American Institute of Mathematical Sciences

June  2012, 7(2): 263-277. doi: 10.3934/nhm.2012.7.263

A semi-discrete approximation for a first order mean field game problem

 1 "Sapienza", Università di Roma, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, 00161 Roma 2 "Sapienza", Università di Roma, Dipartimento di Matematica Guido Castelnuovo, 00185 Rome, Italy

Received  November 2011 Revised  March 2012 Published  June 2012

In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [18]. Its solution $(v,m)$ can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [18]). We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to $(v,m)$ when the discretization parameter tends to zero.
Citation: Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks and Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263
References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. of Control & Optimization, 50 (2012), 77-109. doi: 10.1137/100790069. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477. [3] J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [4] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," With appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. [5] J. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer Series in Operations Research, Springer-Verlag, New York, 2000. [6] P. Cannarsa and C. Sinestrari, "Semiconcave functions, Hamilton-Jacobi equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004. [7] Pierre Cardaliaguet, "Notes on Mean Field Games: From P.-L. Lions' Lectures at Collège de France," Lecture Notes given at Tor Vergata, 2010. [8] I. Capuzzo-Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim., 10 (1983), 367-377. doi: 10.1007/BF01448394. [9] I. Capuzzo-Dolcetta and M. Falcone, Discrete dynamic programming and viscosity solutions of the Bellman equation, Analyse Non Linéaire (Perpignan, 1987), Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 161-183. [10] I. Capuzzo-Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. Optim., 11 (1984), 161-181. doi: 10.1007/BF01442176. [11] M. Falcone and R. Ferretti, "Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations," MOS-SIAM Series on Optimization, to appear. [12] D. A. Gomes, Viscosity solution methods and the discrete Aubry-Mather problem, Discrete Contin. Dyn. Syst., 13 (2005), 103-116. doi: 10.3934/dcds.2005.13.103. [13] D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328. [14] O. Guéant, Mean field games equations with quadratic hamiltonian: a specific approach, arXiv:1106.3269v1, 2011. [15] A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349. [16] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. [17] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.019. [18] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. [19] P.-L. Lions, Cours du Collège de France. Available from: http://www.college-de-france.fr.

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References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. of Control & Optimization, 50 (2012), 77-109. doi: 10.1137/100790069. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477. [3] J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [4] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," With appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. [5] J. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer Series in Operations Research, Springer-Verlag, New York, 2000. [6] P. Cannarsa and C. Sinestrari, "Semiconcave functions, Hamilton-Jacobi equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004. [7] Pierre Cardaliaguet, "Notes on Mean Field Games: From P.-L. Lions' Lectures at Collège de France," Lecture Notes given at Tor Vergata, 2010. [8] I. Capuzzo-Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim., 10 (1983), 367-377. doi: 10.1007/BF01448394. [9] I. Capuzzo-Dolcetta and M. Falcone, Discrete dynamic programming and viscosity solutions of the Bellman equation, Analyse Non Linéaire (Perpignan, 1987), Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 161-183. [10] I. Capuzzo-Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. Optim., 11 (1984), 161-181. doi: 10.1007/BF01442176. [11] M. Falcone and R. Ferretti, "Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations," MOS-SIAM Series on Optimization, to appear. [12] D. A. Gomes, Viscosity solution methods and the discrete Aubry-Mather problem, Discrete Contin. Dyn. Syst., 13 (2005), 103-116. doi: 10.3934/dcds.2005.13.103. [13] D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328. [14] O. Guéant, Mean field games equations with quadratic hamiltonian: a specific approach, arXiv:1106.3269v1, 2011. [15] A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349. [16] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. [17] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.019. [18] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. [19] P.-L. Lions, Cours du Collège de France. Available from: http://www.college-de-france.fr.
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