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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

Fabio Camilli 1, and  Francisco Silva 2,

1. 

"Sapienza", Università di Roma, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, 00161 Roma
2. 

"Sapienza", Università 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.
Keywords: optimal control, convex duality, Mean field games, numerical methods..
Mathematics Subject Classification: Primary: 91A13; Secondary: 65M25, 49L2.
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.

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, 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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