# American Institute of Mathematical Sciences

doi: 10.3934/eect.2022016
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## Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour

 1 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa 2 Institute of Mathematics, Łodź University of Technology, Łodź, Poland

Received  November 2021 Revised  March 2022 Early access March 2022

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 Łodź University of Technology, Poland

In this paper we present an explicit formula for the semigroup governing the solution to hyperbolic systems on a metric graph, satisfying general linear Kirchhoff's type boundary conditions. Further, we use this representation to establish the long term behaviour of the solutions. The crucial role is played by the spectral decomposition of the boundary matrix.

Citation: Jacek Banasiak, Adam Błoch. Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour. Evolution Equations and Control Theory, doi: 10.3934/eect.2022016
##### References:
 [1] F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994,171 pp. [2] J. Banasiak, Explicit formulae for limit periodic flows on networks, Linear Algebra and its Applications, 500 (2016), 30-42.  doi: 10.1016/j.laa.2016.03.010. [3] J. Banasiak and A. Błoch, Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness, Evol. Eq. Control Th., (2021). doi: 10.3934/eect.2021046. [4] J. Banasiak and A. Błoch, Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability, Netw. Heterog. Media, 17 (2022), 73-99.  doi: 10.3934/nhm.2021024. [5] 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. [6] 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. [7] J. Banasiak and A. Puchalska, Transport on networks–a playground of continuous and discrete mathematics in population dynamics, Mathematics Applied to Engineering, Modelling, and Social Issues, Stud. Syst. Decis. Control, Springer, Cham, 200 (2019), 439-487. [8] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control, Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-32062-5. [9] B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356.  doi: 10.1007/s00233-007-9036-2. [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] K.-J. Engel and M. Kramar Fijavž, Waves and diffusion on metric graphs with general vertex conditions, Evol. Eq. Control Th., 8 (2019), 633-661.  doi: 10.3934/eect.2019030. [12] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. [13] 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. [14] B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances and Applications, 223. Linear Operators and Linear Systems, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1. [15] M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162.  doi: 10.1007/s00209-004-0695-3. [16] M. Kramar Fijavž, D. Mugnolo and S. Nicaise, Linear hyperbolic systems on networks: Well-posedness and qualitative properties, ESAIM Control Optim. Calc. Var., 27 (2021), 46 pp. doi: 10.1051/cocv/2020091. [17] P. Kuchment, Quantum graphs: An introduction and a brief survey, Analysis on Graphs and Its Applications, Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 77 (2008), 291-312.  doi: 10.1090/pspum/077/2459876. [18] T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.  doi: 10.1515/FORUM.2007.018. [19] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512. [20] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1. [21] 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. [22] A. Puchalska, Dynamical Systems on Networks. Well-posedness, Asymptotics and the Network's Structure Impact on Their Properties, PhD thesis, Institute of Mathematics, Łodź University of Technology, 2018. [23] O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197. [24] 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, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994,171 pp. [2] J. Banasiak, Explicit formulae for limit periodic flows on networks, Linear Algebra and its Applications, 500 (2016), 30-42.  doi: 10.1016/j.laa.2016.03.010. [3] J. Banasiak and A. Błoch, Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness, Evol. Eq. Control Th., (2021). doi: 10.3934/eect.2021046. [4] J. Banasiak and A. Błoch, Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability, Netw. Heterog. Media, 17 (2022), 73-99.  doi: 10.3934/nhm.2021024. [5] 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. [6] 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. [7] J. Banasiak and A. Puchalska, Transport on networks–a playground of continuous and discrete mathematics in population dynamics, Mathematics Applied to Engineering, Modelling, and Social Issues, Stud. Syst. Decis. Control, Springer, Cham, 200 (2019), 439-487. [8] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control, Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-32062-5. [9] B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356.  doi: 10.1007/s00233-007-9036-2. [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] K.-J. Engel and M. Kramar Fijavž, Waves and diffusion on metric graphs with general vertex conditions, Evol. Eq. Control Th., 8 (2019), 633-661.  doi: 10.3934/eect.2019030. [12] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. [13] 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. [14] B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances and Applications, 223. Linear Operators and Linear Systems, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1. [15] M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162.  doi: 10.1007/s00209-004-0695-3. [16] M. Kramar Fijavž, D. Mugnolo and S. Nicaise, Linear hyperbolic systems on networks: Well-posedness and qualitative properties, ESAIM Control Optim. Calc. Var., 27 (2021), 46 pp. doi: 10.1051/cocv/2020091. [17] P. Kuchment, Quantum graphs: An introduction and a brief survey, Analysis on Graphs and Its Applications, Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 77 (2008), 291-312.  doi: 10.1090/pspum/077/2459876. [18] T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.  doi: 10.1515/FORUM.2007.018. [19] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512. [20] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1. [21] 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. [22] A. Puchalska, Dynamical Systems on Networks. Well-posedness, Asymptotics and the Network's Structure Impact on Their Properties, PhD thesis, Institute of Mathematics, Łodź University of Technology, 2018. [23] O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197. [24] 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.
The graph for the problem (30) with the directions of the flows
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