Non coercive unbounded first order Mean Field Games: The Heisenberg example

In this paper we study evolutive first order Mean Field Games in the Heisenberg group; each agent can move in the whole space but it has to follow “horizontal” trajectories which are given in terms of the vector fields generating the group and the kinetic part of the cost depends only on the horizon...

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Bibliographic Details
Published in:Journal of Differential Equations 2022-02, Vol.309, p.809-840
Main Authors: Mannucci, Paola, Marchi, Claudio, Tchou, Nicoletta
Format: Article
Language:English
Subjects:
Analysis of PDEs
Continuity equation
Degenerate optimal control problem
First order Hamilton-Jacobi equations
Fokker-Planck equation
Heisenberg group
Heisenberg-type groups
Mathematics
Mean Field Games
Noncoercive Hamiltonian
Optimization and Control
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Summary:In this paper we study evolutive first order Mean Field Games in the Heisenberg group; each agent can move in the whole space but it has to follow “horizontal” trajectories which are given in terms of the vector fields generating the group and the kinetic part of the cost depends only on the horizontal velocity. The Hamiltonian is not coercive in the gradient term and the coefficients of the first order term in the continuity equation may have a quadratic growth at infinity. The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game systems while the latter is to represent this solution following the Lagrangian formulation of the Mean Field Games. We also provide some generalizations to Heisenberg-type structures.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2021.11.029