# 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, (). [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: \url{http://www.college-de-france.fr}., ().

show all references

##### 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, (). [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: \url{http://www.college-de-france.fr}., ().
 [1] Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks and Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315 [2] Lucio Boccardo, Luigi Orsina. The duality method for mean field games systems. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1343-1360. doi: 10.3934/cpaa.2022021 [3] Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control and Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026 [4] Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics and Games, 2021, 8 (4) : 467-486. doi: 10.3934/jdg.2021014 [5] Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131 [6] Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations and Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 [7] Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092 [8] Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009 [9] Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053 [10] Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks and Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279 [11] Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics and Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016 [12] Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks and Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021 [13] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173 [14] Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311 [15] Adriano Festa, Diogo Gomes, Francisco J. Silva, Daniela Tonon. Preface: Mean field games: New trends and applications. Journal of Dynamics and Games, 2021, 8 (4) : i-ii. doi: 10.3934/jdg.2021025 [16] Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics and Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006 [17] Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 [18] Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control and Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 [19] Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3 [20] Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks and Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243

2020 Impact Factor: 1.213