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Optima and equilibria for traffic flow on networks with backward propagating queues
Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks
1. | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department Mathematik, Chair of Applied Mathematics 2, Cauerstraße 11, 91058 Erlangen, Germany, Germany, Germany |
2. | School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China |
References:
[1] |
R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM Journal on Numerical Analysis, 53 (2015), 963-983.
doi: 10.1137/140975255. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions Of Bounded Variation And Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. |
[4] |
D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920.
doi: 10.1137/040604625. |
[5] |
D. Armbruster, D. E. Marthaler, C. A. Ringhofer, K. G. Kempf and T.-C. Jo, A continuum model for a re-entrant factory, Operations Research, 54 (2006), 933-950.
doi: 10.1287/opre.1060.0321. |
[6] |
A. A. Assad, Multicommodity network flows - a survey, Networks, 8 (1978), 37-91.
doi: 10.1002/net.3230080107. |
[7] |
H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, vol. 6 of MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.
doi: 10.1137/1.9781611973488. |
[8] |
S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numerische Mathematik, Springer Berlin Heidelberg, (2015), 1-25.
doi: 10.1007/s00211-015-0717-6. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
doi: 10.1007/978-0-387-70914-7. |
[10] |
R. M. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379.
doi: 10.1051/cocv/2010007. |
[11] |
J.-M. Coron, M. Kawski and Z. Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1337-1359.
doi: 10.3934/dcdsb.2010.14.1337. |
[12] |
L. R. Ford Jr. and D. R. Fulkerson, Flows in Networks, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1962. |
[13] |
A. Freno and E. Trentin, Hybrid Random Fields: A Scalable Approach to Structure and Parameter Learning in Probabilistic Graphical Models, Intelligent Systems Reference Library, Springer, 2011.
doi: 10.1007/978-3-642-20308-4. |
[14] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Vol. 80 of Monographs in Mathematics, Birkhäuser Boston, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[15] |
M. Gröschel, A. Keimer, G. Leugering and Z. Wang, Regularity theory and adjoint based optimality conditions for a nonlinear transport equation with nonlocal velocity, SIAM J. Control Optim., 52 (2014), 2141-2163.
doi: 10.1137/120873832. |
[16] |
M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Networks and Heterogeneous Media, 10 (2015), 295-320.
doi: 10.3934/nhm.2015.10.295. |
[17] |
M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, Journal of Optimization Theory and Applications, 126 (2005), 589-616.
doi: 10.1007/s10957-005-5499-z. |
[18] |
M. Gugat, M. Herty, A. Klar, G. Leugering and V. Schleper, Well-posedness of networked hyperbolic systems of balance laws, in Constrained optimization and optimal control for partial differential equations, vol. 160 of Internat. Ser. Numer. Math., Birkhäuser/Springer Basel AG, Basel, 2012, 123-146.
doi: 10.1007/978-3-0348-0133-1_7. |
[19] |
J. L. Kennington, A survey of linear cost multicommodity network flows, Operations Res., 26 (1978), 209-236.
doi: 10.1287/opre.26.2.209. |
[20] |
M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Contr., 55 (2010), 2511-2526.
doi: 10.1109/TAC.2010.2046925. |
[21] |
G. Leoni, A First Course in Sobolev Spaces, vol. 105 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009.
doi: 10.1090/gsm/105. |
[22] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96
doi: 10.1007/BF01762360. |
[23] |
D. W. Stroock, Essentials of Integration Theory for Analysis, vol. 262, Springer, 2011.
doi: 10.1007/978-1-4614-1135-2. |
[24] |
W. W.-Y. Wong, Compactness in $L^{2}$, 2013, Personal Communication.,, , ().
|
[25] |
J. J. Yeh, Lectures On Real Analysis, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.
doi: 10.1142/9789812799531_0003. |
show all references
References:
[1] |
R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM Journal on Numerical Analysis, 53 (2015), 963-983.
doi: 10.1137/140975255. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions Of Bounded Variation And Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. |
[4] |
D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920.
doi: 10.1137/040604625. |
[5] |
D. Armbruster, D. E. Marthaler, C. A. Ringhofer, K. G. Kempf and T.-C. Jo, A continuum model for a re-entrant factory, Operations Research, 54 (2006), 933-950.
doi: 10.1287/opre.1060.0321. |
[6] |
A. A. Assad, Multicommodity network flows - a survey, Networks, 8 (1978), 37-91.
doi: 10.1002/net.3230080107. |
[7] |
H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, vol. 6 of MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.
doi: 10.1137/1.9781611973488. |
[8] |
S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numerische Mathematik, Springer Berlin Heidelberg, (2015), 1-25.
doi: 10.1007/s00211-015-0717-6. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
doi: 10.1007/978-0-387-70914-7. |
[10] |
R. M. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379.
doi: 10.1051/cocv/2010007. |
[11] |
J.-M. Coron, M. Kawski and Z. Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1337-1359.
doi: 10.3934/dcdsb.2010.14.1337. |
[12] |
L. R. Ford Jr. and D. R. Fulkerson, Flows in Networks, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1962. |
[13] |
A. Freno and E. Trentin, Hybrid Random Fields: A Scalable Approach to Structure and Parameter Learning in Probabilistic Graphical Models, Intelligent Systems Reference Library, Springer, 2011.
doi: 10.1007/978-3-642-20308-4. |
[14] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Vol. 80 of Monographs in Mathematics, Birkhäuser Boston, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[15] |
M. Gröschel, A. Keimer, G. Leugering and Z. Wang, Regularity theory and adjoint based optimality conditions for a nonlinear transport equation with nonlocal velocity, SIAM J. Control Optim., 52 (2014), 2141-2163.
doi: 10.1137/120873832. |
[16] |
M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Networks and Heterogeneous Media, 10 (2015), 295-320.
doi: 10.3934/nhm.2015.10.295. |
[17] |
M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, Journal of Optimization Theory and Applications, 126 (2005), 589-616.
doi: 10.1007/s10957-005-5499-z. |
[18] |
M. Gugat, M. Herty, A. Klar, G. Leugering and V. Schleper, Well-posedness of networked hyperbolic systems of balance laws, in Constrained optimization and optimal control for partial differential equations, vol. 160 of Internat. Ser. Numer. Math., Birkhäuser/Springer Basel AG, Basel, 2012, 123-146.
doi: 10.1007/978-3-0348-0133-1_7. |
[19] |
J. L. Kennington, A survey of linear cost multicommodity network flows, Operations Res., 26 (1978), 209-236.
doi: 10.1287/opre.26.2.209. |
[20] |
M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Contr., 55 (2010), 2511-2526.
doi: 10.1109/TAC.2010.2046925. |
[21] |
G. Leoni, A First Course in Sobolev Spaces, vol. 105 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009.
doi: 10.1090/gsm/105. |
[22] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96
doi: 10.1007/BF01762360. |
[23] |
D. W. Stroock, Essentials of Integration Theory for Analysis, vol. 262, Springer, 2011.
doi: 10.1007/978-1-4614-1135-2. |
[24] |
W. W.-Y. Wong, Compactness in $L^{2}$, 2013, Personal Communication.,, , ().
|
[25] |
J. J. Yeh, Lectures On Real Analysis, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.
doi: 10.1142/9789812799531_0003. |
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