September  2015, 35(9): 3879-3900. doi: 10.3934/dcds.2015.35.3879

On the system of partial differential equations arising in mean field type control

1. 

Université Paris Diderot, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, Sorbonne Paris Cité F-75205 Paris, France

Received  May 2014 Revised  September 2014 Published  April 2015

We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations arising from the finite horizon control of McKean-Vlasov dynamics. We give examples of existence and uniqueness results. Finally, we propose some simple models for the motion of pedestrians and report about numerical simulations in which we compare mean filed games and mean field type control.
Citation: Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879
References:
[1]

Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi equations: Approximations, numerical analysis and applications (eds. P. Loreti and N. A. Tchou), vol. 2074 of Lecture Notes in Math., Springer, Heidelberg, (2013), 1-47. doi: 10.1007/978-3-642-36433-4_1.

[2]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612. doi: 10.1137/120882421.

[3]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.

[4]

A. Bensoussan and J. Frehse, Control and Nash games with mean field effect, Chin. Ann. Math. Ser. B, 34 (2013), 161-192. doi: 10.1007/s11401-013-0767-y.

[5]

A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.

[6]

P. Cardaliaguet, J.-M. Lasry, P.-L. Lions and A. Porretta, Long time average of mean field games, Netw. Heterog. Media, 7 (2012), 279-301. doi: 10.3934/nhm.2012.7.279.

[7]

R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 15pp. doi: 10.1214/ECP.v18-2446.

[8]

R. Carmona, F. Delarue and A. Lachapelle, Control of McKean-Vlasov dynamics versus mean field games, Math. Financ. Econ., 7 (2013), 131-166. doi: 10.1007/s11579-012-0089-y.

[9]

D. A. Gomes and J. Saúde, Mean field games models-a brief survey, Dyn. Games Appl., 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2.

[10]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[11]

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.

[12]

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.2006.09.018.

[13]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[14]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[15]

P.-L. Lions, Cours du Collège de France, 2007-2011, http://www.college-de-france.fr/default/EN/all/equ$_-$der/.

[16]

H. P. McKean Jr., A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911. doi: 10.1073/pnas.56.6.1907.

[17]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Archive for Rational Mechanics and Analysis, 216 (2015), 1-62. doi: 10.1007/s00205-014-0799-9.

[18]

A. Porretta, On the planning problem for a class of mean field games, C. R. Math. Acad. Sci. Paris, 351 (2013), 457-462. doi: 10.1016/j.crma.2013.07.004.

[19]

A. Porretta, On the planning problem for the mean field games system, Dyn. Games Appl., 4 (2014), 231-256. doi: 10.1007/s13235-013-0080-0.

[20]

A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, (1991), 165-251. doi: 10.1007/BFb0085169.

show all references

References:
[1]

Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi equations: Approximations, numerical analysis and applications (eds. P. Loreti and N. A. Tchou), vol. 2074 of Lecture Notes in Math., Springer, Heidelberg, (2013), 1-47. doi: 10.1007/978-3-642-36433-4_1.

[2]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612. doi: 10.1137/120882421.

[3]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.

[4]

A. Bensoussan and J. Frehse, Control and Nash games with mean field effect, Chin. Ann. Math. Ser. B, 34 (2013), 161-192. doi: 10.1007/s11401-013-0767-y.

[5]

A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.

[6]

P. Cardaliaguet, J.-M. Lasry, P.-L. Lions and A. Porretta, Long time average of mean field games, Netw. Heterog. Media, 7 (2012), 279-301. doi: 10.3934/nhm.2012.7.279.

[7]

R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 15pp. doi: 10.1214/ECP.v18-2446.

[8]

R. Carmona, F. Delarue and A. Lachapelle, Control of McKean-Vlasov dynamics versus mean field games, Math. Financ. Econ., 7 (2013), 131-166. doi: 10.1007/s11579-012-0089-y.

[9]

D. A. Gomes and J. Saúde, Mean field games models-a brief survey, Dyn. Games Appl., 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2.

[10]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[11]

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.

[12]

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.2006.09.018.

[13]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[14]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[15]

P.-L. Lions, Cours du Collège de France, 2007-2011, http://www.college-de-france.fr/default/EN/all/equ$_-$der/.

[16]

H. P. McKean Jr., A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911. doi: 10.1073/pnas.56.6.1907.

[17]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Archive for Rational Mechanics and Analysis, 216 (2015), 1-62. doi: 10.1007/s00205-014-0799-9.

[18]

A. Porretta, On the planning problem for a class of mean field games, C. R. Math. Acad. Sci. Paris, 351 (2013), 457-462. doi: 10.1016/j.crma.2013.07.004.

[19]

A. Porretta, On the planning problem for the mean field games system, Dyn. Games Appl., 4 (2014), 231-256. doi: 10.1007/s13235-013-0080-0.

[20]

A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, (1991), 165-251. doi: 10.1007/BFb0085169.

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