Convergence of some Mean Field Games systems to aggregation and flocking models. (March 2021)

• Record Type: Journal Article
• Authors: Bardi, Martino; Cardaliaguet, Pierre
• Is Part Of: Nonlinear analysis. Volume 204(2021)
• Publication Date: 2021-03
• Subjects: 35Q89 -- 35Q70 -- 45K05 -- 92DXX; Mean Field Games -- Agent based models -- Aggregation equation -- Kinetic equations -- Swarming -- Flocking -- Crowd motion -- Weighted energy-dissipation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.112199
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modelling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the evolution of the mass density. The first model is a 2nd order MFG system with vanishing viscosity, and the limit is an Aggregation Equation. The result has an interpretation for models of collective animal behaviour and of crowd dynamics. The second class of problems are 1st order MFGs of acceleration and the limit is the kinetic equation associated to the Cucker–Smale model. The first problem is analysed by PDE methods, whereas the second is studied by variational methods in the space of probability measures on trajectories.