# American Institute of Mathematical Sciences

February  2022, 17(1): 73-99. doi: 10.3934/nhm.2021024

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

 1 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa 2 Institute of Mathematics, Lódź University of Technology Lódź, Poland 3 International Scientific Laboratory of Applied Semigroup Research South Ural State University, Chelyabinsk, Russia

Received  March 2021 Revised  October 2021 Published  February 2022 Early access  December 2021

Fund Project: 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

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 $\Gamma$.

Citation: Jacek Banasiak, Adam Błoch. Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability. Networks and Heterogeneous Media, 2022, 17 (1) : 73-99. doi: 10.3934/nhm.2021024
##### References:
 [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. Banasiak, A. 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. Brualdi, F. Harary and Z. Miller, Bigraphs versus digraphs via matrices, J. Graph Theory, 4 (1980), 51-73.  doi: 10.1002/jgt.3190040107. [10] B. Dorn, M. Kramar Fijavž, R. 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. Jacob, K. 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. Zwart, Y. Le Gorrec, B. 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.

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##### References:
 [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. Banasiak, A. 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. Brualdi, F. Harary and Z. Miller, Bigraphs versus digraphs via matrices, J. Graph Theory, 4 (1980), 51-73.  doi: 10.1002/jgt.3190040107. [10] B. Dorn, M. Kramar Fijavž, R. 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. Jacob, K. 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. Zwart, Y. Le Gorrec, B. 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.
Starlike network of channels
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
The reconstructed multi digraph $\boldsymbol{\Gamma}$ for (46), (47)
A network $\Gamma$ realizing the flow (48), (49)
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
The line digraph for both $G_1$ and $G_2$
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