Well-posedness of systems of 1-D hyperbolic partial differential equations

Authors

Abstract

We consider the well-posedness of a class of hyperbolic partial differential equations on a one-dimensional spatial domain. This class includes in particular infinite networks of transport, wave and beam equations, or even combinations of these. Equivalent conditions for contraction semigroup generation are derived. We consider these equations on a finite interval as well as on a semi-axis.

Citation

Journal: Journal of Evolution Equations
Year: 2019
Volume: 19
Issue: 1
Pages: 91–109
Publisher: Springer Science and Business Media LLC
DOI: 10.1007/s00028-018-0470-2

BibTeX

@article{Jacob_2018,
  title={{Well-posedness of systems of 1-D hyperbolic partial differential equations}},
  volume={19},
  ISSN={1424-3202},
  DOI={10.1007/s00028-018-0470-2},
  number={1},
  journal={Journal of Evolution Equations},
  publisher={Springer Science and Business Media LLC},
  author={Jacob, Birgit and Kaiser, Julia T.},
  year={2018},
  pages={91--109}
}

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References

B. Augner, Stabilisation of Infinite-dimensional Port-Hamiltonian Systems, Ph.D. thesis, University of Wuppertal, 2016. Available from: http://elpub.bib.uni-wuppertal.de/edocs/dokumente/fbc/mathematik/diss2016/augner/dc1613.pdf
```
            .
```
Augner B, Jacob B (2014) Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. EECT 3(2):207–229. https://doi.org/10.3934/eect.2014.3.207 – 10.3934/eect.2014.3.207
Bastin G, Coron J-M (2016) Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Springer International Publishing, Cham – 10.1007/978-3-319-32062-5
Berkolaiko G, Kuchment P (2012) Introduction to Quantum Graphs. American Mathematical Society, Providence, Rhode Island – 10.1090/surv/186
Engel K-J (2013) Generator property and stability for generalized difference operators. J Evol Equ 13(2):311–334. https://doi.org/10.1007/s00028-013-0179-1 – 10.1007/s00028-013-0179-1
K. J. Engel and M. Fijavz, Waves and diffusion on metric graphs with general vertex conditions, https://arxiv.org/abs/1712.03030
```
            .
```
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, (1999), Springer Science & Business Media, Berlin Heidelberg.
K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, (2006), Springer Science & Business Media, Berlin Heidelberg.
Hinrichsen D, Pritchard AJ (2005) Mathematical Systems Theory I. Springer Berlin Heidelberg, Berlin, Heidelberg – 10.1007/b137541
Jacob B, Morris K, Zwart H (2015) C 0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. J Evol Equ 15(2):493–502. https://doi.org/10.1007/s00028-014-0271-1 – 10.1007/s00028-014-0271-1
B. Jacob and H.J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Operator Theory: Advances and Applications, 223 (2012), Birkhäuser, Basel.
V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs In: Analysis on graphs and its applications, Proc. Sympos. Pure Math., 77 (2008), 42-458.
Kostrykin V, Schrader R (1999) Kirchhoff’s rule for quantum wires. J Phys A: Math Gen 32(4):595–630. https://doi.org/10.1088/0305-4470/32/4/006 – 10.1088/0305-4470/32/4/006

M. Kurula and H. Zwart, Linear wave systems on

              $$n$$
                
                  n
                  
            -D spatial domains, Internat. J. Control, 88 (5) (2015), 1063–1077.

Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J Control Optim 44(5):1864–1892. https://doi.org/10.1137/040611677 – 10.1137/040611677
Mugnolo D (2014) Semigroup Methods for Evolution Equations on Networks. Springer International Publishing, Cham – 10.1007/978-3-319-04621-1
D. Mugnolo, D. Noja, and C. Seifert, Airy-type evolution equations on star graphs, Preprint, (2016), Available from: https://arxiv.org/pdf/1608.01461.pdf
```
            .
```
Schubert C, Seifert C, Voigt J, Waurick M (2015) Boundary systems and (skew‐)self‐adjoint operators on infinite metric graphs. Mathematische Nachrichten 288(14-15):1776–1785. https://doi.org/10.1002/mana.201500054 – 10.1002/mana.201500054
van der Schaft AJ, Maschke BM (2002) Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42(1-2):166–194. https://doi.org/10.1016/s0393-0440(01)00083-3 – 10.1016/s0393-0440(01)00083-3
J.A. Villegas, A port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D. thesis, Universiteit Twente in Enschede, 2007. Available from: http://doc.utwente.nl/57842/1/thesis_Villegas.pdf
```
            .
```
Zwart H, Le Gorrec Y, Maschke B, Villegas J (2009) Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16(4):1077–1093. https://doi.org/10.1051/cocv/2009036 – 10.1051/cocv/2009036