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 & Continuous Dynamical Systems - A, 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, (2013), 1.  doi: 10.1007/978-3-642-36433-4_1.  Google Scholar

[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.  doi: 10.1137/120882421.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  doi: 10.3934/nhm.2012.7.279.  Google Scholar

[7]

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

[8]

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

[9]

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

[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, (1968).   Google Scholar

[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.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[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.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[13]

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

[14]

G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).  doi: 10.1142/3302.  Google Scholar

[15]

P.-L. Lions, Cours du Collège de France, 2007-2011,, , ().   Google Scholar

[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.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

A.-S. Sznitman, Topics in propagation of chaos,, in École d'Été de Probabilités de Saint-Flour XIX-1989, (1991), 165.  doi: 10.1007/BFb0085169.  Google Scholar

show all references

References:
[1]

Y. Achdou, Finite difference methods for mean field games,, in Hamilton-Jacobi equations: Approximations, (2013), 1.  doi: 10.1007/978-3-642-36433-4_1.  Google Scholar

[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.  doi: 10.1137/120882421.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  doi: 10.3934/nhm.2012.7.279.  Google Scholar

[7]

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

[8]

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

[9]

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

[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, (1968).   Google Scholar

[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.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[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.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[13]

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

[14]

G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).  doi: 10.1142/3302.  Google Scholar

[15]

P.-L. Lions, Cours du Collège de France, 2007-2011,, , ().   Google Scholar

[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.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

A.-S. Sznitman, Topics in propagation of chaos,, in École d'Été de Probabilités de Saint-Flour XIX-1989, (1991), 165.  doi: 10.1007/BFb0085169.  Google Scholar

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