Loading [MathJax]/jax/output/SVG/jax.js

Lyapunov stability analysis of networks of scalar conservation laws

  • Received: 01 June 2007 Revised: 01 August 2007
  • Primary: 34D20, 35L65.

  • It is shown how an entropy-based Lyapunov function can be used for the stability analysis of equilibria in networks of scalar conservation laws. The analysis gives a sufficient stability condition which is weaker than the condition which was previously known in the literature. Various extensions and generalisations are briefly discussed. The approach is illustrated with an application to ramp-metering control of road traffic networks.

    Citation: Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws[J]. Networks and Heterogeneous Media, 2007, 2(4): 751-759. doi: 10.3934/nhm.2007.2.751

    Related Papers:

    [1] 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
    [2] Alexandre M. Bayen, Alexander Keimer, Nils Müller . A proof of Kirchhoff's first law for hyperbolic conservation laws on networks. Networks and Heterogeneous Media, 2023, 18(4): 1799-1819. doi: 10.3934/nhm.2023078
    [3] 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
    [4] Michael Herty, Niklas Kolbe, Siegfried Müller . Central schemes for networked scalar conservation laws. Networks and Heterogeneous Media, 2023, 18(1): 310-340. doi: 10.3934/nhm.2023012
    [5] Gabriella Bretti, Roberto Natalini, Benedetto Piccoli . Numerical approximations of a traffic flow model on networks. Networks and Heterogeneous Media, 2006, 1(1): 57-84. doi: 10.3934/nhm.2006.1.57
    [6] Qing Li, Steinar Evje . Learning the nonlinear flux function of a hidden scalar conservation law from data. Networks and Heterogeneous Media, 2023, 18(1): 48-79. doi: 10.3934/nhm.2023003
    [7] 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
    [8] Darko Mitrovic . Existence and stability of a multidimensional scalar conservation law with discontinuous flux. Networks and Heterogeneous Media, 2010, 5(1): 163-188. doi: 10.3934/nhm.2010.5.163
    [9] 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
    [10] Maria Laura Delle Monache, Paola Goatin . Stability estimates for scalar conservation laws with moving flux constraints. Networks and Heterogeneous Media, 2017, 12(2): 245-258. doi: 10.3934/nhm.2017010
  • It is shown how an entropy-based Lyapunov function can be used for the stability analysis of equilibria in networks of scalar conservation laws. The analysis gives a sufficient stability condition which is weaker than the condition which was previously known in the literature. Various extensions and generalisations are briefly discussed. The approach is illustrated with an application to ramp-metering control of road traffic networks.


  • This article has been cited by:

    1. Edward Canepa, Christian Claudel, 2013, Exact solutions to robust control problems involving scalar hyperbolic conservation laws using Mixed Integer Linear Programming, 978-1-4799-3410-2, 478, 10.1109/Allerton.2013.6736563
    2. Nicolás Espitia, Antoine Girard, Nicolas Marchand, Christophe Prieur, Fluid-flow modeling and stability analysis of communication networks, 2017, 50, 24058963, 4534, 10.1016/j.ifacol.2017.08.727
    3. Michael Herty, Hui Yu, Feedback boundary control of linear hyperbolic equations with stiff source term, 2018, 91, 0020-7179, 230, 10.1080/00207179.2016.1276635
    4. Jorge A. Laval, Ludovic Leclercq, Continuum Approximation for Congestion Dynamics Along Freeway Corridors, 2010, 44, 0041-1655, 87, 10.1287/trsc.1090.0294
    5. Michael Herty, Wen-An Yong, Feedback boundary control of linear hyperbolic systems with relaxation, 2016, 69, 00051098, 12, 10.1016/j.automatica.2016.02.016
    6. Fabio Ancona, Annalisa Cesaroni, Giuseppe M. Coclite, Mauro Garavello, On the Optimization of Conservation Law Models at a Junction with Inflow and Flow Distribution Controls, 2018, 56, 0363-0129, 3370, 10.1137/18M1176233
    7. Andre F. Caldeira, Christophe Prieur, Daniel Coutinho, Valter J. S. Leite, 2015, Modeling and control of flow with dynamical boundary actions, 978-1-4799-7787-1, 1579, 10.1109/CCA.2015.7320835
    8. Graziano Guerra, Michael Herty, Francesca Marcellini, Modeling and analysis of pooled stepped chutes, 2011, 6, 1556-181X, 665, 10.3934/nhm.2011.6.665
    9. Michael Herty, Mohammed Seaïd, Assessment of coupling conditions in water way intersections, 2013, 71, 02712091, 1438, 10.1002/fld.3719
    10. Mapundi K. Banda, Michael Herty, Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws, 2013, 3, 2156-8499, 121, 10.3934/mcrf.2013.3.121
    11. Michael Herty, Hui Yu, 2016, Boundary stabilization of hyperbolic conservation laws using conservative finite volume schemes, 978-1-5090-1837-6, 5577, 10.1109/CDC.2016.7799126
    12. Mamadou Diagne, Shu-Xia Tang, Ababacar Diagne, Miroslav Krstic, Control of shallow waves of two unmixed fluids by backstepping, 2017, 44, 13675788, 211, 10.1016/j.arcontrol.2017.09.003
    13. Y.E. Ge, Chong-Feng Xu, W.Y. Szeto, B.R. Sun, H.M. Zhang, L.W. Zhang, Investigating freeway traffic hypercongestion between an on-ramp and its immediate upstream off-ramp, 2015, 11, 2324-9935, 187, 10.1080/23249935.2014.945509
    14. Thang V. Pham, Didier Georges, Gildas Besançon, Receding horizon boundary control of nonlinear conservation laws with shock avoidance, 2012, 48, 00051098, 2244, 10.1016/j.automatica.2012.06.025
    15. M. Dick, M. Gugat, M. Herty, S. Steffensen, On the relaxation approximation of boundary control of the isothermal Euler equations, 2012, 85, 0020-7179, 1766, 10.1080/00207179.2012.703787
    16. Nicolás Espitia, Antoine Girard, Nicolas Marchand, Christophe Prieur, Event-based control of linear hyperbolic systems of conservation laws, 2016, 70, 00051098, 275, 10.1016/j.automatica.2016.04.009
    17. Paola Goatin, Simone Göttlich, Oliver Kolb, Speed limit and ramp meter control for traffic flow networks, 2016, 48, 0305-215X, 1121, 10.1080/0305215X.2015.1097099
    18. Simone Göttlich, Michael Herty, Peter Schillen, Electric transmission lines: Control and numerical discretization, 2016, 37, 01432087, 980, 10.1002/oca.2219
    19. R. M. Colombo, M. Herty, V. Sachers, On 2×2 Conservation Laws at a Junction, 2008, 40, 0036-1410, 605, 10.1137/070690298
    20. Stephan Gerster, Felix Nagel, Aleksey Sikstel, Giuseppe Visconti, Numerical boundary control for semilinear hyperbolic systems, 2022, 0, 2156-8472, 0, 10.3934/mcrf.2022040
    21. Sebastien Blandin, Xavier Litrico, Maria Laura Delle Monache, Benedetto Piccoli, Alexandre Bayen, Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws, 2017, 62, 0018-9286, 1620, 10.1109/TAC.2016.2590598
    22. Yacine Chitour, Guilherme Mazanti, Mario Sigalotti, Persistently damped transport on a network of circles, 2016, 369, 0002-9947, 3841, 10.1090/tran/6778
    23. Simone Göttlich, Peter Schillen, Numerical Feedback Stabilization with Applications to Networks, 2017, 2017, 1026-0226, 1, 10.1155/2017/6896153
    24. Dong-Xia Zhao, Jun-Min Wang, 2013, On the stabilization of an irrigation channel with a cascade of 2 pools: A linearized case, 978-1-4673-5769-2, 1, 10.1109/ASCC.2013.6606160
    25. Ghada Ben Belgacem, Chaker Jammazi, On the finite-time boundary dissipative for a class of hyperbolic systems. The networks example, 2016, 49, 24058963, 186, 10.1016/j.ifacol.2016.07.435
    26. Van Thang Pham, Didier Georges, Gildas Besancon, 2012, Predictive Control with terminal constraint for 2×2 hyperbolic systems of conservation laws, 978-1-4673-2066-5, 6412, 10.1109/CDC.2012.6426538
    27. Yanning Li, Edward Canepa, Christian Claudel, Efficient robust control of first order scalar conservation laws using semi-analytical solutions, 2014, 7, 1937-1179, 525, 10.3934/dcdss.2014.7.525
    28. Thang V. Pham, Didier Georges, Gildas Besançon, Receding Optimal Boundary Control of Non-linear Hyperbolic Systems of Conservation Laws, 2011, 44, 14746670, 8601, 10.3182/20110828-6-IT-1002.01027
    29. Simone Göttlich, Peter Schillen, Numerical discretization of boundary control problems for systems of balance laws: Feedback stabilization, 2017, 35, 09473580, 11, 10.1016/j.ejcon.2017.02.002
    30. Eduardo Cerpa, Emmanuelle Crépeau, Claudia Moreno, On the boundary controllability of the Korteweg–de Vries equation on a star-shaped network, 2019, 0265-0754, 10.1093/imamci/dny047
    31. Charlotte Rodriguez, Networks of geometrically exact beams: Well-posedness and stabilization, 2022, 12, 2156-8472, 49, 10.3934/mcrf.2021002
    32. Min Ni, Xisheng Dai, Jiaming Zhang, Qiqi Wu, Huang Zuo, 2022, Stability of Linear Hyperbolic Distributed Parameter Systems Based on Event-Triggered Control, 978-1-6654-9675-9, 1358, 10.1109/DDCLS55054.2022.9858478
    33. Yacine Chitour, Guilherme Mazanti, Mario Sigalotti, Stability of non-autonomous difference equations with applications to transport and wave propagation on networks, 2016, 11, 1556-1801, 563, 10.3934/nhm.2016010
    34. Jan Friedrich, Simone Göttlich, Michael Herty, Lyapunov Stabilization for Nonlocal Traffic Flow Models, 2023, 61, 0363-0129, 2849, 10.1137/22M152181X
    35. Alexandre M. Bayen, Alexander Keimer, Nils Müller, A proof of Kirchhoff's first law for hyperbolic conservation laws on networks, 2023, 18, 1556-1801, 1799, 10.3934/nhm.2023078
  • Reader Comments
  • © 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4896) PDF downloads(213) Cited by(34)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog