September  2011, 6(3): 485-519. doi: 10.3934/nhm.2011.6.485

Handling congestion in crowd motion modeling

1. 

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex, France

2. 

Laboratoire de Mathématiques d'Orsay, UMR CNRS 8628, Faculté des Sciences, Université Paris-Sud XI, 91405 Orsay Cedex

3. 

LAMAV, Université de Valenciennes et du Hainaut-Cambrésis Mont Houy, 59313 Valenciennes Cedex 9, France

Received  December 2010 Revised  July 2011 Published  August 2011

We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the non-smooth character of the models. We consider that individuals tend to move according to a desired, or spontaneous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the non-smooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description.
Citation: Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio, Juliette Venel. Handling congestion in crowd motion modeling. Networks & Heterogeneous Media, 2011, 6 (3) : 485-519. doi: 10.3934/nhm.2011.6.485
References:
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A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications,, Trudy Mat. Inst. Steklov., 38 (1951), 5. Google Scholar

[2]

L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur., 19 (1995), 191. Google Scholar

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L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Lectures in Mathematics, (2005). Google Scholar

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N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scalling to hyperbolic macroscopic models,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1317. doi: 10.1142/S0218202508003054. Google Scholar

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F. Bernicot and J. Venel, Differential inclusions with proximal normal cones in Banach spaces,, J. Convex Anal., 17 (2010), 451. Google Scholar

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F. Bernicot and J. Venel, Convergence order of a numerical scheme for sweeping process, submitted., Available from: \url{http://arxiv.org/abs/1009.2837}., (). Google Scholar

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

[8]

A. Borgers and H. Timmermans, A model of pedestrian route choice and demand for retail facilities within inner-cityshopping areas,, Geographycal Analysis, 18 (1986), 115. doi: 10.1111/j.1538-4632.1986.tb00086.x. Google Scholar

[9]

A. Borgers and H. Timmermans, City centre entry points, store location patterns and pedestrian route choice behavior: A microlevel simulation model,, Socio-Economic Planning Sciences, 20 (1986), 25. doi: 10.1016/0038-0121(86)90023-6. Google Scholar

[10]

M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space,, J. Nonlinear Convex Anal., 6 (2005), 359. Google Scholar

[11]

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

[12]

A. Canino, On $p$-convex sets and geodesics,, J. Differential Equations, 75 (1988), 118. Google Scholar

[13]

C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows,, SIAM J. Sci. Comput., 29 (2007), 539. doi: 10.1137/050641211. Google Scholar

[14]

C. Chalons, "Transport-Equilibrium Schemes for Pedestrian Flows with Nonclassical Shocks,", Traffic and Granular Flows'05, (2007), 347. doi: 10.1007/978-3-540-47641-2_31. Google Scholar

[15]

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G. Colombo and V. V. Goncharov, The sweeping processes without convexity,, Set-Valued Anal., 7 (1999), 357. doi: 10.1023/A:1008774529556. Google Scholar

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G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set,, J. Differential Equations, 187 (2003), 46. Google Scholar

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R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

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V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

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J. Dambrine, B. Maury, N. Meunier and A. Roudneff-Chupin, A congestion model for cell migration,, to appear in Communications in Pure and Applied Analysis., (). Google Scholar

[21]

E. De Giorgi, New problems on minimizing movements,, in, 29 (1993), 81. Google Scholar

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P. Degond, L. Navoret, R. Bon and D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness,, J. Stat. Phys., 138 (2010), 85. doi: 10.1007/s10955-009-9879-x. Google Scholar

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[24]

C. Dogbé, On the numerical solutions of second order macroscopic models of pedestrian flows,, Comput. Appl. Math., 56 (2008), 1884. doi: 10.1016/j.camwa.2008.04.028. Google Scholar

[25]

A. Donev, S. Torquato, F. H. Stillinger and Robert Connelly, Jamming in hard sphere and disk packings,, J. Appl. Phys., 95 (2004). doi: 10.1063/1.1633647. Google Scholar

[26]

J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984). Google Scholar

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J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program, 104 (2005), 347. doi: 10.1007/s10107-005-0619-y. Google Scholar

[28]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation,, J. Differential Equations, 226 (2006), 135. Google Scholar

[29]

H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418. doi: 10.1090/S0002-9947-1959-0110078-1. Google Scholar

[30]

P. G. Gipps and B. Marksjö, A micro-simulation model for pedestrian flows,, Mathematics and Computers in Simulation, 27 (1985), 95. doi: 10.1016/0378-4754(85)90027-8. Google Scholar

[31]

B. Gustafsson and M. Sakai, Properties of some balayage operators, with applications to quadrature domains and moving boundary problems,, Nonlinear Analysis, 22 (1994), 1221. doi: 10.1016/0362-546X(94)90107-4. Google Scholar

[32]

S. Gwynne, E. R. Galea, P. J. Lawrence and L. Filippidis, Modelling occupant interaction with fire conditions using the buildingEXODUS evacuation model,, Fire Safety Journal, 36 (2001), 327. doi: 10.1016/S0379-7112(00)00060-6. Google Scholar

[33]

D. Helbing, A fluid dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. Google Scholar

[34]

D. Helbing, P. Molnar and F. Schweitzer, "Computer Simulations of Pedestrian Dynamics and Trail Formation,", Evolution of Natural Structures, 230 (1994), 229. Google Scholar

[35]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[36]

R. L. Hughes, A continuum theory for the flow of pedestrian,, Transport. Res. Part B, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[37]

R. L. Hughes, "The Flow of Human Crowds,", Ann. Rev. Fluid Mech., 35 (2003), 169. Google Scholar

[38]

A. D. Ioffe and J. V. Outrata, On metric and calmness qualification conditions in subdifferential calculus,, Set-Valued Anal., 16 (2008), 199. doi: 10.1007/s11228-008-0076-x. Google Scholar

[39]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[40]

L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources,, J. Anal. Math., 11 (2010), 151. doi: 10.1007/s11854-010-0015-2. Google Scholar

[41]

G. G. Løvås, Modelling and simulation of pedestrian traffic flow,, Transportation Research B, 28 (1994), 429. doi: 10.1016/0191-2615(94)90013-2. Google Scholar

[42]

B. Maury, A time-stepping scheme for inelastic collisions. Numerical handling of the nonoverlapping constraint,, Numerische Mathematik, 102 (2006), 649. doi: 10.1007/s00211-005-0666-6. Google Scholar

[43]

B. Maury and J. Venel, A discrete contact model for crowd motion,, ESAIM Mathematical Modelling and Numerical Analysis, 45 (2011), 145. doi: 10.1051/m2an/2010035. Google Scholar

[44]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 1787. doi: 10.1142/S0218202510004799. Google Scholar

[45]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 346. Google Scholar

[46]

J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires,, C. R. Acad. Sci. Paris, 255 (1962), 238. Google Scholar

[47]

K. Nagel, From particle hopping models to traffic flow theory,, Transportation Research Record, 1644 (1998), 1. doi: 10.3141/1644-01. Google Scholar

[48]

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

[49]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. Google Scholar

[50]

R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis,, Trans. Amer. Math. Soc., 348 (1996), 1805. doi: 10.1090/S0002-9947-96-01544-9. Google Scholar

[51]

R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions,, Trans. Amer. Math. Soc., 352 (2000), 5231. doi: 10.1090/S0002-9947-00-02550-2. Google Scholar

[52]

R. T. Rockafellar and R. Wets, "Variational Analysis,", Grundlehren der Mathematischen, 317 (1998). Google Scholar

[53]

A. Roudneff-Chupin, "Modélisation Macroscopique des Mouvements de Foules,", Ph.D thesis, (). Google Scholar

[54]

Y. Saisho and H. Tanaka, Stochastic differential equations for mutually reflecting Brownian balls,, Osaka J. Math., 23 (1986), 725. Google Scholar

[55]

A. Schadschneider, Cellular automaton approach to pedestrian dynamics-theory,, in, (2001), 75. Google Scholar

[56]

A. Schadschneider, A. Kirchner and K. Nishinari, From ant trails to pedestrian dynamics,, Applied Bionics and Biomechanics, 1 (2003), 11. doi: 10.1533/abib.2003.1.1.11. Google Scholar

[57]

L. Thibault, Sweeping process with regular and nonregular sets,, J. Differential Equations, 193 (2003), 1. Google Scholar

[58]

S. Torquato and F. H. Stillinger, "Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond,", Reviews of Modern Physics, 82 (2010). Google Scholar

[59]

S. Torquato, T. M. Truskett and P. G. Debenedetti, Is random close packing of spheres well defined?,, Phys. Rev. Lett., 84 (2000), 2064. doi: 10.1103/PhysRevLett.84.2064. Google Scholar

[60]

J. Venel, A numerical scheme for a class of sweeping processes,, Numerische Mathematik, 118 (2011), 367. doi: 10.1007/s00211-010-0329-0. Google Scholar

[61]

J. Venel, "Integrating Strategies in Numerical Modelling of Crowd Motion,", Pedestrian and Evacuation Dynamics '08, (2010), 641. doi: 10.1007/978-3-642-04504-2_59. Google Scholar

[62]

J. Venel, "Modélisation Mathématique et Numérique des Mouvements de Foule,", Ph.D thesis, (2008). Google Scholar

[63]

C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003). Google Scholar

[64]

S. J. Yuhaski and J. M. Smith, Modeling circulation systems in buildings using state dependent queueing models,, Queueing Systems Theory Appl., 4 (1989), 319. doi: 10.1007/BF01159471. Google Scholar

show all references

References:
[1]

A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications,, Trudy Mat. Inst. Steklov., 38 (1951), 5. Google Scholar

[2]

L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur., 19 (1995), 191. Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Lectures in Mathematics, (2005). Google Scholar

[4]

N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scalling to hyperbolic macroscopic models,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1317. doi: 10.1142/S0218202508003054. Google Scholar

[5]

F. Bernicot and J. Venel, Differential inclusions with proximal normal cones in Banach spaces,, J. Convex Anal., 17 (2010), 451. Google Scholar

[6]

F. Bernicot and J. Venel, Convergence order of a numerical scheme for sweeping process, submitted., Available from: \url{http://arxiv.org/abs/1009.2837}., (). Google Scholar

[7]

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

[8]

A. Borgers and H. Timmermans, A model of pedestrian route choice and demand for retail facilities within inner-cityshopping areas,, Geographycal Analysis, 18 (1986), 115. doi: 10.1111/j.1538-4632.1986.tb00086.x. Google Scholar

[9]

A. Borgers and H. Timmermans, City centre entry points, store location patterns and pedestrian route choice behavior: A microlevel simulation model,, Socio-Economic Planning Sciences, 20 (1986), 25. doi: 10.1016/0038-0121(86)90023-6. Google Scholar

[10]

M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space,, J. Nonlinear Convex Anal., 6 (2005), 359. Google Scholar

[11]

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

[12]

A. Canino, On $p$-convex sets and geodesics,, J. Differential Equations, 75 (1988), 118. Google Scholar

[13]

C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows,, SIAM J. Sci. Comput., 29 (2007), 539. doi: 10.1137/050641211. Google Scholar

[14]

C. Chalons, "Transport-Equilibrium Schemes for Pedestrian Flows with Nonclassical Shocks,", Traffic and Granular Flows'05, (2007), 347. doi: 10.1007/978-3-540-47641-2_31. Google Scholar

[15]

F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property,, J. Convex Anal., 2 (1995), 117. Google Scholar

[16]

G. Colombo and V. V. Goncharov, The sweeping processes without convexity,, Set-Valued Anal., 7 (1999), 357. doi: 10.1023/A:1008774529556. Google Scholar

[17]

G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set,, J. Differential Equations, 187 (2003), 46. Google Scholar

[18]

R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

[19]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

[20]

J. Dambrine, B. Maury, N. Meunier and A. Roudneff-Chupin, A congestion model for cell migration,, to appear in Communications in Pure and Applied Analysis., (). Google Scholar

[21]

E. De Giorgi, New problems on minimizing movements,, in, 29 (1993), 81. Google Scholar

[22]

P. Degond, L. Navoret, R. Bon and D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness,, J. Stat. Phys., 138 (2010), 85. doi: 10.1007/s10955-009-9879-x. Google Scholar

[23]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model of pedestrian flow: The one-dimensional case,, J. Diff. Eq., 250 (2011), 1334. Google Scholar

[24]

C. Dogbé, On the numerical solutions of second order macroscopic models of pedestrian flows,, Comput. Appl. Math., 56 (2008), 1884. doi: 10.1016/j.camwa.2008.04.028. Google Scholar

[25]

A. Donev, S. Torquato, F. H. Stillinger and Robert Connelly, Jamming in hard sphere and disk packings,, J. Appl. Phys., 95 (2004). doi: 10.1063/1.1633647. Google Scholar

[26]

J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984). Google Scholar

[27]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program, 104 (2005), 347. doi: 10.1007/s10107-005-0619-y. Google Scholar

[28]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation,, J. Differential Equations, 226 (2006), 135. Google Scholar

[29]

H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418. doi: 10.1090/S0002-9947-1959-0110078-1. Google Scholar

[30]

P. G. Gipps and B. Marksjö, A micro-simulation model for pedestrian flows,, Mathematics and Computers in Simulation, 27 (1985), 95. doi: 10.1016/0378-4754(85)90027-8. Google Scholar

[31]

B. Gustafsson and M. Sakai, Properties of some balayage operators, with applications to quadrature domains and moving boundary problems,, Nonlinear Analysis, 22 (1994), 1221. doi: 10.1016/0362-546X(94)90107-4. Google Scholar

[32]

S. Gwynne, E. R. Galea, P. J. Lawrence and L. Filippidis, Modelling occupant interaction with fire conditions using the buildingEXODUS evacuation model,, Fire Safety Journal, 36 (2001), 327. doi: 10.1016/S0379-7112(00)00060-6. Google Scholar

[33]

D. Helbing, A fluid dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. Google Scholar

[34]

D. Helbing, P. Molnar and F. Schweitzer, "Computer Simulations of Pedestrian Dynamics and Trail Formation,", Evolution of Natural Structures, 230 (1994), 229. Google Scholar

[35]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[36]

R. L. Hughes, A continuum theory for the flow of pedestrian,, Transport. Res. Part B, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[37]

R. L. Hughes, "The Flow of Human Crowds,", Ann. Rev. Fluid Mech., 35 (2003), 169. Google Scholar

[38]

A. D. Ioffe and J. V. Outrata, On metric and calmness qualification conditions in subdifferential calculus,, Set-Valued Anal., 16 (2008), 199. doi: 10.1007/s11228-008-0076-x. Google Scholar

[39]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[40]

L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources,, J. Anal. Math., 11 (2010), 151. doi: 10.1007/s11854-010-0015-2. Google Scholar

[41]

G. G. Løvås, Modelling and simulation of pedestrian traffic flow,, Transportation Research B, 28 (1994), 429. doi: 10.1016/0191-2615(94)90013-2. Google Scholar

[42]

B. Maury, A time-stepping scheme for inelastic collisions. Numerical handling of the nonoverlapping constraint,, Numerische Mathematik, 102 (2006), 649. doi: 10.1007/s00211-005-0666-6. Google Scholar

[43]

B. Maury and J. Venel, A discrete contact model for crowd motion,, ESAIM Mathematical Modelling and Numerical Analysis, 45 (2011), 145. doi: 10.1051/m2an/2010035. Google Scholar

[44]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 1787. doi: 10.1142/S0218202510004799. Google Scholar

[45]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 346. Google Scholar

[46]

J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires,, C. R. Acad. Sci. Paris, 255 (1962), 238. Google Scholar

[47]

K. Nagel, From particle hopping models to traffic flow theory,, Transportation Research Record, 1644 (1998), 1. doi: 10.3141/1644-01. Google Scholar

[48]

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

[49]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. Google Scholar

[50]

R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis,, Trans. Amer. Math. Soc., 348 (1996), 1805. doi: 10.1090/S0002-9947-96-01544-9. Google Scholar

[51]

R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions,, Trans. Amer. Math. Soc., 352 (2000), 5231. doi: 10.1090/S0002-9947-00-02550-2. Google Scholar

[52]

R. T. Rockafellar and R. Wets, "Variational Analysis,", Grundlehren der Mathematischen, 317 (1998). Google Scholar

[53]

A. Roudneff-Chupin, "Modélisation Macroscopique des Mouvements de Foules,", Ph.D thesis, (). Google Scholar

[54]

Y. Saisho and H. Tanaka, Stochastic differential equations for mutually reflecting Brownian balls,, Osaka J. Math., 23 (1986), 725. Google Scholar

[55]

A. Schadschneider, Cellular automaton approach to pedestrian dynamics-theory,, in, (2001), 75. Google Scholar

[56]

A. Schadschneider, A. Kirchner and K. Nishinari, From ant trails to pedestrian dynamics,, Applied Bionics and Biomechanics, 1 (2003), 11. doi: 10.1533/abib.2003.1.1.11. Google Scholar

[57]

L. Thibault, Sweeping process with regular and nonregular sets,, J. Differential Equations, 193 (2003), 1. Google Scholar

[58]

S. Torquato and F. H. Stillinger, "Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond,", Reviews of Modern Physics, 82 (2010). Google Scholar

[59]

S. Torquato, T. M. Truskett and P. G. Debenedetti, Is random close packing of spheres well defined?,, Phys. Rev. Lett., 84 (2000), 2064. doi: 10.1103/PhysRevLett.84.2064. Google Scholar

[60]

J. Venel, A numerical scheme for a class of sweeping processes,, Numerische Mathematik, 118 (2011), 367. doi: 10.1007/s00211-010-0329-0. Google Scholar

[61]

J. Venel, "Integrating Strategies in Numerical Modelling of Crowd Motion,", Pedestrian and Evacuation Dynamics '08, (2010), 641. doi: 10.1007/978-3-642-04504-2_59. Google Scholar

[62]

J. Venel, "Modélisation Mathématique et Numérique des Mouvements de Foule,", Ph.D thesis, (2008). Google Scholar

[63]

C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003). Google Scholar

[64]

S. J. Yuhaski and J. M. Smith, Modeling circulation systems in buildings using state dependent queueing models,, Queueing Systems Theory Appl., 4 (1989), 319. doi: 10.1007/BF01159471. Google Scholar

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