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Complex dynamics of a forced discretized version of the Mackey-Glass delay differential equation
Optimal inflow control of production systems with finite buffers
1. | University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany, Germany |
References:
[1] |
D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Applied Mathematics, 66 (2006), 896-920.
doi: 10.1137/040604625. |
[2] |
D. Armbruster, S. Göttlich and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Applied Mathematics, 71 (2011), 1070-1087.
doi: 10.1137/100809374. |
[3] |
A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000. |
[4] | |
[5] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
[6] |
C. D'Apice, G. Bretti, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Netw. Heterog. Media, 2 (2007), 661-694.
doi: 10.3934/nhm.2007.2.661. |
[7] |
C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, 2010.
doi: 10.1137/1.9780898717600. |
[8] |
P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Applied Mathematics, 68 (2007), 59-79.
doi: 10.1137/060674302. |
[9] |
A. Fügenschuh, S. Göttlich, M. Herty, A. Klar and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations, SIAM J. Sci. Comput., 30 (2008), 1490-1507.
doi: 10.1137/060663799. |
[10] |
M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915-1938.
doi: 10.3934/dcds.2012.32.1915. |
[11] |
T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Math. Anal., 24 (1993), 279-289.
doi: 10.1137/0524018. |
[12] |
S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559.
doi: 10.4310/CMS.2005.v3.n4.a5. |
[13] |
S. Göttlich, A. Klar and P. Schindler, Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138.
doi: 10.1137/120882573. |
[14] |
S. Göttlich, O. Kolb and S. Kühn, Optimization for a special class of traffic flow models: Combinatorial and continuous approaches, Netw. Heterog. Media, 9 (2014), 315-334.
doi: 10.3934/nhm.2014.9.315. |
[15] |
S. Göttlich, S. Martin and T. Sickenberger, Time-continuous production networks with random breakdowns, Netw. Heterog. Media, 6 (2011), 695-714.
doi: 10.3934/nhm.2011.6.695. |
[16] |
M. Gugat, M. Herty, A. Klar and G. Leugering, Conservation law constrained optimization based upon Front-Tracking, ESAIM, Math. Model. Numer. Anal., 40 (2006), 939-960.
doi: 10.1051/m2an:2006037. |
[17] |
D. Helbing, S. Lämmer, T. Seidel, P. Seba and T. Platkowski, Physics, stability and dynamics of supply networks, Physical Review E, 70 (2004), 066116.
doi: 10.1103/PhysRevE.70.066116. |
[18] |
M. Herty, C. Joerres and B. Piccoli, Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Analysis and Applications, 401 (2013), 510-517.
doi: 10.1016/j.jmaa.2012.12.002. |
[19] |
M. Herty, A. Kurganov and D. Kurochkin, Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs, Commun. Math. Sci., 13 (2015), 15-48.
doi: 10.4310/CMS.2015.v13.n1.a2. |
[20] |
H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, 2002.
doi: 10.1007/978-3-642-56139-9. |
[21] |
C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models, Netw. Heterog. Media, 1 (2006), 675-688.
doi: 10.3934/nhm.2006.1.675. |
[22] |
M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Control, 55 (2010), 2511-2526.
doi: 10.1109/TAC.2010.2046925. |
[23] |
R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
[24] |
Y. Lu, S. Wong, M. Wang and C.-W. Shu, The entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a discontinuous flow-density relationship, Transportation Science, 43 (2009), 511-530.
doi: 10.1287/trsc.1090.0277. |
[25] |
S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41 (2002), 740-797.
doi: 10.1137/S0363012900370764. |
[26] |
S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws, Syst. Control Lett., 48 (2003), 313-328.
doi: 10.1016/S0167-6911(02)00275-X. |
[27] |
J. Wiens, J. Stockie and J. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.
doi: 10.1016/j.jcp.2013.02.024. |
show all references
References:
[1] |
D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Applied Mathematics, 66 (2006), 896-920.
doi: 10.1137/040604625. |
[2] |
D. Armbruster, S. Göttlich and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Applied Mathematics, 71 (2011), 1070-1087.
doi: 10.1137/100809374. |
[3] |
A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000. |
[4] | |
[5] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
[6] |
C. D'Apice, G. Bretti, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Netw. Heterog. Media, 2 (2007), 661-694.
doi: 10.3934/nhm.2007.2.661. |
[7] |
C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, 2010.
doi: 10.1137/1.9780898717600. |
[8] |
P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Applied Mathematics, 68 (2007), 59-79.
doi: 10.1137/060674302. |
[9] |
A. Fügenschuh, S. Göttlich, M. Herty, A. Klar and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations, SIAM J. Sci. Comput., 30 (2008), 1490-1507.
doi: 10.1137/060663799. |
[10] |
M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915-1938.
doi: 10.3934/dcds.2012.32.1915. |
[11] |
T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Math. Anal., 24 (1993), 279-289.
doi: 10.1137/0524018. |
[12] |
S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559.
doi: 10.4310/CMS.2005.v3.n4.a5. |
[13] |
S. Göttlich, A. Klar and P. Schindler, Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138.
doi: 10.1137/120882573. |
[14] |
S. Göttlich, O. Kolb and S. Kühn, Optimization for a special class of traffic flow models: Combinatorial and continuous approaches, Netw. Heterog. Media, 9 (2014), 315-334.
doi: 10.3934/nhm.2014.9.315. |
[15] |
S. Göttlich, S. Martin and T. Sickenberger, Time-continuous production networks with random breakdowns, Netw. Heterog. Media, 6 (2011), 695-714.
doi: 10.3934/nhm.2011.6.695. |
[16] |
M. Gugat, M. Herty, A. Klar and G. Leugering, Conservation law constrained optimization based upon Front-Tracking, ESAIM, Math. Model. Numer. Anal., 40 (2006), 939-960.
doi: 10.1051/m2an:2006037. |
[17] |
D. Helbing, S. Lämmer, T. Seidel, P. Seba and T. Platkowski, Physics, stability and dynamics of supply networks, Physical Review E, 70 (2004), 066116.
doi: 10.1103/PhysRevE.70.066116. |
[18] |
M. Herty, C. Joerres and B. Piccoli, Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Analysis and Applications, 401 (2013), 510-517.
doi: 10.1016/j.jmaa.2012.12.002. |
[19] |
M. Herty, A. Kurganov and D. Kurochkin, Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs, Commun. Math. Sci., 13 (2015), 15-48.
doi: 10.4310/CMS.2015.v13.n1.a2. |
[20] |
H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, 2002.
doi: 10.1007/978-3-642-56139-9. |
[21] |
C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models, Netw. Heterog. Media, 1 (2006), 675-688.
doi: 10.3934/nhm.2006.1.675. |
[22] |
M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Control, 55 (2010), 2511-2526.
doi: 10.1109/TAC.2010.2046925. |
[23] |
R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
[24] |
Y. Lu, S. Wong, M. Wang and C.-W. Shu, The entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a discontinuous flow-density relationship, Transportation Science, 43 (2009), 511-530.
doi: 10.1287/trsc.1090.0277. |
[25] |
S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41 (2002), 740-797.
doi: 10.1137/S0363012900370764. |
[26] |
S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws, Syst. Control Lett., 48 (2003), 313-328.
doi: 10.1016/S0167-6911(02)00275-X. |
[27] |
J. Wiens, J. Stockie and J. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.
doi: 10.1016/j.jcp.2013.02.024. |
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