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Efficient robust control of first order scalar conservation laws using semi-analytical solutions
Free-congested and micro-macro descriptions of traffic flow
1. | Università di Milano-Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano, Italy |
References:
[1] |
A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278 (electronic).
doi: 10.1137/S0036139900380955. |
[2] |
A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic).
doi: 10.1137/S0036139997332099. |
[3] |
P. Bagnerini, R. M. Colombo and A. Corli, On the role of source terms in continuum traffic flow models, Math. Comput. Modelling, 44 (2006), 917-930.
doi: 10.1016/j.mcm.2006.02.019. |
[4] |
P. Bagnerini and M. Rascle, A multiclass homogenized hyperbolic model of traffic flow, SIAM J. Math. Anal., 35 (2003), 949-973 (electronic).
doi: 10.1137/S0036141002411490. |
[5] |
S. Benzoni Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Europ. J. Appl. Math., 14 (2003), 587-612.
doi: 10.1017/S0956792503005266. |
[6] |
S. Benzoni-Gavage, R. M. Colombo and P. Gwiazda, Measure valued solutions to conservation laws motivated by traffic modelling, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 1791-1803.
doi: 10.1098/rspa.2005.1649. |
[7] |
S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.
doi: 10.1137/090754467. |
[8] |
R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.
doi: 10.1137/S0036139901393184. |
[9] |
R. M. Colombo, P. Goatin and F. S. Priuli, Global well posedness of traffic flow models with phase transitions, Nonlinear Anal., 66 (2007), 2413-2426.
doi: 10.1016/j.na.2006.03.029. |
[10] |
R. M. Colombo and F. Marcellini, A mixed ode-pde model for vehicular traffic, {preprint}, (2013). |
[11] |
R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666.
doi: 10.1137/090752468. |
[12] |
R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772.
doi: 10.1017/S0308210500002663. |
[13] |
L. C. Edie, Car-following and steady-state theory for noncongested traffic, Operations Res., 9 (1961), 66-76.
doi: 10.1287/opre.9.1.66. |
[14] |
P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.
doi: 10.1016/j.mcm.2006.01.016. |
[15] |
M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, J. Hyperbolic Differ. Equ., 5 (2008), 45-63.
doi: 10.1142/S0219891608001428. |
[16] |
D. Helbing and M. Treiber, Critical discussion of synchronized flow, Cooper@tive Tr@nsport@tion Dyn@mics, 1 (2002). |
[17] |
B. S. Kerner, Phase transitions in traffic flow, in Traffic and Granular Flow '99, (eds., D. Helbing, H. Hermann, M. Schreckenberg and D. Wolf), Springer Verlag, 2000, 253-283.
doi: 10.1007/978-3-642-59751-0_25. |
[18] |
B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal., 72 (1979/80), 219-241.
doi: 10.1007/BF00281590. |
[19] |
K. M. Kockelman, Modeling traffics flow-density relation: Accommodation of multiple flow regimes and traveler types, Transportation, 28 (2001), 363-374. |
[20] |
C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370.
doi: 10.1142/S0218202510004945. |
[21] |
J. P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling, in Transportation and Traffic Theory: Proceedings of the 17th International Symposium on Transportation and Traffic Theory, 2007. |
[22] |
R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[23] |
M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[24] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[25] |
B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.
doi: 10.1090/S0002-9947-1983-0716850-2. |
[26] |
H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
show all references
References:
[1] |
A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278 (electronic).
doi: 10.1137/S0036139900380955. |
[2] |
A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic).
doi: 10.1137/S0036139997332099. |
[3] |
P. Bagnerini, R. M. Colombo and A. Corli, On the role of source terms in continuum traffic flow models, Math. Comput. Modelling, 44 (2006), 917-930.
doi: 10.1016/j.mcm.2006.02.019. |
[4] |
P. Bagnerini and M. Rascle, A multiclass homogenized hyperbolic model of traffic flow, SIAM J. Math. Anal., 35 (2003), 949-973 (electronic).
doi: 10.1137/S0036141002411490. |
[5] |
S. Benzoni Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Europ. J. Appl. Math., 14 (2003), 587-612.
doi: 10.1017/S0956792503005266. |
[6] |
S. Benzoni-Gavage, R. M. Colombo and P. Gwiazda, Measure valued solutions to conservation laws motivated by traffic modelling, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 1791-1803.
doi: 10.1098/rspa.2005.1649. |
[7] |
S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.
doi: 10.1137/090754467. |
[8] |
R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.
doi: 10.1137/S0036139901393184. |
[9] |
R. M. Colombo, P. Goatin and F. S. Priuli, Global well posedness of traffic flow models with phase transitions, Nonlinear Anal., 66 (2007), 2413-2426.
doi: 10.1016/j.na.2006.03.029. |
[10] |
R. M. Colombo and F. Marcellini, A mixed ode-pde model for vehicular traffic, {preprint}, (2013). |
[11] |
R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666.
doi: 10.1137/090752468. |
[12] |
R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772.
doi: 10.1017/S0308210500002663. |
[13] |
L. C. Edie, Car-following and steady-state theory for noncongested traffic, Operations Res., 9 (1961), 66-76.
doi: 10.1287/opre.9.1.66. |
[14] |
P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.
doi: 10.1016/j.mcm.2006.01.016. |
[15] |
M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, J. Hyperbolic Differ. Equ., 5 (2008), 45-63.
doi: 10.1142/S0219891608001428. |
[16] |
D. Helbing and M. Treiber, Critical discussion of synchronized flow, Cooper@tive Tr@nsport@tion Dyn@mics, 1 (2002). |
[17] |
B. S. Kerner, Phase transitions in traffic flow, in Traffic and Granular Flow '99, (eds., D. Helbing, H. Hermann, M. Schreckenberg and D. Wolf), Springer Verlag, 2000, 253-283.
doi: 10.1007/978-3-642-59751-0_25. |
[18] |
B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal., 72 (1979/80), 219-241.
doi: 10.1007/BF00281590. |
[19] |
K. M. Kockelman, Modeling traffics flow-density relation: Accommodation of multiple flow regimes and traveler types, Transportation, 28 (2001), 363-374. |
[20] |
C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370.
doi: 10.1142/S0218202510004945. |
[21] |
J. P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling, in Transportation and Traffic Theory: Proceedings of the 17th International Symposium on Transportation and Traffic Theory, 2007. |
[22] |
R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[23] |
M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[24] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[25] |
B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.
doi: 10.1090/S0002-9947-1983-0716850-2. |
[26] |
H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
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