October  2018, 11(5): 901-914. doi: 10.3934/dcdss.2018054

One-dimensional, forward-forward mean-field games with congestion

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

* Corresponding author: Diogo Gomes

Received  March 2017 Revised  July 2017 Published  June 2018

Fund Project: D. Gomes was partially supported by KAUST baseline funds and KAUSTOSR-CRG2017-3452. M. Sedjro was supported by KAUST baseline and start-up funds.

Here, we consider one-dimensional forward-forward mean-field games (MFGs) with congestion, which were introduced to approximate stationary MFGs. We use methods from the theory of conservation laws to examine the qualitative properties of these games. First, by computing Riemann invariants and corresponding invariant regions, we develop a method to prove lower bounds for the density. Next, by combining the lower bound with an entropy function, we prove the existence of global solutions for parabolic forward-forward MFGs. Finally, we construct traveling-wave solutions, which settles in a negative way the convergence problem for forward-forward MFGs. A similar technique gives the existence of time-periodic solutions for non-monotonic MFGs.

Citation: Diogo Gomes, Marc Sedjro. One-dimensional, forward-forward mean-field games with congestion. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 901-914. doi: 10.3934/dcdss.2018054
References:
[1]

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

[2]

Y. AchdouM. Cirant and M. Bardi, Mean field games models of segregation, Math. Models Methods Appl. Sci., 27 (2017), 75-113.  doi: 10.1142/S0218202517400036.  Google Scholar

[3]

M. BurgerM. Di FrancescoP. A. Markowich and M. -T Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.  doi: 10.3934/dcdsb.2014.19.1311.  Google Scholar

[4]

P. Cardaliaguet, Weak solutions for first order mean-field games with local coupling, Analysis and Geometry in Control Theory and Its Applications, 111-158, Springer INdAM Ser., 11, Springer, Cham, 2015. doi: 10.1007/978-3-319-06917-3_5.  Google Scholar

[5]

P. Cardaliaguet and P. J. Graber, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21 (2015), 690-722.  doi: 10.1051/cocv/2014044.  Google Scholar

[6]

M. Cirant, Nonlinear Pdes in Ergodic Control, Mean-field Games and Prescribed Curvature Problems, Thesis, 2013. Google Scholar

[7]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[8]

D. Evangelista, D. A. Gomes and L. Nurbekyan, Radially symmetric mean-field-games with congestion, Decision and Control (CDC), 2017 IEEE 56th Annual Conference on, 2017. arXiv: 1703.07594v1 [math. AP]. doi: 10.1109/CDC.2017.8264121.  Google Scholar

[9]

D. A. Gomes and H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1897-1910.  doi: 10.1007/s00030-015-0349-7.  Google Scholar

[10]

D. A. GomesL. Nurbekyan and M. Prazeres, Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion, 2016 IEEE 55th Conference on Decision and Control, CDC 2016, (2016), 4534-4539.  doi: 10.1109/CDC.2016.7798959.  Google Scholar

[11]

D. A. GomesL. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dyn. Games Appl., 8 (2018), 315-351.   Google Scholar

[12]

D. A. Gomes and S. Patrizi, Obstacle mean-field game problem, Interfaces Free Bound., 17 (2015), 55-68.  doi: 10.4171/IFB/333.  Google Scholar

[13]

D. A. GomesS. Patrizi and V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99 (2014), 49-79.  doi: 10.1016/j.na.2013.12.016.  Google Scholar

[14]

D. A. Gomes and E. Pimentel, Regularity for mean-field games with initial-initial boundary conditions, In J. P. Bourguignon, R. Jeltsch, A. Pinto, and M. Viana, editors, Dynamics, Games and Science Ⅲ, CIM-MS. Springer, 1 (2015), 291-304.  Google Scholar

[15]

D. A. Gomes and E. Pimentel, Time dependent mean-field games with logarithmic nonlinearities, SIAM J. Math. Anal., 47 (2015), 3798-3812.  doi: 10.1137/140984622.  Google Scholar

[16]

D. A. Gomes and E. Pimentel, Local regularity for mean-field games in the whole space, Minimax Theory and its Applications, 1 (2016), 65-82.   Google Scholar

[17]

D. A. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76.  doi: 10.1080/03605302.2014.903574.  Google Scholar

[18]

D. A. Gomes and V. Voskanyan, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92 (2015), 778-799.  doi: 10.1112/jlms/jdv052.  Google Scholar

[19]

D. A. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the superquadratic case, ESAIM Control Optim. Calc. Var., 22 (2016), 562-580.  doi: 10.1051/cocv/2015029.  Google Scholar

[20]

D. A. GomesL. Nurbekyan and M. Sedjro, One-dimensional forward-forward mean-field games, Appl. Math. Optim., 74 (2016), 619-642.  doi: 10.1007/s00245-016-9384-y.  Google Scholar

[21]

J. Graber, Weak solutions for mean field games with congestion, Preprint, 2015. Google Scholar

[22]

J.-M. Lasry and P.-L. Lions, Jeux á champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[23]

J.-M. Lasry and P.-L. Lions, Jeux á champ moyen. Ⅱ. Horizon fini et contrȏle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[24]

J. -M. Lasry, P. -L. Lions, and O. Guéant, Mean Field Games and Applications, Paris-Princeton lectures on Mathematical Finance, 2010. Google Scholar

[25]

P. G. LeFloch, Hyperbolic Systems of Conservation Laws, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002. The theory of classical and nonclassical shock waves. doi: 10.1007/978-3-0348-8150-0.  Google Scholar

[26]

P. -L. Lions, Collége de France course on mean-field games, 2007-2011. Google Scholar

[27]

A. R. Mészáros and F. J. Silva, A variational approach to second order mean field games with density constraints: The stationary case, J. Math. Pures Appl. (9), 104 (2015), 1135-1159.  doi: 10.1016/j.matpur.2015.07.008.  Google Scholar

[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.  Google Scholar

[29]

F. Santambrogio, A modest proposal for MFG with density constraints, Netw. Heterog. Media, 7 (2012), 337-347.  doi: 10.3934/nhm.2012.7.337.  Google Scholar

show all references

References:
[1]

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

[2]

Y. AchdouM. Cirant and M. Bardi, Mean field games models of segregation, Math. Models Methods Appl. Sci., 27 (2017), 75-113.  doi: 10.1142/S0218202517400036.  Google Scholar

[3]

M. BurgerM. Di FrancescoP. A. Markowich and M. -T Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.  doi: 10.3934/dcdsb.2014.19.1311.  Google Scholar

[4]

P. Cardaliaguet, Weak solutions for first order mean-field games with local coupling, Analysis and Geometry in Control Theory and Its Applications, 111-158, Springer INdAM Ser., 11, Springer, Cham, 2015. doi: 10.1007/978-3-319-06917-3_5.  Google Scholar

[5]

P. Cardaliaguet and P. J. Graber, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21 (2015), 690-722.  doi: 10.1051/cocv/2014044.  Google Scholar

[6]

M. Cirant, Nonlinear Pdes in Ergodic Control, Mean-field Games and Prescribed Curvature Problems, Thesis, 2013. Google Scholar

[7]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[8]

D. Evangelista, D. A. Gomes and L. Nurbekyan, Radially symmetric mean-field-games with congestion, Decision and Control (CDC), 2017 IEEE 56th Annual Conference on, 2017. arXiv: 1703.07594v1 [math. AP]. doi: 10.1109/CDC.2017.8264121.  Google Scholar

[9]

D. A. Gomes and H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1897-1910.  doi: 10.1007/s00030-015-0349-7.  Google Scholar

[10]

D. A. GomesL. Nurbekyan and M. Prazeres, Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion, 2016 IEEE 55th Conference on Decision and Control, CDC 2016, (2016), 4534-4539.  doi: 10.1109/CDC.2016.7798959.  Google Scholar

[11]

D. A. GomesL. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dyn. Games Appl., 8 (2018), 315-351.   Google Scholar

[12]

D. A. Gomes and S. Patrizi, Obstacle mean-field game problem, Interfaces Free Bound., 17 (2015), 55-68.  doi: 10.4171/IFB/333.  Google Scholar

[13]

D. A. GomesS. Patrizi and V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99 (2014), 49-79.  doi: 10.1016/j.na.2013.12.016.  Google Scholar

[14]

D. A. Gomes and E. Pimentel, Regularity for mean-field games with initial-initial boundary conditions, In J. P. Bourguignon, R. Jeltsch, A. Pinto, and M. Viana, editors, Dynamics, Games and Science Ⅲ, CIM-MS. Springer, 1 (2015), 291-304.  Google Scholar

[15]

D. A. Gomes and E. Pimentel, Time dependent mean-field games with logarithmic nonlinearities, SIAM J. Math. Anal., 47 (2015), 3798-3812.  doi: 10.1137/140984622.  Google Scholar

[16]

D. A. Gomes and E. Pimentel, Local regularity for mean-field games in the whole space, Minimax Theory and its Applications, 1 (2016), 65-82.   Google Scholar

[17]

D. A. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76.  doi: 10.1080/03605302.2014.903574.  Google Scholar

[18]

D. A. Gomes and V. Voskanyan, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92 (2015), 778-799.  doi: 10.1112/jlms/jdv052.  Google Scholar

[19]

D. A. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the superquadratic case, ESAIM Control Optim. Calc. Var., 22 (2016), 562-580.  doi: 10.1051/cocv/2015029.  Google Scholar

[20]

D. A. GomesL. Nurbekyan and M. Sedjro, One-dimensional forward-forward mean-field games, Appl. Math. Optim., 74 (2016), 619-642.  doi: 10.1007/s00245-016-9384-y.  Google Scholar

[21]

J. Graber, Weak solutions for mean field games with congestion, Preprint, 2015. Google Scholar

[22]

J.-M. Lasry and P.-L. Lions, Jeux á champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[23]

J.-M. Lasry and P.-L. Lions, Jeux á champ moyen. Ⅱ. Horizon fini et contrȏle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[24]

J. -M. Lasry, P. -L. Lions, and O. Guéant, Mean Field Games and Applications, Paris-Princeton lectures on Mathematical Finance, 2010. Google Scholar

[25]

P. G. LeFloch, Hyperbolic Systems of Conservation Laws, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002. The theory of classical and nonclassical shock waves. doi: 10.1007/978-3-0348-8150-0.  Google Scholar

[26]

P. -L. Lions, Collége de France course on mean-field games, 2007-2011. Google Scholar

[27]

A. R. Mészáros and F. J. Silva, A variational approach to second order mean field games with density constraints: The stationary case, J. Math. Pures Appl. (9), 104 (2015), 1135-1159.  doi: 10.1016/j.matpur.2015.07.008.  Google Scholar

[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.  Google Scholar

[29]

F. Santambrogio, A modest proposal for MFG with density constraints, Netw. Heterog. Media, 7 (2012), 337-347.  doi: 10.3934/nhm.2012.7.337.  Google Scholar

Figure 1.  Domain $\mathcal{S}_1$
Figure 2.  Level sets of the Riemann invariats for (37) with $\alpha = \frac 3 2$
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