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doi: 10.3934/eect.2021046
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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 and 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.

[2]

F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994.

[3]

W. Arendt, Resolvent positive operators, Proc. London Math. Soc. (3), 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.

[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.

[5]

J. Banasiak and A. Błoch, Telegraph systems on networks and port-{H}amiltonians. II. {G}raph realizability, arXiv: 2103.06651, 2021.

[6]

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.

[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.

[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.

[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.

[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.

[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.

[12]

A. DiagneG. 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.

[13]

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.

[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.

[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.

[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.

[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.

[18]

R. Fitzpatrick, Maxwells Equations and the Principles of Electromagnetism, Laxmi Publications, Ltd., 2010.

[19]

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.

[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.

[21]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.

[22]

B. Klöss, Difference operators as semigroup generators, Semigroup Forum, 81 (2010), 461-482.  doi: 10.1007/s00233-010-9232-3.

[23]

B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128.  doi: 10.7153/oam-06-08.

[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.

[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.

[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.

[27]

P. Kuchment, Quantum graphs. I. Some basic structures, Special Section on Quantum Graphs. Waves Random Media, 14 (2004), S107–S128.

[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.

[29]

X. Litrico and V. Fromion, Modeling and Control of Hydrosystems, Springer Science & Business Media, 2009. doi: 10.1007/978-1-84882-624-3.

[30]

G. Lumer, Espaces ramifiés, et diffusions sur les réseaux topologiques, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A627–A630.

[31]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[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.

[33]

O. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.

[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.

[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.

[36]

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.

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.

[2]

F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994.

[3]

W. Arendt, Resolvent positive operators, Proc. London Math. Soc. (3), 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.

[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.

[5]

J. Banasiak and A. Błoch, Telegraph systems on networks and port-{H}amiltonians. II. {G}raph realizability, arXiv: 2103.06651, 2021.

[6]

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.

[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.

[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.

[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.

[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.

[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.

[12]

A. DiagneG. 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.

[13]

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.

[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.

[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.

[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.

[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.

[18]

R. Fitzpatrick, Maxwells Equations and the Principles of Electromagnetism, Laxmi Publications, Ltd., 2010.

[19]

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.

[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.

[21]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.

[22]

B. Klöss, Difference operators as semigroup generators, Semigroup Forum, 81 (2010), 461-482.  doi: 10.1007/s00233-010-9232-3.

[23]

B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128.  doi: 10.7153/oam-06-08.

[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.

[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.

[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.

[27]

P. Kuchment, Quantum graphs. I. Some basic structures, Special Section on Quantum Graphs. Waves Random Media, 14 (2004), S107–S128.

[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.

[29]

X. Litrico and V. Fromion, Modeling and Control of Hydrosystems, Springer Science & Business Media, 2009. doi: 10.1007/978-1-84882-624-3.

[30]

G. Lumer, Espaces ramifiés, et diffusions sur les réseaux topologiques, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A627–A630.

[31]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[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.

[33]

O. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.

[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.

[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.

[36]

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.

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