July  2014, 19(5): 1311-1333. doi: 10.3934/dcdsb.2014.19.1311

Mean field games with nonlinear mobilities in pedestrian dynamics

1. 

Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany

2. 

Department of Mathematical Sciences, 4W, 1.14, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom

3. 

King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

4. 

Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria

Received  April 2013 Revised  November 2013 Published  April 2014

In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results.
Citation: Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311
References:
[1]

D. Amadori and M. Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 259-280. doi: 10.1016/S0252-9602(12)60016-2.

[2]

L. Ambrosio, S. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps, Manuscripta Math., 121 (2006), 1-50. doi: 10.1007/s00229-006-0003-0.

[3]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

[4]

C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.

[5]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.

[6]

V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B: Methodological, 35 (2001), 293-312. doi: 10.1016/S0191-2615(99)00052-1.

[7]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.

[8]

C. Brune, 4D Imaging in Tomography and Optimal Nanoscopy, PhD thesis, University of Münster, 2010.

[9]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries, Nonlinearity, 25 (2012), 961-990. doi: 10.1088/0951-7715/25/4/961.

[10]

M. Burger, M. Di Francesco, J.-F. Pietschmann and B. Schlake, Nonlinear cross-diffusion with size exclusion, SIAM J. Math. Anal., 42 (2010), 2842-2871. doi: 10.1137/100783674.

[11]

M. Burger, P. A. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations, Kinet. Relat. Models, 4 (2011), 1025-1047. doi: 10.3934/krm.2011.4.1025.

[12]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8.

[13]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734. doi: 10.1137/120883499.

[14]

M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics, Netw. Heterog. Media, 6 (2011), 425-442. doi: 10.3934/nhm.2011.6.425.

[15]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34p. doi: 10.1142/S0218202511500230.

[16]

R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, in Current advances in nonlinear analysis and related topics, vol. 32 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 2010, 255-272.

[17]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362. doi: 10.1016/j.jde.2010.10.015.

[18]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520. doi: 10.1016/j.mcm.2010.06.012.

[19]

L. Dyson, P. Maini and R. Baker, Macroscopic limits of individual-based models for motile cell populations with volume exclusion, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 86 (2012), 031903. doi: 10.1103/PhysRevE.86.031903.

[20]

H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems, IMA J. Numer. Anal., 30 (2010), 1206-1234. doi: 10.1093/imanum/drn083.

[21]

N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes' model for pedestrian motion, Z. Angew. Math. Phys., 64 (2013), 223-251. doi: 10.1007/s00033-012-0232-x.

[22]

P. Goatin and M. Mimault, The wave-front tracking algorithm for Hughes' model of pedestrian motion, SIAM J. Sci. Comput., 35 (2013), B606-B622. doi: 10.1137/120898863.

[23]

D. Gomes and J. Saúde, Mean Field Games - a Brief Survey, Technical report, submitted, 2013.

[24]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, vol. 2003 of Lecture Notes in Math., Springer, Berlin, 2011, 205-266. doi: 10.1007/978-3-642-14660-2_3.

[25]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.

[26]

D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[27]

S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transportation Research Part B: Methodological, 38 (2004), 169-190. doi: 10.1016/S0191-2615(03)00007-9.

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535, URL http://www.sciencedirect.com/science/article/pii/S0191261501000157. doi: 10.1016/S0191-2615(01)00015-7.

[29]

H. Ishii, Asymptotic solutions for large time of hamilton-jacobi equations in euclidean n space, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 25 (2008), 231-266, URL http://www.sciencedirect.com/science/article/pii/S0294144907000054. doi: 10.1016/j.anihpc.2006.09.002.

[30]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics), 2nd edition, Springer, 1991, URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20 &path=ASIN/0387976558. doi: 10.1007/978-1-4612-0949-2.

[31]

A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transportation Research Part B: Methodological, 45 (2011), 1572-1589, URL http://www.sciencedirect.com/science/article/pii/S0191261511001111. doi: 10.1016/j.trb.2011.07.011.

[32]

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

[33]

M. Moussad, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettr, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds, PLoS Comput. Biol., 8 (2012), e1002442. doi: 10.1371/journal.pcbi.1002442.

[34]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.

[35]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.

[36]

M. Simpson, B. Hughes and K. Landman, Diffusing populations: Ghosts or folks, Australasian Journal of Engineering Education, 15 (2009), 59-68, URL http://eprints.qut.edu.au/44917/.

[37]

J. van den Berg, S. Patil, J. Sewall, D. Manocha and M. Lin, Interactive navigation of multiple agents in crowded environments, in Proceedings of the 2008 symposium on Interactive 3D graphics and games, ACM, (2008), 139-147.

show all references

References:
[1]

D. Amadori and M. Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 259-280. doi: 10.1016/S0252-9602(12)60016-2.

[2]

L. Ambrosio, S. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps, Manuscripta Math., 121 (2006), 1-50. doi: 10.1007/s00229-006-0003-0.

[3]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

[4]

C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.

[5]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.

[6]

V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B: Methodological, 35 (2001), 293-312. doi: 10.1016/S0191-2615(99)00052-1.

[7]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.

[8]

C. Brune, 4D Imaging in Tomography and Optimal Nanoscopy, PhD thesis, University of Münster, 2010.

[9]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries, Nonlinearity, 25 (2012), 961-990. doi: 10.1088/0951-7715/25/4/961.

[10]

M. Burger, M. Di Francesco, J.-F. Pietschmann and B. Schlake, Nonlinear cross-diffusion with size exclusion, SIAM J. Math. Anal., 42 (2010), 2842-2871. doi: 10.1137/100783674.

[11]

M. Burger, P. A. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations, Kinet. Relat. Models, 4 (2011), 1025-1047. doi: 10.3934/krm.2011.4.1025.

[12]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8.

[13]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734. doi: 10.1137/120883499.

[14]

M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics, Netw. Heterog. Media, 6 (2011), 425-442. doi: 10.3934/nhm.2011.6.425.

[15]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34p. doi: 10.1142/S0218202511500230.

[16]

R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, in Current advances in nonlinear analysis and related topics, vol. 32 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 2010, 255-272.

[17]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362. doi: 10.1016/j.jde.2010.10.015.

[18]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520. doi: 10.1016/j.mcm.2010.06.012.

[19]

L. Dyson, P. Maini and R. Baker, Macroscopic limits of individual-based models for motile cell populations with volume exclusion, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 86 (2012), 031903. doi: 10.1103/PhysRevE.86.031903.

[20]

H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems, IMA J. Numer. Anal., 30 (2010), 1206-1234. doi: 10.1093/imanum/drn083.

[21]

N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes' model for pedestrian motion, Z. Angew. Math. Phys., 64 (2013), 223-251. doi: 10.1007/s00033-012-0232-x.

[22]

P. Goatin and M. Mimault, The wave-front tracking algorithm for Hughes' model of pedestrian motion, SIAM J. Sci. Comput., 35 (2013), B606-B622. doi: 10.1137/120898863.

[23]

D. Gomes and J. Saúde, Mean Field Games - a Brief Survey, Technical report, submitted, 2013.

[24]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, vol. 2003 of Lecture Notes in Math., Springer, Berlin, 2011, 205-266. doi: 10.1007/978-3-642-14660-2_3.

[25]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.

[26]

D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[27]

S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transportation Research Part B: Methodological, 38 (2004), 169-190. doi: 10.1016/S0191-2615(03)00007-9.

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535, URL http://www.sciencedirect.com/science/article/pii/S0191261501000157. doi: 10.1016/S0191-2615(01)00015-7.

[29]

H. Ishii, Asymptotic solutions for large time of hamilton-jacobi equations in euclidean n space, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 25 (2008), 231-266, URL http://www.sciencedirect.com/science/article/pii/S0294144907000054. doi: 10.1016/j.anihpc.2006.09.002.

[30]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics), 2nd edition, Springer, 1991, URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20 &path=ASIN/0387976558. doi: 10.1007/978-1-4612-0949-2.

[31]

A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transportation Research Part B: Methodological, 45 (2011), 1572-1589, URL http://www.sciencedirect.com/science/article/pii/S0191261511001111. doi: 10.1016/j.trb.2011.07.011.

[32]

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

[33]

M. Moussad, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettr, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds, PLoS Comput. Biol., 8 (2012), e1002442. doi: 10.1371/journal.pcbi.1002442.

[34]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.

[35]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.

[36]

M. Simpson, B. Hughes and K. Landman, Diffusing populations: Ghosts or folks, Australasian Journal of Engineering Education, 15 (2009), 59-68, URL http://eprints.qut.edu.au/44917/.

[37]

J. van den Berg, S. Patil, J. Sewall, D. Manocha and M. Lin, Interactive navigation of multiple agents in crowded environments, in Proceedings of the 2008 symposium on Interactive 3D graphics and games, ACM, (2008), 139-147.

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