# American Institute of Mathematical Sciences

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

 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

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.
Citation: Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks & 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, (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. doi: 10.1137/140975255. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions Of Bounded Variation And Free Discontinuity Problems,, Oxford Mathematical Monographs, (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. 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. doi: 10.1287/opre.1060.0321. [6] A. A. Assad, Multicommodity network flows - a survey,, Networks, 8 (1978), 37. 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), (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, (2015), 1. doi: 10.1007/s00211-015-0717-6. [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (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. 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. doi: 10.3934/dcdsb.2010.14.1337. [12] L. R. Ford Jr. and D. R. Fulkerson, Flows in Networks,, Princeton Landmarks in Mathematics, (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, (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. 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. 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. 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, (2012), 123. 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. 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. 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, (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. 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., (2000). doi: 10.1142/9789812799531_0003.

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##### References:
 [1] R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam),, 2nd edition, (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. doi: 10.1137/140975255. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions Of Bounded Variation And Free Discontinuity Problems,, Oxford Mathematical Monographs, (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. 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. doi: 10.1287/opre.1060.0321. [6] A. A. Assad, Multicommodity network flows - a survey,, Networks, 8 (1978), 37. 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), (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, (2015), 1. doi: 10.1007/s00211-015-0717-6. [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (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. 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. doi: 10.3934/dcdsb.2010.14.1337. [12] L. R. Ford Jr. and D. R. Fulkerson, Flows in Networks,, Princeton Landmarks in Mathematics, (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, (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. 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. 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. 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, (2012), 123. 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. 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. 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, (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. 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., (2000). doi: 10.1142/9789812799531_0003.
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