Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks

Previous Article
Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics
NHM Home
This Issue
Next Article
Optima and equilibria for traffic flow on networks with backward propagating queues

December 2015, 10(4): 749-785. doi: 10.3934/nhm.2015.10.749

Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks

Martin Gugat ^1, , Alexander Keimer ^1, , Günter Leugering ^1, and Zhiqiang Wang ^2,

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

Received October 2014 Revised May 2015 Published October 2015

Abstract
Related Papers

We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semi-conductor production. Every single commodity is modeled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the dynamics for two commodities. In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered. On every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side boundary data to the following edges and PDEs. For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of mass. We provide results of regularity, existence and well-posedness of the multi-commodity network model for $L^{p}$-, $BV$- and $W^{1,p}$-data. Moreover, we define an $L^{2}$-tracking type objective and show the existence of minimizers that solve the corresponding optimal control problem.

Keywords: nonlocal conservation laws, optimal control of conservation laws on networks, optimal nodal control, Existence of minimizers, conservation laws, systems of hyperbolic PDEs., multi-commodity model, conservation laws on networks.

Mathematics Subject Classification: 49J20, 93C20, 49N60, 35F61, 35L65, 35L5.

Citation: Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks and Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749

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.

[1]	Mauro Garavello. A review of conservation laws on networks. Networks and Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[2]	Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks and Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717
[3]	Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[4]	Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks and Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393
[5]	Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks and Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[6]	Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673
[7]	Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011
[8]	Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51
[9]	Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15
[10]	Avner Friedman. Conservation laws in mathematical biology. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[11]	Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[12]	Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[13]	Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17 (1) : 101-128. doi: 10.3934/nhm.2021025
[14]	Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185
[15]	Dmitry V. Zenkov. Linear conservation laws of nonholonomic systems with symmetry. Conference Publications, 2003, 2003 (Special) : 967-976. doi: 10.3934/proc.2003.2003.967
[16]	Valérie Dos Santos, Bernhard Maschke, Yann Le Gorrec. A Hamiltonian perspective to the stabilization of systems of two conservation laws. Networks and Heterogeneous Media, 2009, 4 (2) : 249-266. doi: 10.3934/nhm.2009.4.249
[17]	Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[18]	Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure and Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[19]	Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[20]	Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control and Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121

2020 Impact Factor: 1.213

Tools

Metrics

Other articles
by authors

on AIMS
Martin Gugat Alexander Keimer Günter Leugering Zhiqiang Wang
on Google Scholar
Martin Gugat Alexander Keimer Günter Leugering Zhiqiang Wang

[Back to Top]