September  2015, 35(9): 4173-4192. doi: 10.3934/dcds.2015.35.4173

A model problem for Mean Field Games on networks

1. 

"Sapienza" Università di Roma, Dip. di Scienze di Base e Applicate per l'Ingegneria, via Scarpa 16, 0161 Roma

2. 

"Sapienza" Università di Roma, Dipartimento di Matematica, P.le A. Moro 5, 00185 Roma, Italy

3. 

Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova

Received  March 2014 Revised  September 2014 Published  April 2015

In [14], Guéant, Lasry and Lions considered the model problem ``What time does meeting start?'' as a prototype for a general class of optimization problems with a continuum of players, called Mean Field Games problems. In this paper we consider a similar model, but with the dynamics of the agents defined on a network. We discuss appropriate transition conditions at the vertices which give a well posed problem and we present some numerical results.
Citation: Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173
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]

J. von Below, Classical solvability of linear parabolic equations on networks,, J. Differential Equations, 72 (1988), 316.  doi: 10.1016/0022-0396(88)90158-1.  Google Scholar

[3]

J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion,, Comm. Partial Differential Equations, 21 (1996), 255.  doi: 10.1080/03605309608821184.  Google Scholar

[4]

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

[5]

F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equation on networks,, J. Differential Equations, 254 (2013), 4122.  doi: 10.1016/j.jde.2013.02.013.  Google Scholar

[6]

P. Cardaliaguet, Notes on Mean Field Games: from P.-L. Lions' lectures at Collège de France,, Lecture Notes given at Tor Vergata, (2010).   Google Scholar

[7]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks,, SIAM J. Math. Anal., 42 (2010), 1761.  doi: 10.1137/090771417.  Google Scholar

[8]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Math. Comput. Modelling, 52 (2010), 1506.  doi: 10.1016/j.mcm.2010.06.012.  Google Scholar

[9]

M. Freidlin and S. J. Sheu, Diffusion processes on graphs: Stochastic differential equations, large deviation principle,, Probab. Theory Related Fields, 116 (2000), 181.  doi: 10.1007/PL00008726.  Google Scholar

[10]

M. Freidlin and A. Wentzell, Diffusion processes on graphs and the averaging principle,, Ann. Probab., 21 (1993), 2215.  doi: 10.1214/aop/1176989018.  Google Scholar

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).   Google Scholar

[12]

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

[13]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks,, Appl. Math. Optim., 55 (2007), 219.  doi: 10.1007/s00245-006-0887-9.  Google Scholar

[14]

O. Guéant, J-M. Lasry and P-L. Lions, Mean field games and applications,, in Paris-Princeton Lectures on Mathematical Finance 2010, (2011), 205.  doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[15]

O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach,, M3AS Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202512500224.  Google Scholar

[16]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[17]

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

[18]

D. Mugnolo, Gaussian estimates for a heat equation on a network,, Netw. Het. Media, 2 (2007), 55.  doi: 10.3934/nhm.2007.2.55.  Google Scholar

[19]

D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations,, Math. Methods Appl. Sci., 30 (2007), 681.  doi: 10.1002/mma.805.  Google Scholar

[20]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707.  doi: 10.1007/s00205-010-0366-y.  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]

J. von Below, Classical solvability of linear parabolic equations on networks,, J. Differential Equations, 72 (1988), 316.  doi: 10.1016/0022-0396(88)90158-1.  Google Scholar

[3]

J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion,, Comm. Partial Differential Equations, 21 (1996), 255.  doi: 10.1080/03605309608821184.  Google Scholar

[4]

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

[5]

F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equation on networks,, J. Differential Equations, 254 (2013), 4122.  doi: 10.1016/j.jde.2013.02.013.  Google Scholar

[6]

P. Cardaliaguet, Notes on Mean Field Games: from P.-L. Lions' lectures at Collège de France,, Lecture Notes given at Tor Vergata, (2010).   Google Scholar

[7]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks,, SIAM J. Math. Anal., 42 (2010), 1761.  doi: 10.1137/090771417.  Google Scholar

[8]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Math. Comput. Modelling, 52 (2010), 1506.  doi: 10.1016/j.mcm.2010.06.012.  Google Scholar

[9]

M. Freidlin and S. J. Sheu, Diffusion processes on graphs: Stochastic differential equations, large deviation principle,, Probab. Theory Related Fields, 116 (2000), 181.  doi: 10.1007/PL00008726.  Google Scholar

[10]

M. Freidlin and A. Wentzell, Diffusion processes on graphs and the averaging principle,, Ann. Probab., 21 (1993), 2215.  doi: 10.1214/aop/1176989018.  Google Scholar

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).   Google Scholar

[12]

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

[13]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks,, Appl. Math. Optim., 55 (2007), 219.  doi: 10.1007/s00245-006-0887-9.  Google Scholar

[14]

O. Guéant, J-M. Lasry and P-L. Lions, Mean field games and applications,, in Paris-Princeton Lectures on Mathematical Finance 2010, (2011), 205.  doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[15]

O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach,, M3AS Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202512500224.  Google Scholar

[16]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[17]

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

[18]

D. Mugnolo, Gaussian estimates for a heat equation on a network,, Netw. Het. Media, 2 (2007), 55.  doi: 10.3934/nhm.2007.2.55.  Google Scholar

[19]

D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations,, Math. Methods Appl. Sci., 30 (2007), 681.  doi: 10.1002/mma.805.  Google Scholar

[20]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[1]

Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315

[2]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021

[3]

Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026

[4]

Matt Barker. From mean field games to the best reply strategy in a stochastic framework. Journal of Dynamics & Games, 2019, 6 (4) : 291-314. doi: 10.3934/jdg.2019020

[5]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[6]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020026

[7]

Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3

[8]

Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131

[9]

Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092

[10]

Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009

[11]

Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053

[12]

Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279

[13]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016

[14]

Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311

[15]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[16]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

[17]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303

[18]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243

[19]

Yves Achdou, Victor Perez. Iterative strategies for solving linearized discrete mean field games systems. Networks & Heterogeneous Media, 2012, 7 (2) : 197-217. doi: 10.3934/nhm.2012.7.197

[20]

Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (103)
  • HTML views (0)
  • Cited by (6)

[Back to Top]