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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

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 & Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749
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show all references

References:
[1]

2nd edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

SIAM Journal on Numerical Analysis, 53 (2015), 963-983. doi: 10.1137/140975255.  Google Scholar

[3]

Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. Google Scholar

[4]

SIAM J. Appl. Math., 66 (2006), 896-920. doi: 10.1137/040604625.  Google Scholar

[5]

Operations Research, 54 (2006), 933-950. doi: 10.1287/opre.1060.0321.  Google Scholar

[6]

Networks, 8 (1978), 37-91. doi: 10.1002/net.3230080107.  Google Scholar

[7]

Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9781611973488.  Google Scholar

[8]

Numerische Mathematik, Springer Berlin Heidelberg, (2015), 1-25. doi: 10.1007/s00211-015-0717-6.  Google Scholar

[9]

Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[10]

ESAIM Control Optim. Calc. Var., 17 (2011), 353-379. doi: 10.1051/cocv/2010007.  Google Scholar

[11]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1337-1359. doi: 10.3934/dcdsb.2010.14.1337.  Google Scholar

[12]

Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1962.  Google Scholar

[13]

Intelligent Systems Reference Library, Springer, 2011. doi: 10.1007/978-3-642-20308-4.  Google Scholar

[14]

Birkhäuser Boston, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[15]

SIAM J. Control Optim., 52 (2014), 2141-2163. doi: 10.1137/120873832.  Google Scholar

[16]

Networks and Heterogeneous Media, 10 (2015), 295-320. doi: 10.3934/nhm.2015.10.295.  Google Scholar

[17]

Journal of Optimization Theory and Applications, 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.  Google Scholar

[18]

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.  Google Scholar

[19]

Operations Res., 26 (1978), 209-236. doi: 10.1287/opre.26.2.209.  Google Scholar

[20]

IEEE Trans. Automat. Contr., 55 (2010), 2511-2526. doi: 10.1109/TAC.2010.2046925.  Google Scholar

[21]

American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/105.  Google Scholar

[22]

Ann. Mat. Pura Appl. (4), 146 (1987), 65-96 doi: 10.1007/BF01762360.  Google Scholar

[23]

Springer, 2011. doi: 10.1007/978-1-4614-1135-2.  Google Scholar

[24]

W. W.-Y. Wong, Compactness in $L^{2}$, 2013, Personal Communication.,, , ().   Google Scholar

[25]

World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/9789812799531_0003.  Google Scholar

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