Figure 1.
One step of the catching-up algorithm. Here $ t = 2k\tau $ , the dark rounded rectangle represents an obstacle
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2021, 41(11): 5009-5036.
doi: 10.3934/dcds.2021066
Modeling of crowds in regions with moving obstacles
Matrosov Institute for System Dynamics and Control Theory, 134 Lermontov St., Irkutsk, 664033, Russia |
We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to $ r $-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.
Keywords:
Crowd dynamics,
sweeping process,
Wasserstein space,
well-posedness,
catching-up scheme,
moving boundary problem.
Mathematics Subject Classification:
Primary: 35F31; Secondary: 35R37, 49J53, 49Q20.
Citation:
Nadezhda Maltugueva, Nikolay Pogodaev. Modeling of crowds in regions with moving obstacles. Discrete and Continuous Dynamical Systems,
2021, 41
(11)
: 5009-5036.
doi: 10.3934/dcds.2021066
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References:
| [1] |
L. Ambrosio and G. Crippa,
Continuity equations and ODE flows with non-smooth velocity, Proc. R. Soc. Edinb., Sect. A, Math., 144 (2014), 1191-1244.
doi: 10.1017/S0308210513000085. |
| [2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Basel: Birkhäuser, 2005. |
| [3] |
V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
| [4] |
D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, vol. 33, Providence, RI: American Mathematical Society (AMS), 2001.
doi: 10.1090/gsm/033. |
| [5] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Boston, MA: Birkhäuser, 2010,297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
| [6] |
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics, Springer London, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
| [7] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier,
Non-local crowd dynamics, Comptes Rendus Mathematique, 349 (2011), 769-772.
doi: 10.1016/j.crma.2011.07.005. |
| [8] |
R. M. Colombo and E. Rossi,
Nonlocal conservation laws in bounded domains, SIAM J. Math. Anal., 50 (2018), 4041-4065.
doi: 10.1137/18M1171783. |
| [9] |
R. Colombo, Control of the continuity equation with a non local flow, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 353–379, http://journals.cambridge.org/abstract{\_}S1292811910000072.
doi: 10.1051/cocv/2010007. |
| [10] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, vol. 8, Basel: Birkhäuser, 1993.
doi: 10.1007/978-1-4612-0327-8. |
| [11] |
S. Di Marino, B. Maury and F. Santambrogio,
Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601.
|
| [12] |
H. Federer,
Curvature measures, Trans. Am. Math. Soc., 93 (1959), 418-491.
doi: 10.1090/S0002-9947-1959-0110078-1. |
| [13] |
A. Figalli and N. Gigli,
A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions, Journal des Mathematiques Pures et Appliquees, 94 (2010), 107-130.
doi: 10.1016/j.matpur.2009.11.005. |
| [14] |
D. Helbing, I. J. Farkas and T. Vicsek, Crowd disasters and simulation of panic situations, The Science of Disasters, Springer Berlin Heidelberg, Berlin, Heidelberg, 2002,330–350
doi: 10.1007/978-3-642-56257-0\_11. |
| [15] |
R. L. Hughes,
The flow of human crowds, Annual Review of Fluid Mechanics, 35 (2003), 169-182.
doi: 10.1146/annurev.fluid.35.101101.161136. |
| [16] |
B. Maury and S. Faure, Crowds in Equations. An Introduction to the Microscopic Modeling of Crowds, Hackensack, NJ: World Scientific, 2019. |
| [17] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
| [18] |
B. Piccoli and F. Rossi,
Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105.
doi: 10.1007/s10440-012-9771-6. |
| [19] |
B. Piccoli and F. Rossi, Measure-theoretic models for crowd dynamics, in Crowd Dynamics, Volume 1. Theory, Models, and Safety Problems, Cham: Birkhäuser, 2018,137–165. |
| [20] |
R. A. Poliquin, R. T. Rockafellar and L. Thibault,
Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.
doi: 10.1090/S0002-9947-00-02550-2. |
| [21] |
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
| [22] |
F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, vol. 87, Cham: Birkhäuser/Springer, 2015.
doi: 10.1007/978-3-319-20828-2. |
| [23] |
M. Sene and L. Thibault,
Regularization of dynamical systems associated with prox-regular moving sets., J. Nonlinear Convex Anal., 15 (2014), 647-663.
|
| [24] |
L. Thibault,
Regularization of nonconvex sweeping process in Hilbert space, Set-Valued Analysis, 16 (2008), 319-333.
doi: 10.1007/s11228-008-0083-y. |
| [25] |
C. Villani, Optimal Transport. Old and New, vol. 338, Berlin: Springer, 2009.
doi: 10.1007/978-3-540-71050-9. |
show all references
References:
| [1] |
L. Ambrosio and G. Crippa,
Continuity equations and ODE flows with non-smooth velocity, Proc. R. Soc. Edinb., Sect. A, Math., 144 (2014), 1191-1244.
doi: 10.1017/S0308210513000085. |
| [2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Basel: Birkhäuser, 2005. |
| [3] |
V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
| [4] |
D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, vol. 33, Providence, RI: American Mathematical Society (AMS), 2001.
doi: 10.1090/gsm/033. |
| [5] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Boston, MA: Birkhäuser, 2010,297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
| [6] |
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics, Springer London, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
| [7] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier,
Non-local crowd dynamics, Comptes Rendus Mathematique, 349 (2011), 769-772.
doi: 10.1016/j.crma.2011.07.005. |
| [8] |
R. M. Colombo and E. Rossi,
Nonlocal conservation laws in bounded domains, SIAM J. Math. Anal., 50 (2018), 4041-4065.
doi: 10.1137/18M1171783. |
| [9] |
R. Colombo, Control of the continuity equation with a non local flow, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 353–379, http://journals.cambridge.org/abstract{\_}S1292811910000072.
doi: 10.1051/cocv/2010007. |
| [10] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, vol. 8, Basel: Birkhäuser, 1993.
doi: 10.1007/978-1-4612-0327-8. |
| [11] |
S. Di Marino, B. Maury and F. Santambrogio,
Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601.
|
| [12] |
H. Federer,
Curvature measures, Trans. Am. Math. Soc., 93 (1959), 418-491.
doi: 10.1090/S0002-9947-1959-0110078-1. |
| [13] |
A. Figalli and N. Gigli,
A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions, Journal des Mathematiques Pures et Appliquees, 94 (2010), 107-130.
doi: 10.1016/j.matpur.2009.11.005. |
| [14] |
D. Helbing, I. J. Farkas and T. Vicsek, Crowd disasters and simulation of panic situations, The Science of Disasters, Springer Berlin Heidelberg, Berlin, Heidelberg, 2002,330–350
doi: 10.1007/978-3-642-56257-0\_11. |
| [15] |
R. L. Hughes,
The flow of human crowds, Annual Review of Fluid Mechanics, 35 (2003), 169-182.
doi: 10.1146/annurev.fluid.35.101101.161136. |
| [16] |
B. Maury and S. Faure, Crowds in Equations. An Introduction to the Microscopic Modeling of Crowds, Hackensack, NJ: World Scientific, 2019. |
| [17] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
| [18] |
B. Piccoli and F. Rossi,
Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105.
doi: 10.1007/s10440-012-9771-6. |
| [19] |
B. Piccoli and F. Rossi, Measure-theoretic models for crowd dynamics, in Crowd Dynamics, Volume 1. Theory, Models, and Safety Problems, Cham: Birkhäuser, 2018,137–165. |
| [20] |
R. A. Poliquin, R. T. Rockafellar and L. Thibault,
Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.
doi: 10.1090/S0002-9947-00-02550-2. |
| [21] |
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
| [22] |
F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, vol. 87, Cham: Birkhäuser/Springer, 2015.
doi: 10.1007/978-3-319-20828-2. |
| [23] |
M. Sene and L. Thibault,
Regularization of dynamical systems associated with prox-regular moving sets., J. Nonlinear Convex Anal., 15 (2014), 647-663.
|
| [24] |
L. Thibault,
Regularization of nonconvex sweeping process in Hilbert space, Set-Valued Analysis, 16 (2008), 319-333.
doi: 10.1007/s11228-008-0083-y. |
| [25] |
C. Villani, Optimal Transport. Old and New, vol. 338, Berlin: Springer, 2009.
doi: 10.1007/978-3-540-71050-9. |
Figure 2.
Solutions to the attraction/repulsion model (Example 1) at time moments $ t = 2, 6, 10, 14, 16, 18 $ . The initial measure $ \vartheta $ is the Gaussian probability distribution on $ \mathbb R^{2} $ with mean $ (4, 0) $ and variance $ \mathbf{id} $ . The obstacle is the blue ellipse moving from the bottom left to the top right corner. Parameters of the model: $ A_{a} = 4 $ , $ A_{r} = 7 $ , $ a = 1/\sqrt 2 $ , $ r = 0.5 $ , $ w\equiv -0.3 $ , parameters of discretization: $ \tau = 0.01 $ , $ N = 300 $
Figure 3.
Solutions to the congestion model (Example 2) at time moments $ t = 2, 6, 10, 14, 20 $ . First column: no obstacle ($ c = (100, 100) $ , $ a = (0.9, 0.16) $ , $ \omega = 0 $ ), second column: a stationary obstacle ($ c = (1.1, 0) $ , $ a = (0.9, 0.16) $ , $ \omega = 0 $ ), third column: a moving obstacle ($ c = (1.1, 0) $ , $ a = (0.9, 0.1) $ , $ \omega = 1 $ ). The initial measure $ \vartheta $ is absolutely continuous with density $\frac{1}{32} {\mathit{\boldsymbol{1}}}_{[2, 6]\times[-4, 4]} $ . Parameters of the model: $ b = 0.6 $ , $ \delta = 0.1 $ , parameters of discretization: $ \tau = 0.01 $ , $ N = 300 $
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