September  2021, 11(3): 681-713. doi: 10.3934/mcrf.2021018

Uncertainty damping in kinetic traffic models by driver-assist controls

1. 

Politecnico di Torino, Department of Mathematical Sciences "G. L. Lagrange", C.so Duca degli Abruzzi 24, 10129 Torino, Italy

2. 

University of Pavia, Department of Mathematics "F. Casorati", Via A. Ferrata 5, 27100 Pavia, Italy

* Corresponding author: Mattia Zanella

Received  March 2019 Revised  February 2020 Published  September 2021 Early access  March 2021

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

Citation: Andrea Tosin, Mattia Zanella. Uncertainty damping in kinetic traffic models by driver-assist controls. Mathematical Control & Related Fields, 2021, 11 (3) : 681-713. doi: 10.3934/mcrf.2021018
References:
[1]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Commun. Math. Sci., 6 (2015), 1407-1429.  doi: 10.4310/CMS.2015.v13.n6.a3.  Google Scholar

[2]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. End. Sci., 372 (2014), 20140138, 18 pp. doi: 10.1098/rsta.2014.0138.  Google Scholar

[3]

E. AriaJ. Olstam and C. Schwietering, Investigation of automated vehicle effects on driver's behavior and traffic performance, Transp. Res. Procedia, 15 (2016), 761-770.  doi: 10.1016/j.trpro.2016.06.063.  Google Scholar

[4]

S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612.  doi: 10.1017/S0956792503005266.  Google Scholar

[5]

S. BoscarinoF. Filbet and G. Russo, High order semi-implicit schemes for time dependent partial differential equations, J. Sci. Comput., 68 (2016), 975-1001.  doi: 10.1007/s10915-016-0168-y.  Google Scholar

[6]

J. A. Carrillo and M. Zanella, Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties, Vitenam J. Math., 47 (2019), 931-954.  doi: 10.1007/s10013-019-00374-2.  Google Scholar

[7]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.  Google Scholar

[8]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, 106 of Applied Mathematical Sciences, Springer, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[9]

R. M. Colombo, C. Klingenberg and M.-C. Meltzer, A multispecies traffic model based on the Lighthill-Whitham and Richards model, In C. Klingenberg and M. Westdickenberg, editors, Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016, 236 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2018,375-394.  Google Scholar

[10]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[11]

A. I. DelisI. K. Nikolos and M. Papageorgiou, Macroscopic traffic flow modeling with adaptive cruise control: Development and numerical solution, Comput. Math. Appl., 70 (2015), 1921-1947.  doi: 10.1016/j.camwa.2015.08.002.  Google Scholar

[12]

A. I. Delis, I. K. Nikolos and M. Papageorgiou, A macroscopic multi-lane traffic flow model for ACC/CACC traffic dynamics, Transp. Res. Record, 2018. doi: 10.1177/0361198118786823.  Google Scholar

[13]

G. Dimarco, L. Pareschi and M. Zanella, Uncertainty quantification for kinetic models in socio-economic and life sciences, In S. Jin and L. Pareschi, editors, Uncertainty quantification for Hyperbolic and Kinetic Equations, 14 of SEMA-SIMAI Springer Series, Springer, Cham, 2017,151-191.  Google Scholar

[14]

P. Freguglia and A. Tosin, Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach, Commun. Math. Sci., 15 (2017), 213-236.  doi: 10.4310/CMS.2017.v15.n1.a10.  Google Scholar

[15]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[16]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Mod., 3 (2010), 165-179.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[17]

M. HertyA. TosinG. Visconti and M. Zanella, Hybrid stochastic kinetic description of two-dimensional traffic dynamics, SIAM J. Appl. Math., 78 (2018), 2737-2762.  doi: 10.1137/17M1155909.  Google Scholar

[18]

M. Herty, A. Tosin, G. Visconti and M. Zanella, Reconstruction of traffic speed distributions from kinetic models with uncertainties, In G. Puppo, A. Tosin editors, Mathematical Descriptions of Traffic Flow: Micro, Macro and Kinetic Models, SEMA-SIMAI Springer Series, to appear. Google Scholar

[19]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, In S. Jin and L. Pareschi, editors, Uncertainty Quantification for Hyperbolic and Kinetic Equations, 14 of SEMA-SIMAI Springer Series, Springer, Cham, 2017,193-229. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[20]

A. H. JamsonN. MeratO. M. J. Carsten and F. C. H. Lai, Behavioural changes in drivers experiencing highly-automated vehicle control in varying traffic conditions, Transport. Res. C, 30 (2013), 116-125.  doi: 10.1016/j.trc.2013.02.008.  Google Scholar

[21]

S. Jin and L. Pareschi, editors, Uncertainty quantification for hyperbolic and kinetic equations, 14 of SEMA-SIMAI Springer Series., Springer, 2017. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[22]

B. S. Kerner, The Physics of Traffic, Understanding Complex Systems, Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1.  Google Scholar

[23]

A. Klar and R. Wegener, Enskog-like models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114.  doi: 10.1007/BF02181481.  Google Scholar

[24]

Y. Marzouk and D. Xiu, A stochastic collocation approach to Bayesian inference in inverse problems, Commun. Comput. Phys., 6 (2009), 826-847.  doi: 10.4208/cicp.2009.v6.p826.  Google Scholar

[25]

A. D. Mason and A. W. Woods, Car-following model of multispecies systems of road traffic, Phys. Rev. E, 55 (1997), 2203-2214.  doi: 10.1103/PhysRevE.55.2203.  Google Scholar

[26]

A. K. MauryaS. DasS. Dey and S. Nama, Study on speed and time-headway distributions on two-lane bidirectional road in heterogeneous traffic condition, Transp. Res. Proc., 17 (2016), 428-437.  doi: 10.1016/j.trpro.2016.11.084.  Google Scholar

[27]

T. Nagatani, Traffic behavior in a mixture of different vehicles, Phys. A, 284 (2000), 405-420.  doi: 10.1016/S0378-4371(00)00263-6.  Google Scholar

[28]

D. NiH. K. Hsieh and T. Jiang, Modeling phase diagrams as stochastic processes with application in vehicular traffic flow, Appl. Math. Model., 53 (2018), 106-117.  doi: 10.1016/j.apm.2017.08.029.  Google Scholar

[29]

I. A. NtousakisI. K. Nikolos and M. Papageorgiou, On microscopic modelling of adaptive cruise control systems, Transp. Res. Proc., 6 (2015), 111-127.  doi: 10.1016/j.trpro.2015.03.010.  Google Scholar

[30]

L. Pareschi and T. Rey, Residual equilibrium schemes for time dependent partial differential equations, Comput. Fluids, 156 (2017), 329-342.  doi: 10.1016/j.compfluid.2017.07.013.  Google Scholar

[31]

L. Pareschi and G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM: Proc., 10 (2001), 35-75.  doi: 10.1051/proc:2001004.  Google Scholar

[32] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013.   Google Scholar
[33]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600.  doi: 10.1007/s10915-017-0510-z.  Google Scholar

[34]

S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[35]

B. Piccoli, A. Tosin and M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory, Z. Angew. Math. Phys., 71 (2020), No. 152, 25 pp. doi: 10.1007/s00033-020-01383-9.  Google Scholar

[36]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971. Google Scholar

[37]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), 643-669.  doi: 10.4310/CMS.2016.v14.n3.a3.  Google Scholar

[38]

G. PuppoM. SempliceA. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412.  doi: 10.4310/CMS.2017.v15.n2.a5.  Google Scholar

[39]

B. SeiboldM. R. FlynnA. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772.  doi: 10.3934/nhm.2013.8.745.  Google Scholar

[40]

R. E. SternS. CuiM. L. Delle MonacheR. BhadaniM. BuntingM. ChurchillN. HamiltonR. HaulcyH. PohlmannF. WuB. PiccoliB. SeiboldJ. Sprinkle and D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Res. Part C, 89 (2018), 205-221.  doi: 10.1016/j.trc.2018.02.005.  Google Scholar

[41]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.  Google Scholar

[42]

A. Tosin and M. Zanella, Boltzmann-type models with uncertain binary interactions, Commun. Math. Sci., 16 (2018), 963-985.  doi: 10.4310/CMS.2018.v16.n4.a3.  Google Scholar

[43]

A. Tosin and M. Zanella, Control strategies for road risk mitigation in kinetic traffic modelling, IFAC-PapersOnLine, 51 (2018), 67-72.  doi: 10.1016/j.ifacol.2018.07.012.  Google Scholar

[44]

A. Tosin and M. Zanella, Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles, Multiscale Model. Simul., 17 (2019), 716-749.  doi: 10.1137/18M1203766.  Google Scholar

[45]

C. Villani, Contribution à l'étude Mathématique Des Équations de Boltzmann et de Landau en Théorie Cinétique Des Gaz et Des Plasmas, PhD thesis, Paris 9, 1998. Google Scholar

[46]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.  Google Scholar

[47] D. Xiu, Numerical Methods for Stochastic Computations, Princeton University Press, 2010.   Google Scholar
[48]

D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 614-644.  doi: 10.1137/S1064827501387826.  Google Scholar

[49]

M. Zanella, Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions, Math. Comput. Simulation, 168 (2020), 28-47.  doi: 10.1016/j.matcom.2019.07.012.  Google Scholar

[50]

Y. Zhu and S. Jin, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic-preserving method, Multiscale Model. Simul., 15 (2017), 1502-1529.  doi: 10.1137/16M1090028.  Google Scholar

show all references

References:
[1]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Commun. Math. Sci., 6 (2015), 1407-1429.  doi: 10.4310/CMS.2015.v13.n6.a3.  Google Scholar

[2]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. End. Sci., 372 (2014), 20140138, 18 pp. doi: 10.1098/rsta.2014.0138.  Google Scholar

[3]

E. AriaJ. Olstam and C. Schwietering, Investigation of automated vehicle effects on driver's behavior and traffic performance, Transp. Res. Procedia, 15 (2016), 761-770.  doi: 10.1016/j.trpro.2016.06.063.  Google Scholar

[4]

S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612.  doi: 10.1017/S0956792503005266.  Google Scholar

[5]

S. BoscarinoF. Filbet and G. Russo, High order semi-implicit schemes for time dependent partial differential equations, J. Sci. Comput., 68 (2016), 975-1001.  doi: 10.1007/s10915-016-0168-y.  Google Scholar

[6]

J. A. Carrillo and M. Zanella, Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties, Vitenam J. Math., 47 (2019), 931-954.  doi: 10.1007/s10013-019-00374-2.  Google Scholar

[7]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.  Google Scholar

[8]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, 106 of Applied Mathematical Sciences, Springer, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[9]

R. M. Colombo, C. Klingenberg and M.-C. Meltzer, A multispecies traffic model based on the Lighthill-Whitham and Richards model, In C. Klingenberg and M. Westdickenberg, editors, Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016, 236 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2018,375-394.  Google Scholar

[10]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[11]

A. I. DelisI. K. Nikolos and M. Papageorgiou, Macroscopic traffic flow modeling with adaptive cruise control: Development and numerical solution, Comput. Math. Appl., 70 (2015), 1921-1947.  doi: 10.1016/j.camwa.2015.08.002.  Google Scholar

[12]

A. I. Delis, I. K. Nikolos and M. Papageorgiou, A macroscopic multi-lane traffic flow model for ACC/CACC traffic dynamics, Transp. Res. Record, 2018. doi: 10.1177/0361198118786823.  Google Scholar

[13]

G. Dimarco, L. Pareschi and M. Zanella, Uncertainty quantification for kinetic models in socio-economic and life sciences, In S. Jin and L. Pareschi, editors, Uncertainty quantification for Hyperbolic and Kinetic Equations, 14 of SEMA-SIMAI Springer Series, Springer, Cham, 2017,151-191.  Google Scholar

[14]

P. Freguglia and A. Tosin, Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach, Commun. Math. Sci., 15 (2017), 213-236.  doi: 10.4310/CMS.2017.v15.n1.a10.  Google Scholar

[15]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[16]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Mod., 3 (2010), 165-179.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[17]

M. HertyA. TosinG. Visconti and M. Zanella, Hybrid stochastic kinetic description of two-dimensional traffic dynamics, SIAM J. Appl. Math., 78 (2018), 2737-2762.  doi: 10.1137/17M1155909.  Google Scholar

[18]

M. Herty, A. Tosin, G. Visconti and M. Zanella, Reconstruction of traffic speed distributions from kinetic models with uncertainties, In G. Puppo, A. Tosin editors, Mathematical Descriptions of Traffic Flow: Micro, Macro and Kinetic Models, SEMA-SIMAI Springer Series, to appear. Google Scholar

[19]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, In S. Jin and L. Pareschi, editors, Uncertainty Quantification for Hyperbolic and Kinetic Equations, 14 of SEMA-SIMAI Springer Series, Springer, Cham, 2017,193-229. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[20]

A. H. JamsonN. MeratO. M. J. Carsten and F. C. H. Lai, Behavioural changes in drivers experiencing highly-automated vehicle control in varying traffic conditions, Transport. Res. C, 30 (2013), 116-125.  doi: 10.1016/j.trc.2013.02.008.  Google Scholar

[21]

S. Jin and L. Pareschi, editors, Uncertainty quantification for hyperbolic and kinetic equations, 14 of SEMA-SIMAI Springer Series., Springer, 2017. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[22]

B. S. Kerner, The Physics of Traffic, Understanding Complex Systems, Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1.  Google Scholar

[23]

A. Klar and R. Wegener, Enskog-like models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114.  doi: 10.1007/BF02181481.  Google Scholar

[24]

Y. Marzouk and D. Xiu, A stochastic collocation approach to Bayesian inference in inverse problems, Commun. Comput. Phys., 6 (2009), 826-847.  doi: 10.4208/cicp.2009.v6.p826.  Google Scholar

[25]

A. D. Mason and A. W. Woods, Car-following model of multispecies systems of road traffic, Phys. Rev. E, 55 (1997), 2203-2214.  doi: 10.1103/PhysRevE.55.2203.  Google Scholar

[26]

A. K. MauryaS. DasS. Dey and S. Nama, Study on speed and time-headway distributions on two-lane bidirectional road in heterogeneous traffic condition, Transp. Res. Proc., 17 (2016), 428-437.  doi: 10.1016/j.trpro.2016.11.084.  Google Scholar

[27]

T. Nagatani, Traffic behavior in a mixture of different vehicles, Phys. A, 284 (2000), 405-420.  doi: 10.1016/S0378-4371(00)00263-6.  Google Scholar

[28]

D. NiH. K. Hsieh and T. Jiang, Modeling phase diagrams as stochastic processes with application in vehicular traffic flow, Appl. Math. Model., 53 (2018), 106-117.  doi: 10.1016/j.apm.2017.08.029.  Google Scholar

[29]

I. A. NtousakisI. K. Nikolos and M. Papageorgiou, On microscopic modelling of adaptive cruise control systems, Transp. Res. Proc., 6 (2015), 111-127.  doi: 10.1016/j.trpro.2015.03.010.  Google Scholar

[30]

L. Pareschi and T. Rey, Residual equilibrium schemes for time dependent partial differential equations, Comput. Fluids, 156 (2017), 329-342.  doi: 10.1016/j.compfluid.2017.07.013.  Google Scholar

[31]

L. Pareschi and G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM: Proc., 10 (2001), 35-75.  doi: 10.1051/proc:2001004.  Google Scholar

[32] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013.   Google Scholar
[33]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600.  doi: 10.1007/s10915-017-0510-z.  Google Scholar

[34]

S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[35]

B. Piccoli, A. Tosin and M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory, Z. Angew. Math. Phys., 71 (2020), No. 152, 25 pp. doi: 10.1007/s00033-020-01383-9.  Google Scholar

[36]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971. Google Scholar

[37]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), 643-669.  doi: 10.4310/CMS.2016.v14.n3.a3.  Google Scholar

[38]

G. PuppoM. SempliceA. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412.  doi: 10.4310/CMS.2017.v15.n2.a5.  Google Scholar

[39]

B. SeiboldM. R. FlynnA. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772.  doi: 10.3934/nhm.2013.8.745.  Google Scholar

[40]

R. E. SternS. CuiM. L. Delle MonacheR. BhadaniM. BuntingM. ChurchillN. HamiltonR. HaulcyH. PohlmannF. WuB. PiccoliB. SeiboldJ. Sprinkle and D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Res. Part C, 89 (2018), 205-221.  doi: 10.1016/j.trc.2018.02.005.  Google Scholar

[41]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.  Google Scholar

[42]

A. Tosin and M. Zanella, Boltzmann-type models with uncertain binary interactions, Commun. Math. Sci., 16 (2018), 963-985.  doi: 10.4310/CMS.2018.v16.n4.a3.  Google Scholar

[43]

A. Tosin and M. Zanella, Control strategies for road risk mitigation in kinetic traffic modelling, IFAC-PapersOnLine, 51 (2018), 67-72.  doi: 10.1016/j.ifacol.2018.07.012.  Google Scholar

[44]

A. Tosin and M. Zanella, Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles, Multiscale Model. Simul., 17 (2019), 716-749.  doi: 10.1137/18M1203766.  Google Scholar

[45]

C. Villani, Contribution à l'étude Mathématique Des Équations de Boltzmann et de Landau en Théorie Cinétique Des Gaz et Des Plasmas, PhD thesis, Paris 9, 1998. Google Scholar

[46]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.  Google Scholar

[47] D. Xiu, Numerical Methods for Stochastic Computations, Princeton University Press, 2010.   Google Scholar
[48]

D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 614-644.  doi: 10.1137/S1064827501387826.  Google Scholar

[49]

M. Zanella, Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions, Math. Comput. Simulation, 168 (2020), 28-47.  doi: 10.1016/j.matcom.2019.07.012.  Google Scholar

[50]

Y. Zhu and S. Jin, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic-preserving method, Multiscale Model. Simul., 15 (2017), 1502-1529.  doi: 10.1137/16M1090028.  Google Scholar

Figure 1.  The probability of accelerating $ P(\rho;\, z) $ given in (3) plotted for various $ z $
Figure 2.  The case $ z\in\{1, \, 3\} $ with $ \operatorname{Prob}(z = 1) = \alpha_1 $, $ \operatorname{Prob}(z = 3) = \alpha_2 $: fundamental diagram $ \rho\mapsto\rho\bar{V}_\infty(\rho) $ (solid line) and uncertainty lines $ \rho\mapsto\rho\bar{V}_\infty(\rho)\pm\rho\varsigma_\infty(\rho) $ (dash-dotted and dashed lines, respectively). The filled area is (12). The function $ P(\rho;\, z) $ is taken like in (3)
Figure 3.  The case $ z\sim \mathcal{U}([1, \, 3]) $: fundamental diagram $ \rho\mapsto\rho\bar{V}_\infty(\rho) $ (solid line) and uncertainty lines $ \rho\mapsto\rho\bar{V}_\infty(\rho)\pm\rho\varsigma_\infty(\rho) $ (dash-dotted and dashed lines, respectively). The filled area is (12). The function $ P(\rho;\, z) $ is taken like in (3)
Figure 4.  Representation of $ \bar{f}_\infty(v) $ (solid lines) and of its $ z $-standard deviation $ \pm\sqrt{ \operatorname{Var}_z(f_\infty(v;\, z))} $ (dashed lines) for $ z $ such that $ z-1 $ has a binomial distribution. In (a), $ \rho = 0.2 $. In (b), $ \rho = 0.4 $
Figure 5.  Representation of $ \bar{f}_\infty(v) $ (solid lines) and of its $ z $-standard deviation $ \pm\sqrt{ \operatorname{Var}_z(f_\infty(v;\, z))} $ (dashed lines) for different distributions of the uncertain parameter $ z $ and various traffic densities. In (a), $ z $ is uniformly distributed in $ [1, \, 3] $. In (b), $ z $ is such that $ z-2 $ has a gamma distribution, thus it is, in particular, unbounded
Figure 3, which illustrates the fundamental diagram and its scattering in the uncontrolled case">Figure 6.  Action of the uncertain control (19) on the fundamental diagram and its scattering for two possible distributions of the uncertain parameter $ z $ and two values of the effective penetration rate $ p^\ast $. We considered $ v_d(\rho) = 1-\rho $ as optimal speed. For uniformly distributed $ z $ (bottom row), a comparison is possible with Figure 3, which illustrates the fundamental diagram and its scattering in the uncontrolled case
Figure 7.  Representation of $ \bar{f}_\infty(v) $ (solid lines) and of its $ z $-standard deviation $ \pm\sqrt{ \operatorname{Var}_z(f_\infty(v;\, z))} $ (dashed lines) for different distributions of the uncertain parameter $ z $, different effective penetration rates $ p^\ast $ of the ADAS technology and various traffic densities. In (a)-(b), $ z $ is such that $ z-1 $ has a binomial distribution. In (c)-(d), $ z $ is uniformly distributed in $ [1, \, 3] $. We considered the optimal speed $ v_d(\rho) = 1-\rho $ and $ \lambda = 5\cdot 10^{-2} $ in (29)
Figure 8.  Comparison between the large time $ z $-averaged solution to the Boltzmann-type equation (32) with non-constant collision kernel (bulleted lines) and the average equilibrium solution to the Fokker-Planck equation (28) (solid line) for decreasing $ \epsilon $, mimicking the quasi-invariant limit $ \epsilon\to 0^+ $. The relevant parameters are $ \rho = 0.2, \, 0.4, \, 0.6 $, $ p^\ast = 1 $, $ z\sim \mathcal{U}([1, \, 3]) $
Figure 9.  Uncontrolled case. Convergence of the $ L^2 $-numerical error with respect to the exact solution (17) of the Fokker-Planck equation (15) for: (a) $ z\sim \mathcal{U}([1, \, 3]) $ (circular markers); (b) $ z $ such that $ z-1\sim \operatorname{B}\!\left(50, \, \frac{1}{50}\right) $ (triangular markers) and $ z $ with beta distribution in $ I_Z = [1, \, 3] $ with zero mean and variance equal to $ \frac{1}{3} $ (circular markers)
Figure 10.  Uncontrolled case, $ \boldsymbol{z\sim \mathcal{U}([1, \, 3])} $. Contours of $ \bar{f}(t, \, v) = \mathbb{E}_z(f(t, \, v;\, z)) $ (top row) and of $ \operatorname{Var}_z(f(t, \, v;\, z)) $ (bottom row), where $ f $ is the solution to (15) with $ \lambda = 5\cdot 10^{-2} $ issuing from the initial datum (44), for $ t\in [0, \, 20] $ and $ \rho = 0.2, \, 0.4, \, 0.6 $ in the case of uniformly distributed $ z $
Figure 10">Figure 11.  Uncontrolled case, $ \boldsymbol{z-1\sim \operatorname{B}\!\left(50, \, \frac{1}{50}\right)} $. Contours of $ \bar{f}(t, \, v)= \mathbb{E}_z(f(t, \, v;\, z)) $ (top row) and of $ \operatorname{Var}_z(f(t, \, v;\, z)) $ (bottom row), where $ f $ is the solution tok__ge (15), when $ z $ is such that $ z-1 $ has binomial distribution. All the parameters are like in Figure 10
Figure 12.  Controlled case, $ \boldsymbol{z\sim \mathcal{U}([1, \, 3])} $. Contours of $ \operatorname{Var}_z(f(t, \, v;\, z)) $, where $ f $ is the solution to (28) with $ \lambda = 5\cdot 10^{-2} $ issuing from the initial datum (44), for $ t\in [0, \, 20] $ and $ \rho = 0.2, \, 0.4, \, 0.6 $ in the case of uniformly distributed $ z $. Top row: $ p^\ast = 1 $; bottom row: $ p^\ast = 10 $
Figure 12">Figure 13.  Controlled case, $ \boldsymbol{z-1\sim \operatorname{B}\!\left(50, \, \frac{1}{50}\right)} $. Contours of $ \bar{f}(t, \, v)= \mathbb{E}_z(f(t, \, v;\, z)) $ (top row) and of $ \operatorname{Var}_z(f(t, \, v;\, z)) $ (bottom row), where $ f $ is the solution tok__ge (28), when $ z $ is such that $ z-1 $ has binomial distribution. All the parameters are like in Figure 12
Figure 14.  Controlled case. Asymptotic variance $ \operatorname{Var}_z(f_\infty(v;\, z)) $, where $ f_\infty(v;\, z) $ is like in (29), obtained with a uniform and a binomial distribution of the uncertainty for a decreasing penalisation of the control: from $ \kappa = 10^5 $, corresponding to a virtually uncontrolled setting, to $ \kappa = 10^{-1}, \, 10^{-2} $
Table 1.  Choices of the $ \Phi_k $'s depending on the distribution $ \Psi $ of the uncertain parameter
Distribution $ \Psi $ Polynomials $ \Phi_k $ Support of the $ \Phi_k $'s
Uniform Legendre Compact interval
Beta Jacobi Compact interval
Gamma Laguerre $ \mathbb{R}_+ $
Binomial Krawtchouk $ \mathbb{N} $
Distribution $ \Psi $ Polynomials $ \Phi_k $ Support of the $ \Phi_k $'s
Uniform Legendre Compact interval
Beta Jacobi Compact interval
Gamma Laguerre $ \mathbb{R}_+ $
Binomial Krawtchouk $ \mathbb{N} $
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