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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

* Corresponding author

Received  February 2021 Published  March 2021

Fund Project: This work was partially supported by the Russian Foundation for Basic Research under contracts no. 18-31-20030 (the first author) and no. 18-01-00026 (the second author)

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.

Citation: Nadezhda Maltugueva, Nikolay Pogodaev. Modeling of crowds in regions with moving obstacles. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021066
##### 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.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Basel: Birkhäuser, 2005.  Google Scholar [3] V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] G. Dal Maso, An Introduction to $\Gamma$-convergence, vol. 8, Basel: Birkhäuser, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [11] S. Di Marino, B. Maury and F. Santambrogio, Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601.   Google Scholar [12] H. Federer, Curvature measures, Trans. Am. Math. Soc., 93 (1959), 418-491.  doi: 10.1090/S0002-9947-1959-0110078-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] B. Maury and S. Faure, Crowds in Equations. An Introduction to the Microscopic Modeling of Crowds, Hackensack, NJ: World Scientific, 2019.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] M. Sene and L. Thibault, Regularization of dynamical systems associated with prox-regular moving sets., J. Nonlinear Convex Anal., 15 (2014), 647-663.   Google Scholar [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.  Google Scholar [25] C. Villani, Optimal Transport. Old and New, vol. 338, Berlin: Springer, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

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.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Basel: Birkhäuser, 2005.  Google Scholar [3] V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] G. Dal Maso, An Introduction to $\Gamma$-convergence, vol. 8, Basel: Birkhäuser, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [11] S. Di Marino, B. Maury and F. Santambrogio, Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601.   Google Scholar [12] H. Federer, Curvature measures, Trans. Am. Math. Soc., 93 (1959), 418-491.  doi: 10.1090/S0002-9947-1959-0110078-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] B. Maury and S. Faure, Crowds in Equations. An Introduction to the Microscopic Modeling of Crowds, Hackensack, NJ: World Scientific, 2019.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] M. Sene and L. Thibault, Regularization of dynamical systems associated with prox-regular moving sets., J. Nonlinear Convex Anal., 15 (2014), 647-663.   Google Scholar [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.  Google Scholar [25] C. Villani, Optimal Transport. Old and New, vol. 338, Berlin: Springer, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar
One step of the catching-up algorithm. Here $t = 2k\tau$, the dark rounded rectangle represents an obstacle
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: $\tau = 0.01$, $N = 300$
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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