American Institute of Mathematical Sciences

Advanced
December  2017, 12(4): 663-681. doi: 10.3934/nhm.2017027

Capacity drop and traffic control for a second order traffic model

Oliver Kolb 1, , Simone Göttlich 1, and  Paola Goatin 2,

1. 

Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany
2. 

Inria Sophia Antipolis -Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 06902 Sophia Antipolis Cedex, France

Received  November 2016 Revised  February 2017 Published  October 2017

Fund Project: The second author is supported by DFG grant GO 1920/4-1

In this paper, we illustrate how second order traffic flow models, in our case the Aw-Rascle equations, can be used to reproduce empirical observations such as the capacity drop at merges and solve related optimal control problems. To this aim, we propose a model for on-ramp junctions and derive suitable coupling conditions. These are associated to the first order Godunov scheme to numerically study the well-known capacity drop effect, where the outflow of the system is significantly below the expected maximum. Control issues such as speed and ramp meter control are also addressed in a first-discretize-then-optimize framework.

Keywords: Traffic flow, second order model, on-ramp coupling, numerical simulations, optimal control.
Mathematics Subject Classification: 90B20, 35L65.
Citation: Oliver Kolb, Simone Göttlich, Paola Goatin. Capacity drop and traffic control for a second order traffic model. Networks & Heterogeneous Media, 2017, 12 (4) : 663-681. doi: 10.3934/nhm.2017027
References:
[1]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[2]

A. Aw and M. Rascle, Resurrection of ''second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.  Google Scholar

[3]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.  Google Scholar

[4]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Communications in Mathematical Sciences, 5 (2007), 533-551.  doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[5]

M.L. DelleMonache, B. Piccoli and F. Rossi, Traffic Regulation via Controlled Speed Limit, SIAM Journal on Control and Optimization, 55 (2017), 2936-2958.  doi: 10.1137/16M1066038.  Google Scholar

[6]

M.L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin and A.M. Bayen, A PDE-ODE model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), 22-39.  doi: 10.1137/130908993.  Google Scholar

[7]

S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Networks and Heterogeneous Media, 9 (2014), 239-268.  doi: 10.3934/nhm.2014.9.239.  Google Scholar

[8]

M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Communications in Partial Differential Equations, 31 (2006), 243-275.  doi: 10.1080/03605300500358053.  Google Scholar

[9]

M. Garavello and B. Piccoli, Traffic Flow on Networks, Springfield, MO: American Institute of Mathematical Sciences (AIMS), 2006.  Google Scholar

[10]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Mathematical and Computer Modelling, 44 (2006), 287-303.  doi: 10.1016/j.mcm.2006.01.016.  Google Scholar

[11]

P. Goatin, S. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Engineering Optimization, 48 (2016), 1121-1144.  doi: 10.1080/0305215X.2015.1097099.  Google Scholar

[12]

J.M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM Journal on Applied Mathematics, 62 (2001), 729-745.  doi: 10.1137/S0036139900378657.  Google Scholar

[13]

B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic networks, Networks and Heterogeneous Media, 2 (2007), 227-253.  doi: 10.3934/nhm.2007.2.227.  Google Scholar

[14]

A. Hegyi, B.D. Schutter and H. Hellendoorn, Optimal coordination of variable speed limits to suppress shock waves, IEEE Transactions on Intelligent Transportation Systems, 6 (2005), 102-112.  doi: 10.1109/CDC.2003.1273043.  Google Scholar

[15]

M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Networks and Heterogeneous Media, 1 (2006), 275-294.  doi: 10.3934/nhm.2006.1.275.  Google Scholar

[16]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616.  doi: 10.1137/05062617X.  Google Scholar

[17]

W.-L. Jin, Q.-J. Gan and J.-P. Lebacque, A kinematic wave theory of capacity drop, Transportation Research Part B: Methodological, 81 (2015), 316-329.  doi: 10.1016/j.trb.2015.07.020.  Google Scholar

[18]

W. Jin and H. Zhang, On the distribution schemes for determining flows through a merge, Transportation Research Part B: Methodological, 37 (2003), 521-540.  doi: 10.1016/S0191-2615(02)00026-7.  Google Scholar

[19]

O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, PhD thesis, TU Darmstadt, 2011. Google Scholar

[20]

O. Kolb and J. Lang, Simulation and continuous optimization, in "Mathematical Optimization of Water Networks" (eds. A. Martin, K. Klamroth, J. Lang, G. Leugering, A. Morsi, M. Oberlack, M. Ostrowski and R. Rosen), Springer Basel, 162 (2012), 17-33. doi: 10.1007/978-3-0348-0436-3_2.  Google Scholar

[21]

L. Leclercq, V.L. Knoop, F. Marczak and S.P. Hoogendoorn, Capacity drops at merges: New analytical investigations, Transportation Research Part C: Emerging Technologies, 62 (2016), 171-181.  doi: 10.1109/ITSC.2014.6957839.  Google Scholar

[22]

M.J. Lighthill and G.B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[23]

C. Parzani and C. Buisson, Second-order model and capacity drop at merge, Transportation Research Record: Journal of the Transportation Research Board, 2315 (2012), 25-34.  doi: 10.3141/2315-03.  Google Scholar

[24]

B. Piccoli, K. Han, T.L. Friesz, T. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transportation Research Part C: Emerging Technologies, 52 (2015), 32-56.  doi: 10.1016/j.trc.2014.12.013.  Google Scholar

[25]

J. Reilly, S. Samaranayake, M.L. DelleMonache, W. Krichene, P. Goatin and A.M. Bayen, Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering, Journal of Optimization Theory and Applications, 167 (2015), 733-760.  doi: 10.1007/s10957-015-0749-1.  Google Scholar

[26]

F. Siebel, W. Mauser, S. Moutari and M. Rascle, Balanced vehicular traffic at a bottleneck, Mathematical and Computer Modelling, 49 (2009), 689-702.  doi: 10.1016/j.mcm.2008.01.006.  Google Scholar

[27]

P. Spellucci, A new technique for inconsistent QP problems in the SQP method, Mathematical Methods of Operations Research, 47 (1998), 355-400.  doi: 10.1007/BF01198402.  Google Scholar

[28]

P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Mathematical Programming, 82 (1998), 413-448.  doi: 10.1007/BF01580078.  Google Scholar

[29]

A. Srivastava and N. Geroliminis, Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model, Transportation Research Part C: Emerging Technologies, 30 (2013), 161-177.  doi: 10.1016/j.trc.2013.02.006.  Google Scholar

[30]

M. Treiber and A. Kesting, Traffic Flow Dynamics, Data, models and simulation, Translated by Treiber and Christian Thiemann, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.  Google Scholar

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 Journal on Applied Mathematics, 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[2]

A. Aw and M. Rascle, Resurrection of ''second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.  Google Scholar

[3]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.  Google Scholar

[4]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Communications in Mathematical Sciences, 5 (2007), 533-551.  doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[5]

M.L. DelleMonache, B. Piccoli and F. Rossi, Traffic Regulation via Controlled Speed Limit, SIAM Journal on Control and Optimization, 55 (2017), 2936-2958.  doi: 10.1137/16M1066038.  Google Scholar

[6]

M.L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin and A.M. Bayen, A PDE-ODE model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), 22-39.  doi: 10.1137/130908993.  Google Scholar

[7]

S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Networks and Heterogeneous Media, 9 (2014), 239-268.  doi: 10.3934/nhm.2014.9.239.  Google Scholar

[8]

M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Communications in Partial Differential Equations, 31 (2006), 243-275.  doi: 10.1080/03605300500358053.  Google Scholar

[9]

M. Garavello and B. Piccoli, Traffic Flow on Networks, Springfield, MO: American Institute of Mathematical Sciences (AIMS), 2006.  Google Scholar

[10]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Mathematical and Computer Modelling, 44 (2006), 287-303.  doi: 10.1016/j.mcm.2006.01.016.  Google Scholar

[11]

P. Goatin, S. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Engineering Optimization, 48 (2016), 1121-1144.  doi: 10.1080/0305215X.2015.1097099.  Google Scholar

[12]

J.M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM Journal on Applied Mathematics, 62 (2001), 729-745.  doi: 10.1137/S0036139900378657.  Google Scholar

[13]

B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic networks, Networks and Heterogeneous Media, 2 (2007), 227-253.  doi: 10.3934/nhm.2007.2.227.  Google Scholar

[14]

A. Hegyi, B.D. Schutter and H. Hellendoorn, Optimal coordination of variable speed limits to suppress shock waves, IEEE Transactions on Intelligent Transportation Systems, 6 (2005), 102-112.  doi: 10.1109/CDC.2003.1273043.  Google Scholar

[15]

M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Networks and Heterogeneous Media, 1 (2006), 275-294.  doi: 10.3934/nhm.2006.1.275.  Google Scholar

[16]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616.  doi: 10.1137/05062617X.  Google Scholar

[17]

W.-L. Jin, Q.-J. Gan and J.-P. Lebacque, A kinematic wave theory of capacity drop, Transportation Research Part B: Methodological, 81 (2015), 316-329.  doi: 10.1016/j.trb.2015.07.020.  Google Scholar

[18]

W. Jin and H. Zhang, On the distribution schemes for determining flows through a merge, Transportation Research Part B: Methodological, 37 (2003), 521-540.  doi: 10.1016/S0191-2615(02)00026-7.  Google Scholar

[19]

O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, PhD thesis, TU Darmstadt, 2011. Google Scholar

[20]

O. Kolb and J. Lang, Simulation and continuous optimization, in "Mathematical Optimization of Water Networks" (eds. A. Martin, K. Klamroth, J. Lang, G. Leugering, A. Morsi, M. Oberlack, M. Ostrowski and R. Rosen), Springer Basel, 162 (2012), 17-33. doi: 10.1007/978-3-0348-0436-3_2.  Google Scholar

[21]

L. Leclercq, V.L. Knoop, F. Marczak and S.P. Hoogendoorn, Capacity drops at merges: New analytical investigations, Transportation Research Part C: Emerging Technologies, 62 (2016), 171-181.  doi: 10.1109/ITSC.2014.6957839.  Google Scholar

[22]

M.J. Lighthill and G.B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[23]

C. Parzani and C. Buisson, Second-order model and capacity drop at merge, Transportation Research Record: Journal of the Transportation Research Board, 2315 (2012), 25-34.  doi: 10.3141/2315-03.  Google Scholar

[24]

B. Piccoli, K. Han, T.L. Friesz, T. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transportation Research Part C: Emerging Technologies, 52 (2015), 32-56.  doi: 10.1016/j.trc.2014.12.013.  Google Scholar

[25]

J. Reilly, S. Samaranayake, M.L. DelleMonache, W. Krichene, P. Goatin and A.M. Bayen, Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering, Journal of Optimization Theory and Applications, 167 (2015), 733-760.  doi: 10.1007/s10957-015-0749-1.  Google Scholar

[26]

F. Siebel, W. Mauser, S. Moutari and M. Rascle, Balanced vehicular traffic at a bottleneck, Mathematical and Computer Modelling, 49 (2009), 689-702.  doi: 10.1016/j.mcm.2008.01.006.  Google Scholar

[27]

P. Spellucci, A new technique for inconsistent QP problems in the SQP method, Mathematical Methods of Operations Research, 47 (1998), 355-400.  doi: 10.1007/BF01198402.  Google Scholar

[28]

P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Mathematical Programming, 82 (1998), 413-448.  doi: 10.1007/BF01580078.  Google Scholar

[29]

A. Srivastava and N. Geroliminis, Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model, Transportation Research Part C: Emerging Technologies, 30 (2013), 161-177.  doi: 10.1016/j.trc.2013.02.006.  Google Scholar

[30]

M. Treiber and A. Kesting, Traffic Flow Dynamics, Data, models and simulation, Translated by Treiber and Christian Thiemann, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.  Google Scholar

Figure 1.  Demand and supply functions on a fixed road for $\rho^{\max}=1$, $v^{\rm{ref}}=2$, $\gamma=2$ and the given values for $c$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  1-to-1 junction
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  1-to-1 junction with on-ramp
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  On-ramp at origin
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Outflow at a vertex
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  A single road
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Density (top) and velocity (bottom) after 36 seconds for different choices of the relaxation parameter $\delta$ and for the LWR model
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Two roads with an on-ramp in between
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Actual outflow depending on the desired inflow at the on-ramp for the AR and the LWR model
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Road network with an on-ramp at the node ''on-ramp''
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Inflow profiles for the network in Figure 10
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  Queue at the origin ''in'' and the on-ramp with and without optimization
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 13.  Flow at the origin ''in'' of the network (left) and at the node ''out'' (right) with and without optimization
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 14.  Optimal control of $v_i^{\max}(t)$ on road2 (top left) and road3 (top right) and $u(t)$ at the on-ramp (bottom)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 15.  Density (top) and velocity (bottom) behind the on-ramp in the uncontrolled case for different choices of the relaxation parameter $\delta$ and for the LWR model
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Capacity drop effect
inflow at on-ramp in $[\frac{\rm{cars}}{\rm{h}}]$}$\rho_1$in $[\frac{\rm{cars}}{\rm{km}}]$$v_1$in $[\frac{\rm{km}}{\rm{h}}]$$w_1$in $[\frac{\rm{km}}{\rm{h}}]$outflow AR in $[\frac{\rm{cars}}{\rm{h}}]$outflow LWR in $[\frac{\rm{cars}}{\rm{h}}]$
desiredactual
50050047.673.677.140004000
1000100047.673.677.145004500
15001500156.413.150.935544500
20001764160.211.050.635274500
25001764160.211.050.635274500
10001000148.017.851.636294500
500500137.223.852.837624000
Table Options

Download as excel

inflow at on-ramp in $[\frac{\rm{cars}}{\rm{h}}]$}$\rho_1$in $[\frac{\rm{cars}}{\rm{km}}]$$v_1$in $[\frac{\rm{km}}{\rm{h}}]$$w_1$in $[\frac{\rm{km}}{\rm{h}}]$outflow AR in $[\frac{\rm{cars}}{\rm{h}}]$outflow LWR in $[\frac{\rm{cars}}{\rm{h}}]$
desiredactual
50050047.673.677.140004000
1000100047.673.677.145004500
15001500156.413.150.935544500
20001764160.211.050.635274500
25001764160.211.050.635274500
10001000148.017.851.636294500
500500137.223.852.837624000
Table 2.  Properties of the roads in Figure 10
roadlength $[\text{km}]$$\rho^{\max}$ $[\frac{\text{cars}}{\text{km}}]$$v^{\rm{low}}$ $[\frac{\text{km}}{\text{h}}]$$v^{\rm{high}}$ $[\frac{\text{km}}{\text{h}}]$initial density $[\frac{\text{cars}}{\text{km}}]$
road1218010010050
road211805010050
road311805010050
road4218010010050
Table Options

Download as excel

roadlength $[\text{km}]$$\rho^{\max}$ $[\frac{\text{cars}}{\text{km}}]$$v^{\rm{low}}$ $[\frac{\text{km}}{\text{h}}]$$v^{\rm{high}}$ $[\frac{\text{km}}{\text{h}}]$initial density $[\frac{\text{cars}}{\text{km}}]$
road1218010010050
road211805010050
road311805010050
road4218010010050
Table 3.  Optimization results for the network in Figure 10
AR, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$AR, $\frac{\partial p_i}{\partial v_i^{\max}}=0$LWR
no control1871.71871.7834.9
ramp metering only1325.31325.3834.9
speed control only1122.8872.6834.9
both control types814.5818.4834.9
Table Options

Download as excel

AR, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$AR, $\frac{\partial p_i}{\partial v_i^{\max}}=0$LWR
no control1871.71871.7834.9
ramp metering only1325.31325.3834.9
speed control only1122.8872.6834.9
both control types814.5818.4834.9
Table 4.  Total travel time for different choices of the relaxation parameter $\delta$
$\delta$no controlopt. control, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}=0$
$5\cdot10^{-5}$2199.4868.8953.4
$5\cdot10^{-4}$2137.1860.1856.3
$5\cdot10^{-3}$1871.7814.5818.4
$5\cdot10^{-2}$731.4731.4731.4
$5\cdot10^{-1}$725.9725.9725.9
Table Options

Download as excel

$\delta$no controlopt. control, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}=0$
$5\cdot10^{-5}$2199.4868.8953.4
$5\cdot10^{-4}$2137.1860.1856.3
$5\cdot10^{-3}$1871.7814.5818.4
$5\cdot10^{-2}$731.4731.4731.4
$5\cdot10^{-1}$725.9725.9725.9
[1]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Homogenization of second order discrete model with local perturbation and application to traffic flow. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1437-1487. doi: 10.3934/dcds.2017060
[2]

Emmanuel Trélat. Optimal control of a space shuttle, and numerical simulations. Conference Publications, 2003, 2003 (Special) : 842-851. doi: 10.3934/proc.2003.2003.842
[3]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Numerical approximations of a traffic flow model on networks. Networks & Heterogeneous Media, 2006, 1 (1) : 57-84. doi: 10.3934/nhm.2006.1.57
[4]

Michael Burger, Simone Göttlich, Thomas Jung. Derivation of second order traffic flow models with time delays. Networks & Heterogeneous Media, 2019, 14 (2) : 265-288. doi: 10.3934/nhm.2019011
[5]

Bertrand Haut, Georges Bastin. A second order model of road junctions in fluid models of traffic networks. Networks & Heterogeneous Media, 2007, 2 (2) : 227-253. doi: 10.3934/nhm.2007.2.227
[6]

Lino J. Alvarez-Vázquez, Néstor García-Chan, Aurea Martínez, Miguel E. Vázquez-Méndez. Optimal control of urban air pollution related to traffic flow in road networks. Mathematical Control & Related Fields, 2018, 8 (1) : 177-193. doi: 10.3934/mcrf.2018008
[7]

Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012
[8]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
[9]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[10]

Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018
[11]

Marco Di Francesco, Simone Fagioli, Massimiliano D. Rosini. Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Mathematical Biosciences & Engineering, 2017, 14 (1) : 127-141. doi: 10.3934/mbe.2017009
[12]

Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135
[13]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[14]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[15]

Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations & Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024
[16]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929
[17]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003
[18]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445
[19]

Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001
[20]

Martin Burger, Peter Alexander Markowich, Jan-Frederik Pietschmann. Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinetic & Related Models, 2011, 4 (4) : 1025-1047. doi: 10.3934/krm.2011.4.1025

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (90)
  • HTML views (153)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences