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

Authors

Birgit Jacob, Julia T. Kaiser

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}
}

Download the bib file

References

• B. Augner, Stabilisation of Infinite-dimensional Port-Hamiltonian Systems, Ph.D. thesis, University of Wuppertal, 2016.
• Augner, B. & Jacob, B. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory vol. 3 207–229 (2014) – 10.3934/eect.2014.3.207
• 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
• Berkolaiko, G. & Kuchment, P. Introduction to Quantum Graphs. Mathematical Surveys and Monographs (2012) doi:10.1090/surv/186 – 10.1090/surv/186
• 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
• 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. K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, (1999), Springer Science & Business Media, Berlin Heidelberg. (1999)
• K-J Engel. K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, (2006), Springer Science & Business Media, Berlin Heidelberg. (2006)
• Hinrichsen, D. & Pritchard, A. J. Mathematical Systems Theory I. Texts in Applied Mathematics (Springer Berlin Heidelberg, 2005). doi:10.1007/b137541 – 10.1007/b137541
• 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
• 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. 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
• M. Kurula and H. Zwart, Linear wave systems on $n$-D spatial domains, Internat. J. Control, 88 (5) (2015), 1063–1077. (2015)
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• 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
• D. Mugnolo, D. Noja, and C. Seifert, Airy-type evolution equations on star graphs, Preprint, (2016), https://arxiv.org/pdf/1608.01461.pdf
• Schubert, C., Seifert, C., Voigt, J. & Waurick, M. Boundary systems and (skew‐)self‐adjoint operators on infinite metric graphs. Mathematische Nachrichten vol. 288 1776–1785 (2015) – 10.1002/mana.201500054
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 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.
• 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