Authors

Jacek Banasiak, Adam Błoch

Abstract

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

  • Journal: Evolution Equations and Control Theory
  • Year: 2022
  • Volume: 11
  • Issue: 4
  • Pages: 1331
  • Publisher: American Institute of Mathematical Sciences (AIMS)
  • DOI: 10.3934/eect.2021046

BibTeX

@article{Banasiak_2022,
  title={{Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness}},
  volume={11},
  ISSN={2163-2480},
  DOI={10.3934/eect.2021046},
  number={4},
  journal={Evolution Equations and Control Theory},
  publisher={American Institute of Mathematical Sciences (AIMS)},
  author={Banasiak, Jacek and Błoch, Adam},
  year={2022},
  pages={1331}
}

Download the bib file

References

  • Mehmeti, F. A. & Meister, E. Regular solutions of transmission and interaction problems for wave equations. Mathematical Methods in the Applied Sciences vol. 11 665–685 (1989) – 10.1002/mma.1670110507
  • F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994.
  • Arendt, W. Resolvent Positive Operators. Proceedings of the London Mathematical Society vols s3-54 321–349 (1987) – 10.1112/plms/s3-54.2.321
  • Banasiak, J. Acta Applicandae Mathematicae vol. 49 199–228 (1997) – 10.1023/a:1005882912151
  • J. Banasiak and A. Błoch, Telegraph systems on networks and port-{H}amiltonians. II. {G}raph realizability, arXiv: 2103.06651, 2021.
  • Banasiak, J., Falkiewicz, A. & Namayanja, P. Semigroup approach to diffusion and transport problems on networks. Semigroup Forum vol. 93 427–443 (2015) – 10.1007/s00233-015-9730-4
  • Banasiak, J. & Lachowicz, M. Methods of Small Parameter in Mathematical Biology. Modeling and Simulation in Science, Engineering and Technology (Springer International Publishing, 2014). doi:10.1007/978-3-319-05140-6 – 10.1007/978-3-319-05140-6
  • Banasiak, J. & Namayanja, P. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media vol. 9 197–216 (2014) – 10.3934/nhm.2014.9.197
  • Bastin, G. & Coron, J.-M. Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Progress in Nonlinear Differential Equations and Their Applications (Springer International Publishing, 2016). doi:10.1007/978-3-319-32062-5 – 10.1007/978-3-319-32062-5
  • Bobrowski, A. From Diffusions on Graphs to Markov Chains via Asymptotic State Lumping. Annales Henri Poincaré vol. 13 1501–1510 (2012) – 10.1007/s00023-012-0158-z
  • Carlson, R. Linear network models related to blood flow. Contemporary Mathematics 65–80 (2006) doi:10.1090/conm/415/07860 – 10.1090/conm/415/07860
  • Diagne, A., Bastin, G. & Coron, J.-M. Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws. Automatica vol. 48 109–114 (2012) – 10.1016/j.automatica.2011.09.030
  • Dorn, B., Kramar Fijavž, M., Nagel, R. & Radl, A. The semigroup approach to transport processes in networks. Physica D: Nonlinear Phenomena vol. 239 1416–1421 (2010) – 10.1016/j.physd.2009.06.012
  • Engel, K.-J. Generator property and stability for generalized difference operators. Journal of Evolution Equations vol. 13 311–334 (2013) – 10.1007/s00028-013-0179-1
  • Engel, K.-J. & Kramar Fijavž, M. Waves and diffusion on metric graphs with general vertex conditions. Evolution Equations & Control Theory vol. 8 633–661 (2019) – 10.3934/eect.2019030
  • Engel, K.-J. & Nagel, R. One-parameter semigroups for linear evolution equations. Semigroup Forum vol. 63 278–280 (2001) – 10.1007/s002330010042
  • Exner, P. Momentum operators on graphs. Proceedings of Symposia in Pure Mathematics 105–118 (2013) doi:10.1090/pspum/087/01427 – 10.1090/pspum/087/01427
  • R. Fitzpatrick, Maxwells Equations and the Principles of Electromagnetism, Laxmi Publications, Ltd., 2010.
  • Jacob, B., Morris, K. & Zwart, H. C 0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. Journal of Evolution Equations vol. 15 493–502 (2015) – 10.1007/s00028-014-0271-1
  • Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
  • T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.
  • Klöss, B. Difference operators as semigroup generators. Semigroup Forum vol. 81 461–482 (2010) – 10.1007/s00233-010-9232-3
  • Klöss, B. The flow approach for waves in networks. Operators and Matrices 107–128 (2012) doi:10.7153/oam-06-08 – 10.7153/oam-06-08
  • Kostrykin, V. & Schrader, R. Kirchhoff’s rule for quantum wires. Journal of Physics A: Mathematical and General vol. 32 595–630 (1999) – 10.1088/0305-4470/32/4/006
  • Kramar, M. & Sikolya, E. Spectral properties and asymptotic periodicity of flows in networks. Mathematische Zeitschrift vol. 249 139–162 (2004) – 10.1007/s00209-004-0695-3
  • Kramar Fijavž, M., Mugnolo, D. & Nicaise, S. Linear hyperbolic systems on networks: well-posedness and qualitative properties. ESAIM: Control, Optimisation and Calculus of Variations vol. 27 7 (2021) – 10.1051/cocv/2020091
  • Kuchment, P. Quantum graphs: I. Some basic structures. Waves in Random Media vol. 14 S107–S128 (2004) – 10.1088/0959-7174/14/1/014
  • Kuchment, P. Quantum graphs: An introduction and a brief survey. Proceedings of Symposia in Pure Mathematics 291–312 (2008) doi:10.1090/pspum/077/2459876 – 10.1090/pspum/077/2459876
  • Litrico, X. & Fromion, V. Modeling and Control of Hydrosystems. (Springer London, 2009). doi:10.1007/978-1-84882-624-3 – 10.1007/978-1-84882-624-3
  • G. Lumer, Espaces ramifiés, et diffusions sur les réseaux topologiques, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A627–A630.
  • Mugnolo, D. Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems (Springer International Publishing, 2014). doi:10.1007/978-3-319-04621-1 – 10.1007/978-3-319-04621-1
  • Nicaise, S. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields vol. 7 53–72 (2017) – 10.3934/mcrf.2017004
  • Staffans, O. Well-Posed Linear Systems. (2005) doi:10.1017/cbo9780511543197 – 10.1017/cbo9780511543197
  • von Below, J. Classical solvability of linear parabolic equations on networks. Journal of Differential Equations vol. 72 316–337 (1988) – 10.1016/0022-0396(88)90158-1
  • 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.
  • Zwart, H., Le Gorrec, Y., Maschke, B. & Villegas, J. Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: Control, Optimisation and Calculus of Variations vol. 16 1077–1093 (2009) – 10.1051/cocv/2009036