Mean field games systems of first order

Cardaliaguet, Pierre; Graber, P. Jameson

doi:10.1051/cocv/2014044

Cardaliaguet, Pierre ¹ ; Graber, P. Jameson ²

¹ Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
² Commands team (ENSTA ParisTech, INRIA Saclay), 828, Boulevard des Maréchaux, 91762 Palaiseau cedex, France

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 690-722.

Résumé

We consider a first-order system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton−Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem.

MR Zbl | 2 citations dans Numdam

DOI : 10.1051/cocv/2014044

Classification : 35Q91, 49K20
Mots clés : Mean field games, Hamilton−Jacobi equations, optimal control, nonlinear PDE, transport theory, long time average

Affiliations des auteurs :

Cardaliaguet, Pierre ¹ ; Graber, P. Jameson ²

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     title = {Mean field games systems of first order},
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     publisher = {EDP-Sciences},
     volume = {21},
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Cardaliaguet, Pierre; Graber, P. Jameson. Mean field games systems of first order. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 690-722. doi : 10.1051/cocv/2014044. http://www.numdam.org/articles/10.1051/cocv/2014044/

Bibliographie
Cité par

L. Ambrosio and G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, Transport equations and multi-D hyperbolic conservation laws. Springer (2008) 3–57. | MR | Zbl

P. Cardaliaguet, Long time average of first order mean field games and weak KAM theory. Dyn. Games Appl. 3 (2013) 473–488. | DOI | MR | Zbl

P. Cardaliaguet, Weak solutions for first order mean field games with local coupling. Preprint arXiv:1305.7015 (2013). | MR

P. Cardaliaguet and L. Silvestre, Hölder continuity to Hamilton−Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side. Comm. Partial Differ. Eq. 37 (2012) 1668–1688. | DOI | MR | Zbl

P. Cardaliaguet, G. Carlier, and B. Nazaret, Geodesics for a class of distances in the space of probability measures. Calc. Var. Partial Differ. Eq. (2012) 1–26. | MR | Zbl

P. Cardaliaguet, J.-M. Lasry, P.-Louis Lions and A. Porretta, et al., Long time average of mean field games. Networks and Heterogeneous Media 7 (2012) 279–301. | DOI | MR | Zbl

P. Cardaliaguet, J.-Michel Lasry, P.-L. Lions and A. Porretta, Long time average of mean field games with a nonlocal coupling. SIAM J. Control Optim. 51 (2013) 3558–3591. | DOI | MR | Zbl

Benamou, J.-David. and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375–393. | DOI | MR | Zbl

I. Ekeland and R. Temam, Convex analysis and variational problems, vol. 28, SIAM (1976). | MR

L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differ. Eq. 17 (2003) 159–177. | DOI | MR | Zbl

D.A. Gomes D.A., J. Mohr, and R.R. Souza, Discrete time, finite state space mean field games. J. Math. Pures Appl. 93 (2010) 308–328. | DOI | MR | Zbl

D.A. Gomes, E. Pimentel and H. Sánchez-Morgado, Time dependent mean-field games in the superquadratic case. Preprint arXiv:1311.6684 (2013). | MR

D.A. Gomes, E. Pimentel, and H. Sánchez-Morgado, Time dependent mean-field games in the superquadratic case. Commun. Partial Differ. Eq. 40 (2015) 40–76. | DOI | MR | Zbl

P.J. Graber, Optimal control of first-order Hamilton−Jacobi equations with linearly bounded Hamiltonian. Appl. Math. Optimization 70 (2014) 185–224. | DOI | MR | Zbl

M. Huang, P.E. Caines and R.P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $ε$ -Nash equilibria. Automatic Control, IEEE Transactions 52 (2007) 1560–1571. | DOI | MR | Zbl

M. Huang, R.P. Malhamé and P.E. Caines, Large population stochastic dynamic games: closed-loop McKean−Vlasov systems and the Nash certainty equivalence principle. Comm. Inform. Syst. 6 (2006) 221–252. | DOI | MR | Zbl

E. Kosygina and S.R.S. Varadhan, Homogenization of Hamilton−Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium. Comm. Pure Appl. Math. 61 (2008) 816–847. | DOI | MR | Zbl

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. i − le cas stationnaire. C. R. Math. 343 (2006) 619–625. | DOI | MR | Zbl

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. ii − horizon fini et contrôle optimal. C. R. Math. 343 (2006) 679–684. | DOI | MR | Zbl

J.-M. Lasry and P.-L. Lions, Mean field games. Japan. J. Math. 2 (2007) 229–260. | DOI | MR | Zbl

P.-L. Lions, Théorie des jeux de champ moyen et applications (mean field games), Cours du College de France. http://www.college-de-france.fr/site/pierre-louis-lions/ 2009 (2007).

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games. Arch. Ration. Mech. Anal. 216 (2015) 1–62. | DOI | MR | Zbl

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