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Explicit solutions of some linear-quadratic mean field games
A semi-discrete approximation for a first order mean field game problem
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 |
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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