On the existence and uniqueness properties of the Hoogendoorn–Bovy pedestrian flow model

References

• Aghamohammadi, R., and J. A. Laval. 2020. “Dynamic Traffic Assignment Using the Macroscopic Fundamental Diagram: A Review of Vehicular and Pedestrian Flow Models.” Transportation Research Part B: Methodological 137: 99–118.
   Web of Science ®Google Scholar
• Asano, M., T. Iryo, and M. Kuwahara. 2010. “Microscopic Pedestrian Simulation Model Combined with a Tactical Model for Route Choice Behaviour.” Transportation Research Part C: Emerging Technologies18 (6): 842–855.
   Web of Science ®Google Scholar
• Bouchut, F., and F. James. 1998. “One-Dimensional Transport Equations with Discontinuous Coefficients.” Nonlinear Analysis: Theory, Methods & Applications 32 (7): 891–933.
   Web of Science ®Google Scholar
• Bressan, A., and K. Han. 2013. “Existence of Optima and Equilibria for Traffic Flow on Networks.” Networks and Heterogeneous Media 8 (3): 627–648.
   Web of Science ®Google Scholar
• Cannarsa, P., and C. Sinestrari. 2004. 'Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Vol. 58. Boston, MA: Springer Science & Business Media.
   Google Scholar
• Cardaliaguet, P. 2010. “Notes on Mean Field Games (p. 120).” Technical Report.
   Google Scholar
• Conway, E. D.. 1967. “Generalized Solutions of Linear Differential Equations with Discontinuous Coefficients and the Uniqueness Question for Multidimensional Quasilinear Conservation Laws.” Journal of Mathematical Analysis and Applications 18 (2): 238–251.
   Web of Science ®Google Scholar
• Crandall, M. G., L. C. Evans, and P. L. Lions. 1984. “Some Properties of Viscosity Solutions of Hamilton–Jacobi Equations.” Transactions of the American Mathematical Society 282 (2): 487–502.
   Web of Science ®Google Scholar
• Crandall, M. G., and P. L. Lions. 1983. “Viscosity Solutions of Hamilton–Jacobi Equations.” Transactions of the American Mathematical Society 277 (1): 1–42.
   Web of Science ®Google Scholar
• Crociani, L., and G. Laemmel. 2016. “Multidestination Pedestrian Flows in Equilibrium: A Cellular Automaton-Based Approach.” Computer-Aided Civil and Infrastructure Engineering 31 (6): 432–448.
   Web of Science ®Google Scholar
• Dolcetta, I. C., and B. Perthame. 1996. “On Some Analogy Between Different Approaches to First Order PDE's with Non Smooth Coefficients.” Advances in Mathematical Sciences and Applications 6: 689–703.
   Google Scholar
• Du, J., S. C. Wong, C. W. Shu, T. Xiong, M. P. Zhang, and K Choi. 2013. “Revisiting Jiang's Dynamic Continuum Model for Urban Cities.” Transportation Research Part B: Methodological 56: 96–119.
   Web of Science ®Google Scholar
• Evans, L. C.. 1998. Partial Differential Equations. Providence, RI: American Mathematical Society.
   Google Scholar
• Fathi, A.. 2011. “On Existence of Smooth Critical Subsolutions of the Hamilton–Jacobi Equation.” Publicaciones Matemáticas del Uruguay 12: 87–98.
   Google Scholar
• Fleming, W. H.. 1969. “The Cauchy Problem for a Nonlinear First Order Partial Differential Equation.” Journal of Differential Equations 5 (3): 515–530.
   Web of Science ®Google Scholar
• Fosgerau, M., E. Frejinger, and A. Karlstrom. 2013. “A Link Based Network Route Choice Model with Unrestricted Choice Set.” Transportation Research Part B: Methodological 56 (1): 70–80.
   Google Scholar
• Frejinger, E., M. Bierlaire, and M. Ben-Akiva. 2009. “Sampling of Alternatives for Route Choice Modeling.” Transportation Research Part B: Methodological 43 (10): 984–994.
   Web of Science ®Google Scholar
• Friedman, A., and E. Hopf. 1973. “The Cauchy Problem for First Order Partial Differential Equations.” Indiana University Mathematics Journal 23 (1): 27–40.
   Web of Science ®Google Scholar
• Gao, Z., Y. Qu, X. Li, J. Long, and H. J. Huang. 2014. “Simulating the Dynamic Escape Process in Large Public Places.” Operations Research 62 (6): 1344–1357.
   Web of Science ®Google Scholar
• Han, K., T. L. Friesz, and T. Yao. 2013. “Existence of Simultaneous Route and Departure Choice Dynamic User Equilibrium.” Transportation Research Part B: Methodological 53: 17–30.
   Web of Science ®Google Scholar
• Hoogendoorn, S. P., and P. H. Bovy. 2004a. “Pedestrian Route-Choice and Activity Scheduling Theory and Models.” Transportation Research Part B: Methodological 38 (2): 169–190.
   Web of Science ®Google Scholar
• Hoogendoorn, S. P., and P. H. Bovy. 2004b. “Dynamic User-Optimal Assignment in Continuous Time and Space.” Transportation Research Part B: Methodological 38 (7): 571–592.
   Web of Science ®Google Scholar
• Huang, L., S. C. Wong, M. Zhang, C. W. Shu, and W. H. Lam. 2009. “Revisiting Hughes' Dynamic Continuum Model for Pedestrian Flow and the Development of An Efficient Solution Algorithm.” Transportation Research Part B: Methodological 43 (1): 127–141.
   Web of Science ®Google Scholar
• Hughes, R. L.. 2002. “A Continuum Theory for the Flow of Pedestrians.” Transportation Research Part B: Methodological 36 (6): 507–535.
   Web of Science ®Google Scholar
• Lam, W. H., C. Y. Cheung, and C. F. Lam. 1999. “A Study of Crowding Effects At the Hong Kong Light Rail Transit Stations.” Transportation Research Part A: Policy and Practice 33 (5): 401–415.
   Web of Science ®Google Scholar
• Lam, H. K. W., J. F. Morrall, and H. Ho. 1995. “Pedestrian Flow Characteristics in Hong Kong.” Transportation Research Record 1487: 56–62.
   Google Scholar
• Lasry, J. M., and P. L. Lions. 2007. “Mean Field Games.” Japanese Journal of Mathematics 2 (1): 229–260.
   Web of Science ®Google Scholar
• Lax, P. D.. 1954. “Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation.” Communications on Pure and Applied Mathematics 7 (1): 159–193.
   Web of Science ®Google Scholar
• Lax, P. D.. 1971. “Shock Waves and Entropy.” In Contributions to Nonlinear Functional Analysis, 603–634. Academic Press.
   Google Scholar
• Lin, C. T., and E. Tadmor. 2001. “L1-Stability and Error Estimates for Approximate Hamilton–Jacobi Solutions.” Numerische Mathematik 87 (4): 701–735.
   Web of Science ®Google Scholar
• Lin, Z., P. Zhang, and H. Hang. 2022. “A Dynamic Continuum Route Choice Model for Pedestrian Flow with Mixed Crowds.” Transportmetrica A: Transport Science 1–26. doi:10.1080/23249935.2022.2075951.
   Web of Science ®Google Scholar
• Lions, P. L.. 1982. Generalized Solutions of Hamilton–Jacobi Equations. Vol. 69. London: Pitman.
   Google Scholar
• Mai, T., M. Fosgerau, and E. Frejinger. 2015. “A Nested Recursive Logit Model for Route Choice Analysis.” Transportation Research Part B: Methodological 75 (5): 100–112.
   Google Scholar
• Morrall, J. F., L. L. Ratnayake, and P. N. Seneviratne. 1991. “Comparison of Central Business District Pedestrian Characteristics in Canada and Sri Lanka.” Transportation Research Record 1294: 57–61.
   Google Scholar
• Petrova, G., and B. Popov. 1999. “Linear Transport Equations with Discontinuous Coefficients.” Communications in Partial Differential Equations 24 (9–10): 1849–1873.
   Web of Science ®Google Scholar
• Plaue, M., M. Chen, G. B'arwolff, and H. Schwandt. 2011. “Trajectory Extraction and Density Analysis of Intersecting Pedestrian Flows from Video Recordings.” In ISPRS Conference on Photogrammetric Image Analysis, 285–296. Berlin, Heidelberg: Springer.
   Google Scholar
• Plaue, M., M. Chen, G. B'arwolff, and H. Schwandt. 2012. “Multi-View Extraction of Dynamic Pedestrian Density Fields.” Photogrammetrie-Fernerkundung-Geoinformation 2012 (5): 547–555.
   Google Scholar
• Sánchez-Morgado, H., P. Padilla, R. Iturriaga, and N. Anantharaman. 2012. “Physical Solutions of the Hamilton–Jacobi Equation.” Discrete and Continuous Dynamical Systems -- Series B 5 (3): 513–528.
   Google Scholar
• Tadmor, E.. 1991. “Local Error Estimates for Discontinuous Solutions of Nonlinear Hyperbolic Equations.” SIAM Journal on Numerical Analysis 28 (4): 891–906.
   Web of Science ®Google Scholar
• Taherifar, N., H. Hamedmoghadam, S. Sree, and M. Saberi. 2019. “A Macroscopic Approach for Calibration and Validation of a Modified Social Force Model for Bidirectional Pedestrian Streams.” Transportmetrica A: Transport Science 15 (2): 1637–1661.
   Web of Science ®Google Scholar
• Wong, S. C., W. L. Leung, S. H. Chan, W. H. Lam, N. H. Yung, C. Y. Liu, and P. Zhang. 2010. “Bidirectional Pedestrian Stream Model with Oblique Intersecting Angle.” Journal of Transportation Engineering 136 (3): 234–242.
   Google Scholar
• Xie, S., and S. C. Wong. 2015. “A Bayesian Inference Approach to the Development of a Multidirectional Pedestrian Stream Model.” Transportmetrica A: Transport Science 11 (1): 61–73.
   Web of Science ®Google Scholar
• Xie, S., S. C. Wong, W. H. Lam, and A. Chen. 2013. “Development of a Bidirectional Pedestrian Stream Model with an Oblique Intersecting Angle.” Journal of Transportation Engineering 139 (7): 678–685.
   Google Scholar
• Yang, L., T. Li, S. C. Wong, C. W. Shu, and M. Zhang. 2019. “Modeling and Simulation of Urban Air Pollution from the Dispersion of Vehicle Exhaust: A Continuum Modeling Approach.” International Journal of Sustainable Transportation 13 (10): 722–740.
   Web of Science ®Google Scholar
• Yang, L., S. C. Wong, H. W. Ho, M. Zhang, and C. W. Shu. 2022. “Effects of Air Quality on Housing Location: A Predictive Dynamic Continuum User-Optimal Approach.” Transportation Science 56 (5): 1111–1134.
   Google Scholar