Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour
Authors
Abstract
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
- Journal: Evolution Equations and Control Theory
- Year: 2022
- Volume: 11
- Issue: 6
- Pages: 2165
- Publisher: American Institute of Mathematical Sciences (AIMS)
- DOI: 10.3934/eect.2022016
BibTeX
@article{Banasiak_2022,
title={{Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour}},
volume={11},
ISSN={2163-2480},
DOI={10.3934/eect.2022016},
number={6},
journal={Evolution Equations and Control Theory},
publisher={American Institute of Mathematical Sciences (AIMS)},
author={Banasiak, Jacek and Błoch, Adam},
year={2022},
pages={2165}
}
References
- F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994,171 pp.
- Banasiak, J. Explicit formulae for limit periodic flows on networks. Linear Algebra and its Applications vol. 500 30–42 (2016) – 10.1016/j.laa.2016.03.010
- Banasiak, J. & Błoch, A. Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness. Evolution Equations and Control Theory vol. 11 1331 (2022) – 10.3934/eect.2021046
- Banasiak, J. & Błoch, A. Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability. Networks and Heterogeneous Media vol. 17 73 (2022) – 10.3934/nhm.2021024
- 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. & Namayanja, P. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media vol. 9 197–216 (2014) – 10.3934/nhm.2014.9.197
- Banasiak, J. & Puchalska, A. Transport on Networks—A Playground of Continuous and Discrete Mathematics in Population Dynamics. Studies in Systems, Decision and Control 439–487 (2019) doi:10.1007/978-3-030-12232-4_14 – 10.1007/978-3-030-12232-4_14
- 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
- Dorn, B. Semigroups for flows in infinite networks. Semigroup Forum vol. 76 341–356 (2008) – 10.1007/s00233-007-9036-2
- 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. & 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
- K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.
- 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
- 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: 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
- Mátrai, T. & Sikolya, E. Asymptotic behavior of flows in networks. Forum Mathematicum vol. 19 (2007) – 10.1515/forum.2007.018
- Meyer, C. Matrix Analysis and Applied Linear Algebra. (2000) doi:10.1137/1.9780898719512 – 10.1137/1.9780898719512
- 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
- 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.
- Staffans, O. Well-Posed Linear Systems. (2005) doi:10.1017/cbo9780511543197 – 10.1017/cbo9780511543197
- 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