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On optimization of a highly re-entrant production system
A weakly coupled model of differential equations for thief tracking
1. | Department of Mathematics, University of Mannheim, D-68131 Mannheim, Germany |
References:
[1] |
D. A. Andrews and J. Bonta, The Psychology Of Criminal Conduct, Anderson Publishing, Ltd., 2010. |
[2] |
M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
[3] |
G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM Journal on Applied Dynamical Systems, 7 (2008), 510-531.
doi: 10.1137/070697768. |
[4] |
G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886.
doi: 10.1137/S0036141004402683. |
[5] |
R. M. Colombo and A. Marson, Conservation laws and ODEs: A traffic problem, Springer, (2003), 455-461. |
[6] |
R. M. Colombo and A. Marson, A hölder continuous ODE related to traffic flow, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 759-772.
doi: 10.1017/S0308210500002663. |
[7] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp.
doi: 10.1142/S0218202511500230. |
[8] |
J. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11.
doi: 10.1109/TAC.2006.887903. |
[9] |
C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization Of Supply Chains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010.
doi: 10.1137/1.9780898717600. |
[10] |
B. C. Dean, Shortest Paths in FIFO Time-Dependent Networks: Theory and Algorithms, Technical report, MIT Department of Computer Science, 2004. |
[11] |
M. L. Delle Monache and P. Goatin, A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 435-447.
doi: 10.3934/dcdss.2014.7.435. |
[12] |
S. E. Dreyfus, An appraisal of some shortest-path algorithms, Operations Research, 17 (1969), 395-412.
doi: 10.1287/opre.17.3.395. |
[13] |
G. Feichtinger, A differential games solution to a model of competition between a thief and the police, Management Science, 29 (1983), 686-699.
doi: 10.1287/mnsc.29.6.686. |
[14] |
A. F. Filippov and F. M. Arscott, Differential Equations With Discontinuous Righthand Sides: Control Systems, Springer, 1988.
doi: 10.1007/978-94-015-7793-9. |
[15] |
M. Garavello and B. Piccoli, Traffic Flow On Networks, American Institute of Mathematical Sciences Springfield, MO, USA, 2006. |
[16] |
S. Göttlich, S. Kühn, P. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Netw. Heterog. Media, 6 (2011), 443-464.
doi: 10.3934/nhm.2011.6.443. |
[17] |
S. Göttlich and U. Ziegler, Traffic light control: A case study, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 483-501.
doi: 10.3934/dcdss.2014.7.483. |
[18] |
S. Göttlich, M. Herty and U. Ziegler, Modeling and optimizing traffic light settings in road networks, Computers & Operations Research, 55 (2015), 36-51.
doi: 10.1016/j.cor.2014.10.001. |
[19] |
H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995), 999-1017.
doi: 10.1137/S0036141093243289. |
[20] |
G. Jiang, D. Levy, C. Lin, S. Osher and E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 35 (1998), 2147-2168.
doi: 10.1137/S0036142997317560. |
[21] |
C. Lattanzio, A. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM Journal on Mathematical Analysis, 43 (2011), 50-67.
doi: 10.1137/090767224. |
[22] |
M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[23] |
A. K. Misra, Modeling the effect of police deterrence on the prevalence of crime in the society, Applied Mathematics and Computation, 237 (2014), 531-545.
doi: 10.1016/j.amc.2014.03.136. |
[24] |
A. A. Reid, R. Frank, N. Iwanski, V.Dabbaghian and P. Brantingham, Uncovering the spatial patterning of crimes: A criminal movement model (CriMM), Journal of Research in Crime and Delinquency, 51 (2014), 230-255.
doi: 10.1177/0022427813483753. |
[25] |
P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[26] |
N. Rodriguez and A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, Math. Models Methods Appl. Sci., 20 (2010), 1425-1457.
doi: 10.1142/S0218202510004696. |
[27] |
M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behaviour, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.
doi: 10.1142/S0218202508003029. |
[28] |
M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM Journal on Applied Dynamical Systems, 9 (2010), 462-483.
doi: 10.1137/090759069. |
[29] |
J. T. Woodworth, G. O. Mohler, A. L. Bertozzi and P. J. Brantingham, Non-local crime density estimation incorporating housing information, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130403, 15pp.
doi: 10.1098/rsta.2013.0403. |
[30] |
J. R. Zipkin, M. B. Short and A. L. Bertozzi, Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1479-1506.
doi: 10.3934/dcdsb.2014.19.1479. |
show all references
References:
[1] |
D. A. Andrews and J. Bonta, The Psychology Of Criminal Conduct, Anderson Publishing, Ltd., 2010. |
[2] |
M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
[3] |
G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM Journal on Applied Dynamical Systems, 7 (2008), 510-531.
doi: 10.1137/070697768. |
[4] |
G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886.
doi: 10.1137/S0036141004402683. |
[5] |
R. M. Colombo and A. Marson, Conservation laws and ODEs: A traffic problem, Springer, (2003), 455-461. |
[6] |
R. M. Colombo and A. Marson, A hölder continuous ODE related to traffic flow, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 759-772.
doi: 10.1017/S0308210500002663. |
[7] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp.
doi: 10.1142/S0218202511500230. |
[8] |
J. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11.
doi: 10.1109/TAC.2006.887903. |
[9] |
C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization Of Supply Chains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010.
doi: 10.1137/1.9780898717600. |
[10] |
B. C. Dean, Shortest Paths in FIFO Time-Dependent Networks: Theory and Algorithms, Technical report, MIT Department of Computer Science, 2004. |
[11] |
M. L. Delle Monache and P. Goatin, A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 435-447.
doi: 10.3934/dcdss.2014.7.435. |
[12] |
S. E. Dreyfus, An appraisal of some shortest-path algorithms, Operations Research, 17 (1969), 395-412.
doi: 10.1287/opre.17.3.395. |
[13] |
G. Feichtinger, A differential games solution to a model of competition between a thief and the police, Management Science, 29 (1983), 686-699.
doi: 10.1287/mnsc.29.6.686. |
[14] |
A. F. Filippov and F. M. Arscott, Differential Equations With Discontinuous Righthand Sides: Control Systems, Springer, 1988.
doi: 10.1007/978-94-015-7793-9. |
[15] |
M. Garavello and B. Piccoli, Traffic Flow On Networks, American Institute of Mathematical Sciences Springfield, MO, USA, 2006. |
[16] |
S. Göttlich, S. Kühn, P. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Netw. Heterog. Media, 6 (2011), 443-464.
doi: 10.3934/nhm.2011.6.443. |
[17] |
S. Göttlich and U. Ziegler, Traffic light control: A case study, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 483-501.
doi: 10.3934/dcdss.2014.7.483. |
[18] |
S. Göttlich, M. Herty and U. Ziegler, Modeling and optimizing traffic light settings in road networks, Computers & Operations Research, 55 (2015), 36-51.
doi: 10.1016/j.cor.2014.10.001. |
[19] |
H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995), 999-1017.
doi: 10.1137/S0036141093243289. |
[20] |
G. Jiang, D. Levy, C. Lin, S. Osher and E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 35 (1998), 2147-2168.
doi: 10.1137/S0036142997317560. |
[21] |
C. Lattanzio, A. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM Journal on Mathematical Analysis, 43 (2011), 50-67.
doi: 10.1137/090767224. |
[22] |
M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[23] |
A. K. Misra, Modeling the effect of police deterrence on the prevalence of crime in the society, Applied Mathematics and Computation, 237 (2014), 531-545.
doi: 10.1016/j.amc.2014.03.136. |
[24] |
A. A. Reid, R. Frank, N. Iwanski, V.Dabbaghian and P. Brantingham, Uncovering the spatial patterning of crimes: A criminal movement model (CriMM), Journal of Research in Crime and Delinquency, 51 (2014), 230-255.
doi: 10.1177/0022427813483753. |
[25] |
P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[26] |
N. Rodriguez and A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, Math. Models Methods Appl. Sci., 20 (2010), 1425-1457.
doi: 10.1142/S0218202510004696. |
[27] |
M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behaviour, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.
doi: 10.1142/S0218202508003029. |
[28] |
M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM Journal on Applied Dynamical Systems, 9 (2010), 462-483.
doi: 10.1137/090759069. |
[29] |
J. T. Woodworth, G. O. Mohler, A. L. Bertozzi and P. J. Brantingham, Non-local crime density estimation incorporating housing information, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130403, 15pp.
doi: 10.1098/rsta.2013.0403. |
[30] |
J. R. Zipkin, M. B. Short and A. L. Bertozzi, Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1479-1506.
doi: 10.3934/dcdsb.2014.19.1479. |
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