferences :
https://www.rism.it/events/indam-rism-congress-mean-field-models-in-optimal-control
https://sites.google.com/view/pdeoc2026prin/home-page
Postdocs:
Elisa Continelli (Università di Padova)
Davide Redaelli (Università di Roma Tor Vergata)
Related publications
Accepted:
Cirant, Goffi, Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case, to appear in Indiana Univ. Math. J.
Cirant, Redaelli, A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity, to appear in Comm. Pure Appl. Math.
Cirant, Kong, Wei, Zeng, Critical Mass Phenomena and Blow-up behavior of Ground States in stationary second order Mean-Field Games systems with decreasing cost, to appear in J. Math. Pures Appl.
Cirant, Redaelli, Some remarks on Linear-quadratic closed-loop games with many players, Dyn. Games Appl. 15, 558–591 (2025)
Bernardini, Cesaroni, Boundary value problems for Choquard equations. Nonlinear Anal. 254 (2025), Paper No. 113745, 14 pp.
Cesaroni, Cirant, Stationary equilibria and their stability in a Kuramoto MFG with strong interaction. Comm.Partial Differential Equations 49 (2024), no 1-2, 121–147.
Achdou, Mannucci, Marchi, Tchou, Deterministic Mean Filed Games on Networks: a Lagrangian approach, SIAM J. Math Anal. 56, 5 (2024) 6689--6730
Mannucci, Marchi, Mendico, Semi-linear parabolic equations on homogeneous Lie groups arising from mean field games, Math. Ann. vol. 390 (2) 2024, 3077--3108
Cutri, Mannucci, Marchi, Tchou, The continuity equation in the Heisenberg-periodic case: a representation formula and an application to Mean Field Games, NoDEA Nonlinear Differential Equations Appl. vol. 31 (no. 91) 2024, 25 pp.
Camilli, Marchi, Mendico, A note on first order quasi-stationary mean field games, Commun. Math. Sci. 23 (2025), no. 6, 1729--1740.
Camilli, Marchi, A continuous dependence estimate for viscous Hamilton-Jacobi equations on networks with applications, Calc. Var. Partial Differential Equations 63 (2024), no. 1, Paper No. 18, 22 pp.
Submitted:
Bardi, Asymptotic properties of non-coercive Hamiltonians with drift
Cirant, Continelli, Uniqueness of solutions to MFG systems with large discount
Cirant, Jackson, Redaelli, A non-asymptotic approach to stochastic differential games with many players under semi-monotonicity
Cirant, Mészáros, Long Time Behavior and Stabilization for Displacement Monotone Mean Field Games
Chaintron, Conforti, Eichinger, Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations
Conforti, Eichinger, A coupling approach to Lipschitz transport maps
Chiarini, Conforti, Greco, Tamanini, A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm
Cecchin, Conforti, Durmus, Eichinger, The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interactions
Ancona, Cesaroni, Coclite, Garavello , On the structure of optimal solutions of conservation laws at a junction with one incoming and one outgoing arc
