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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 |
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 $.
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. |
show all references
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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