Advanced Search
Article Contents
Article Contents

Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability

J. B. acknowledges partial support from the National Science Centre of Poland Grant 2017/25/B/ST1/00051 and the National Research Foundation of South Africa Grant 82770. The research was completed while A. B. was a doctoral candidate in the Interdisciplinary Doctoral School at L´od´z University of Technology, Poland

Abstract Full Text(HTML) Figure(6) Related Papers Cited by
  • Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of $ 2\times 2 $ hyperbolic equations on a metric graph $ \Gamma $. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of $ \Gamma $ and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of $ \Gamma $.

    Mathematics Subject Classification: Primary: 35R02, 05C50; Secondary: 35F46, 05C90.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Starlike network of channels

    Figure 2.  The reconstructed multi digraph $ \boldsymbol{\Gamma} $. It is seen that it cannot describe a flow on $ \Gamma $ as $ \varpi_5 $ and $ \varpi_6 $ must flow in the same direction

    Figure 3.  The reconstructed multi digraph $ \boldsymbol{\Gamma} $ for (46), (47)

    Figure 4.  A network $ \Gamma $ realizing the flow (48), (49)

    Figure 5.  Multi digraphs $ G_1 $ with 3 sources and two sinks and $ G_2 $ with all sources and all sinks grouped into a single source and a single sink

    Figure 6.  The line digraph for both $ G_1 $ and $ G_2 $

  • [1] F. Ali Mehmeti, Nonlinear Waves in Networks, vol. 80 of Mathematical Research, Akademie-Verlag, Berlin, 1994.
    [2] J. Banasiak and A. Bƚoch, Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posednes, Evol. Eq. Control Th., 2021. doi: 10.3934/eect.2021046.
    [3] J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability, Appl. Math. Lett., 45 (2015), 25-30.  doi: 10.1016/j.aml.2015.01.006.
    [4] J. BanasiakA. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks, Semigroup Forum, 93 (2016), 427-443.  doi: 10.1007/s00233-015-9730-4.
    [5] J. Banasiak and P. Namayanja, Asymptotic behaviour of flows on reducible networks, Netw. Heterog. Media, 9 (2014), 197-216.  doi: 10.3934/nhm.2014.9.197.
    [6] J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer Science & Business Media, London, 2008.
    [7] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, vol. 88, Springer, 2016. doi: 10.1007/978-3-319-32062-5.
    [8] A. Bátkai, M. Kramar Fijavž and A. Rhandi, Positive Operator Semigroups. From Finite to Infinite Dimensions, vol. 257 of Operator Theory: Advances and Applications, Birkhäuser, Cham, 2017. doi: 10.1007/978-3-319-42813-0.
    [9] R. A. BrualdiF. Harary and Z. Miller, Bigraphs versus digraphs via matrices, J. Graph Theory, 4 (1980), 51-73.  doi: 10.1002/jgt.3190040107.
    [10] B. DornM. Kramar FijavžR. Nagel and A. Radl, The semigroup approach to transport processes in networks, Phys. D, 239 (2010), 1416-1421.  doi: 10.1016/j.physd.2009.06.012.
    [11] M. K. Fijavž, D. Mugnolo and S. Nicaise, Linear hyperbolic systems on networks: Well-posedness and qualitative properties, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 7, 46 pp. doi: 10.1051/cocv/2020091.
    [12] F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959.
    [13] R. Hemminger and L. Beineke, Line graphs and line digraphs, in Selected Topics in Graph Theory I (eds. L. Beineke and R. Wilson), Academic Press, London, 1978,271–305.
    [14] B. JacobK. Morris and H. Zwart, $C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.  doi: 10.1007/s00028-014-0271-1.
    [15] B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, vol. 223 of Operator Theory: Advances and Applications, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1.
    [16] B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128.  doi: 10.7153/oam-06-08.
    [17] P. Kuchment, Quantum graphs: An introduction and a brief survey, in Analysis on Graphs and its Applications, vol. 77 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2008,291–312. doi: 10.1090/pspum/077/2459876.
    [18] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, vol. 71, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.
    [19] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.
    [20] S. Nicaise, Control and stabilization of $2\times 2$ hyperbolic systems on graphs, Math. Control Relat. Fields, 7 (2017), 53-72.  doi: 10.3934/mcrf.2017004.
    [21] H. ZwartY. Le GorrecB. Maschke and J. Villegas, Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM Control Optim. Calc. Var., 16 (2010), 1077-1093.  doi: 10.1051/cocv/2009036.
  • 加载中



Article Metrics

HTML views(299) PDF downloads(211) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint