June  2022, 21(6): 2147-2187. doi: 10.3934/cpaa.2022054

A potential approach for planning mean-field games in one dimension

King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia

*Corresponding author

Received  April 2021 Revised  November 2021 Published  June 2022 Early access  March 2022

Fund Project: T. Bakaryan, R. Ferreira, and D. Gomes were partially supported by King Abdullah University of Science and Technology (KAUST) baseline funds and KAUST OSR-CRG2021-4674

This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. Applying Poincaré's Lemma to the Fokker–Planck equation, we deduce the existence of a potential. Rewriting the Hamilton–Jacobi equation in terms of the potential, we obtain a system of Euler–Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation.

We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.

Citation: Tigran Bakaryan, Rita Ferreira, Diogo Gomes. A potential approach for planning mean-field games in one dimension. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2147-2187. doi: 10.3934/cpaa.2022054
References:
[1]

Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math., Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36433-4_1.

[2]

Y. AchdouF. Camilli and I. Capuzzo-Dolcetta, Mean field games: numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109.  doi: 10.1137/100790069.

[3]

T. Bakaryan, R. Ferreira and D. Gomes, Some estimates for the planning problem with potential, NoDEA Nonlinear Differ. Equ. Appl., 28 (2021), 20 pp. doi: 10.1007/s00030-021-00681-z.

[4]

M. Bardi and S. Faggian, Hopf-type estimates and formulas for nonconvex nonconcave hamilton–jacobi equations, SIAM J. Math. Anal., 29 (1998), 1067-1086.  doi: 10.1137/S0036141096309629.

[5]

P. CardaliaguetG. Carlier and B. Nazaret, Geodesics for a class of distances in the space of probability measures, Calc. Var. Partial Differ. Equ., 48 (2013), 395-420.  doi: 10.1007/s00526-012-0555-7.

[6]

G. Csató, B. Dacorogna and O. Kneuss, The Pullback Equation for Differential Forms, Progress in Nonlinear Differential Equations and their Applications. Birkhäuser/Springer, New York, 2012. doi: 10.1007/978-0-8176-8313-9.

[7]

B. DacorognaW. Gangbo and O. Kneuss, Optimal transport of closed differential forms for convex costs, Comptes Rendus Math., 353 (2015), 1099-1104.  doi: 10.1016/j.crma.2015.09.015.

[8]

W. Dacorogna and B. and Gangbo, Transportation of closed differential forms with non-homogeneous convex costs, Calc. Var. Partial Differ. Equ., 57 (2018), 108 pp. doi: 10.1007/s00526-018-1376-0.

[9]

J. DolbeaultB. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differ. Equ., 34 (2009), 193-231.  doi: 10.1007/s00526-008-0182-5.

[10]

D. EvangelistaR. FerreiraD. GomesL. Nurbekyan and V. Voskanyan, First-order, stationary mean-field games with congestion, Nonlinear Anal., 173 (2018), 37-74.  doi: 10.1016/j.na.2018.03.011.

[11]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. American Mathematical Society, 1998.

[12]

R. Ferreira and D. Gomes, Existence of weak solutions to stationary mean-field games through variational inequalities, SIAM J. Math. Anal., 50 (2018), 5969-6006.  doi: 10.1137/16M1106705.

[13]

R. FerreiraD. Gomes and T. Tada, Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions, Proc. Amer. Math. Soc., 147 (2019), 4713-4731.  doi: 10.1090/proc/14475.

[14]

R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to time-dependent mean-field games, Nonlinear Anal., 212 (2021), 31 pp. doi: 10.1016/j.na.2021.112470.

[15]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ spaces, Springer Monographs in Mathematics, Springer, New York, 2007.

[16]

D. Gomes and T. Seneci, Displacement convexity for first-order mean-field games, Minimax Theory Appl., 3 (2018), 261-284. 

[17]

P. J. Graber, A. R. Mészáros, F. J. Silva and D. Tonon, The planning problem in mean field games as regularized mass transport, Calc. Var. Partial Differ. Equ., 58 (2019), 28 pp. doi: 10.1007/s00526-019-1561-9.

[18]

M. HuangR. P Malhamé and P. Caines, Large population stochastic dynamic games: closed-loop mckean-vlasov systems and the nash certainty equivalence principle, Commun. Inform. Syst., 6 (2006), 221-252. 

[19]

D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, volume 31 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.

[20]

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.

[21]

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.

[22]

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

[23]

P. L. Lions, Cours au Collège de France, lectures on November 27th, December 4th-11th, 2009.

[24]

A. R. Mészáros and F. J. Silva, On the variational formulation of some stationary second-order mean field games systems, SIAM J. Math. Anal., 50 (2018), 1255-1277.  doi: 10.1137/17M1125960.

[25]

S. Muñoz, Classical and weak solutions to local first order mean field games through elliptic regularity, arXiv: 2006.07367v2.

[26]

C. OrrieriA. Porretta and G. Savaré, A variational approach to the mean field planning problem, J. Funct. Anal., 277 (2019), 1868-1957.  doi: 10.1016/j.jfa.2019.04.011.

[27]

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.

[28]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Ration. Mech. Anal., 216 (2015), 1-62.  doi: 10.1007/s00205-014-0799-9.

[29]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations: Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2016. doi: 10.1017/CBO9781139095143.

show all references

References:
[1]

Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math., Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36433-4_1.

[2]

Y. AchdouF. Camilli and I. Capuzzo-Dolcetta, Mean field games: numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109.  doi: 10.1137/100790069.

[3]

T. Bakaryan, R. Ferreira and D. Gomes, Some estimates for the planning problem with potential, NoDEA Nonlinear Differ. Equ. Appl., 28 (2021), 20 pp. doi: 10.1007/s00030-021-00681-z.

[4]

M. Bardi and S. Faggian, Hopf-type estimates and formulas for nonconvex nonconcave hamilton–jacobi equations, SIAM J. Math. Anal., 29 (1998), 1067-1086.  doi: 10.1137/S0036141096309629.

[5]

P. CardaliaguetG. Carlier and B. Nazaret, Geodesics for a class of distances in the space of probability measures, Calc. Var. Partial Differ. Equ., 48 (2013), 395-420.  doi: 10.1007/s00526-012-0555-7.

[6]

G. Csató, B. Dacorogna and O. Kneuss, The Pullback Equation for Differential Forms, Progress in Nonlinear Differential Equations and their Applications. Birkhäuser/Springer, New York, 2012. doi: 10.1007/978-0-8176-8313-9.

[7]

B. DacorognaW. Gangbo and O. Kneuss, Optimal transport of closed differential forms for convex costs, Comptes Rendus Math., 353 (2015), 1099-1104.  doi: 10.1016/j.crma.2015.09.015.

[8]

W. Dacorogna and B. and Gangbo, Transportation of closed differential forms with non-homogeneous convex costs, Calc. Var. Partial Differ. Equ., 57 (2018), 108 pp. doi: 10.1007/s00526-018-1376-0.

[9]

J. DolbeaultB. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differ. Equ., 34 (2009), 193-231.  doi: 10.1007/s00526-008-0182-5.

[10]

D. EvangelistaR. FerreiraD. GomesL. Nurbekyan and V. Voskanyan, First-order, stationary mean-field games with congestion, Nonlinear Anal., 173 (2018), 37-74.  doi: 10.1016/j.na.2018.03.011.

[11]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. American Mathematical Society, 1998.

[12]

R. Ferreira and D. Gomes, Existence of weak solutions to stationary mean-field games through variational inequalities, SIAM J. Math. Anal., 50 (2018), 5969-6006.  doi: 10.1137/16M1106705.

[13]

R. FerreiraD. Gomes and T. Tada, Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions, Proc. Amer. Math. Soc., 147 (2019), 4713-4731.  doi: 10.1090/proc/14475.

[14]

R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to time-dependent mean-field games, Nonlinear Anal., 212 (2021), 31 pp. doi: 10.1016/j.na.2021.112470.

[15]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ spaces, Springer Monographs in Mathematics, Springer, New York, 2007.

[16]

D. Gomes and T. Seneci, Displacement convexity for first-order mean-field games, Minimax Theory Appl., 3 (2018), 261-284. 

[17]

P. J. Graber, A. R. Mészáros, F. J. Silva and D. Tonon, The planning problem in mean field games as regularized mass transport, Calc. Var. Partial Differ. Equ., 58 (2019), 28 pp. doi: 10.1007/s00526-019-1561-9.

[18]

M. HuangR. P Malhamé and P. Caines, Large population stochastic dynamic games: closed-loop mckean-vlasov systems and the nash certainty equivalence principle, Commun. Inform. Syst., 6 (2006), 221-252. 

[19]

D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, volume 31 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.

[20]

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.

[21]

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.

[22]

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

[23]

P. L. Lions, Cours au Collège de France, lectures on November 27th, December 4th-11th, 2009.

[24]

A. R. Mészáros and F. J. Silva, On the variational formulation of some stationary second-order mean field games systems, SIAM J. Math. Anal., 50 (2018), 1255-1277.  doi: 10.1137/17M1125960.

[25]

S. Muñoz, Classical and weak solutions to local first order mean field games through elliptic regularity, arXiv: 2006.07367v2.

[26]

C. OrrieriA. Porretta and G. Savaré, A variational approach to the mean field planning problem, J. Funct. Anal., 277 (2019), 1868-1957.  doi: 10.1016/j.jfa.2019.04.011.

[27]

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.

[28]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Ration. Mech. Anal., 216 (2015), 1-62.  doi: 10.1007/s00205-014-0799-9.

[29]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations: Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2016. doi: 10.1017/CBO9781139095143.

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