# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021046
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness

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

Received  February 2021 Revised  July 2021 Early access August 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 Łódź University of Technology, Poland

The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's-type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup-theoretic proof of its well-posedness. A number of examples showing the relation of our results with recent research is also provided.

Citation: Jacek Banasiak, Adam Błoch. Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness. Evolution Equations & Control Theory, doi: 10.3934/eect.2021046
##### References:
 [1] F. Ali Mehmeti, Regular solutions of transmission and interaction problems for wave equations, Math. Methods Appl. Sci., 11 (1989), 665-685.  doi: 10.1002/mma.1670110507.  Google Scholar [2] F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994.  Google Scholar [3] W. Arendt, Resolvent positive operators, Proc. London Math. Soc. (3), 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.  Google Scholar [4] J. Banasiak, Singularly perturbed linear and semilinear hyperbolic systems: Kinetic theory approach to some folk's theorems, Acta Appl. Math., 49 (1997), 199-228.  doi: 10.1023/A:1005882912151.  Google Scholar [5] J. Banasiak and A. Błoch, Telegraph systems on networks and port-{H}amiltonians. II. {G}raph realizability, arXiv: 2103.06651, 2021. Google Scholar [6] 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.  Google Scholar [7] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6.  Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] A. Bobrowski, From diffusions on graphs to Markov chains via asymptotic state lumping, Ann. Henri Poincaré, 13 (2012), 1501–1510. doi: 10.1007/s00023-012-0158-z.  Google Scholar [11] R. Carlson, Linear network models related to blood flow, Quantum Graphs and Their Applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 415 (2006), 65-80.  doi: 10.1090/conm/415/07860.  Google Scholar [12] A. Diagne, G. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.  Google Scholar [13] 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.  Google Scholar [14] K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334.  doi: 10.1007/s00028-013-0179-1.  Google Scholar [15] K.-J. Engel and M. Kramar Fijavž, Waves and diffusion on metric graphs with general vertex conditions, Evol. Equ. Control Theory, 8 (2019), 633-661.  doi: 10.3934/eect.2019030.  Google Scholar [16] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, With Contributions By S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, 2000.  Google Scholar [17] P. Exner, Momentum operators on graphs, In Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th birthday, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 87 (2013), 105–118. doi: 10.1090/pspum/087/01427.  Google Scholar [18] R. Fitzpatrick, Maxwells Equations and the Principles of Electromagnetism, Laxmi Publications, Ltd., 2010. Google Scholar [19] 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.  Google Scholar [20] 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.  Google Scholar [21] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar [22] B. Klöss, Difference operators as semigroup generators, Semigroup Forum, 81 (2010), 461-482.  doi: 10.1007/s00233-010-9232-3.  Google Scholar [23] B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128.  doi: 10.7153/oam-06-08.  Google Scholar [24] V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.  Google Scholar [25] 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.  Google Scholar [26] 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), 46pp. doi: 10.1051/cocv/2020091.  Google Scholar [27] P. Kuchment, Quantum graphs. I. Some basic structures, Special Section on Quantum Graphs. Waves Random Media, 14 (2004), S107–S128.  Google Scholar [28] P. Kuchment, Quantum graphs: An introduction and a brief survey, In Analysis on Graphs and Its Applications, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 77 (2008), 291–312. doi: 10.1090/pspum/077/2459876.  Google Scholar [29] X. Litrico and V. Fromion, Modeling and Control of Hydrosystems, Springer Science & Business Media, 2009. doi: 10.1007/978-1-84882-624-3.  Google Scholar [30] G. Lumer, Espaces ramifiés, et diffusions sur les réseaux topologiques, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A627–A630.  Google Scholar [31] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar [32] 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.  Google Scholar [33] O. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar [34] J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.  doi: 10.1016/0022-0396(88)90158-1.  Google Scholar [35] E. Zauderer, Partial Differential Equations of Applied Mathematics, 2$^{nd}$ edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, A Wiley-Interscience Publication, 1989.  Google Scholar [36] 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.  Google Scholar

show all references

##### References:
 [1] F. Ali Mehmeti, Regular solutions of transmission and interaction problems for wave equations, Math. Methods Appl. Sci., 11 (1989), 665-685.  doi: 10.1002/mma.1670110507.  Google Scholar [2] F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994.  Google Scholar [3] W. Arendt, Resolvent positive operators, Proc. London Math. Soc. (3), 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.  Google Scholar [4] J. Banasiak, Singularly perturbed linear and semilinear hyperbolic systems: Kinetic theory approach to some folk's theorems, Acta Appl. Math., 49 (1997), 199-228.  doi: 10.1023/A:1005882912151.  Google Scholar [5] J. Banasiak and A. Błoch, Telegraph systems on networks and port-{H}amiltonians. II. {G}raph realizability, arXiv: 2103.06651, 2021. Google Scholar [6] 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.  Google Scholar [7] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6.  Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] A. Bobrowski, From diffusions on graphs to Markov chains via asymptotic state lumping, Ann. Henri Poincaré, 13 (2012), 1501–1510. doi: 10.1007/s00023-012-0158-z.  Google Scholar [11] R. Carlson, Linear network models related to blood flow, Quantum Graphs and Their Applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 415 (2006), 65-80.  doi: 10.1090/conm/415/07860.  Google Scholar [12] A. Diagne, G. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.  Google Scholar [13] 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.  Google Scholar [14] K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334.  doi: 10.1007/s00028-013-0179-1.  Google Scholar [15] K.-J. Engel and M. Kramar Fijavž, Waves and diffusion on metric graphs with general vertex conditions, Evol. Equ. Control Theory, 8 (2019), 633-661.  doi: 10.3934/eect.2019030.  Google Scholar [16] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, With Contributions By S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, 2000.  Google Scholar [17] P. Exner, Momentum operators on graphs, In Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th birthday, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 87 (2013), 105–118. doi: 10.1090/pspum/087/01427.  Google Scholar [18] R. Fitzpatrick, Maxwells Equations and the Principles of Electromagnetism, Laxmi Publications, Ltd., 2010. Google Scholar [19] 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.  Google Scholar [20] 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.  Google Scholar [21] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar [22] B. Klöss, Difference operators as semigroup generators, Semigroup Forum, 81 (2010), 461-482.  doi: 10.1007/s00233-010-9232-3.  Google Scholar [23] B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128.  doi: 10.7153/oam-06-08.  Google Scholar [24] V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.  Google Scholar [25] 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.  Google Scholar [26] 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), 46pp. doi: 10.1051/cocv/2020091.  Google Scholar [27] P. Kuchment, Quantum graphs. I. Some basic structures, Special Section on Quantum Graphs. Waves Random Media, 14 (2004), S107–S128.  Google Scholar [28] P. Kuchment, Quantum graphs: An introduction and a brief survey, In Analysis on Graphs and Its Applications, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 77 (2008), 291–312. doi: 10.1090/pspum/077/2459876.  Google Scholar [29] X. Litrico and V. Fromion, Modeling and Control of Hydrosystems, Springer Science & Business Media, 2009. doi: 10.1007/978-1-84882-624-3.  Google Scholar [30] G. Lumer, Espaces ramifiés, et diffusions sur les réseaux topologiques, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A627–A630.  Google Scholar [31] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar [32] 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.  Google Scholar [33] O. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar [34] J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.  doi: 10.1016/0022-0396(88)90158-1.  Google Scholar [35] E. Zauderer, Partial Differential Equations of Applied Mathematics, 2$^{nd}$ edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, A Wiley-Interscience Publication, 1989.  Google Scholar [36] 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.  Google Scholar
Starlike network of channels
 [1] Hassen Arfaoui, Faker Ben Belgacem, Henda El Fekih, Jean-Pierre Raymond. Boundary stabilizability of the linearized viscous Saint-Venant system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 491-511. doi: 10.3934/dcdsb.2011.15.491 [2] Jean-Frédéric Gerbeau, Benoit Perthame. Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 89-102. doi: 10.3934/dcdsb.2001.1.89 [3] Marie-Odile Bristeau, Jacques Sainte-Marie. Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 733-759. doi: 10.3934/dcdsb.2008.10.733 [4] E. Audusse. A multilayer Saint-Venant model: Derivation and numerical validation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 189-214. doi: 10.3934/dcdsb.2005.5.189 [5] Emmanuel Audusse, Fayssal Benkhaldoun, Jacques Sainte-Marie, Mohammed Seaid. Multilayer Saint-Venant equations over movable beds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 917-934. doi: 10.3934/dcdsb.2011.15.917 [6] Georges Bastin, Jean-Michel Coron, Brigitte d'Andréa-Novel. On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks & Heterogeneous Media, 2009, 4 (2) : 177-187. doi: 10.3934/nhm.2009.4.177 [7] Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049 [8] Renato Manfrin. On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 91-106. doi: 10.3934/dcds.1997.3.91 [9] Banavara N. Shashikanth. Kirchhoff's equations of motion via a constrained Zakharov system. Journal of Geometric Mechanics, 2016, 8 (4) : 461-485. doi: 10.3934/jgm.2016016 [10] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [11] Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259 [12] Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207 [13] Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2441-2474. doi: 10.3934/cpaa.2021049 [14] Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069 [15] V. Pata, Sergey Zelik. A result on the existence of global attractors for semigroups of closed operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 481-486. doi: 10.3934/cpaa.2007.6.481 [16] Christian Budde, Marjeta Kramar Fijavž. Bi-Continuous semigroups for flows on infinite networks. Networks & Heterogeneous Media, 2021, 16 (4) : 553-567. doi: 10.3934/nhm.2021017 [17] Huey-Er Lin, Jian-Guo Liu, Wen-Qing Xu. Effects of small viscosity and far field boundary conditions for hyperbolic systems. Communications on Pure & Applied Analysis, 2004, 3 (2) : 267-290. doi: 10.3934/cpaa.2004.3.267 [18] Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic & Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032 [19] Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010 [20] Sebastián Donoso. Enveloping semigroups of systems of order d. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2729-2740. doi: 10.3934/dcds.2014.34.2729

2020 Impact Factor: 1.081

## Tools

Article outline

Figures and Tables

[Back to Top]