On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems

Authors

Antoine Bendimerad-Hohl, Denis Matignon, Ghislain Haine, Laurent Lefèvre

Abstract

Port-Hamiltonian systems were recently extended to include implicitly defined energy and energy ports thanks to a (Stokes-)Lagrange subspace. Here, we study the equivalent port-Hamiltonian representations of two systems with damping, written using either a classical Hamiltonian or a Stokes-Lagrange subspace. Then, we study the Timoshenko beam and Euler-Bernoulli models, the latter being the flow-constrained version of the former, and show how they can be written using either a Stokes-Dirac or Stokes-Lagrange subspace related by a transformation operator. Finally, it is proven that these transformations commute with the flow-constraint projection operator.

Keywords

Distributed parameter systems; Implicit port-Hamiltonian systems; Constrained port-Hamiltonian systems; Dzektser equation; non-local viscous dissipation; Timoshenko beam; Euler-Bernoulli beam

Citation

• Journal: IFAC-PapersOnLine
• Year: 2024
• Volume: 58
• Issue: 17
• Pages: 238–243
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2024.10.174
• Note: 26th International Symposium on Mathematical Theory of Networks and Systems MTNS 2024- Cambridge, United Kingdom, August 19-23, 2024

BibTeX

@article{Bendimerad_Hohl_2024,
  title={{On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems}},
  volume={58},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2024.10.174},
  number={17},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Bendimerad-Hohl, Antoine and Matignon, Denis and Haine, Ghislain and Lefèvre, Laurent},
  year={2024},
  pages={238--243}
}

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References

• Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0223-3
• Bendimerad-Hohl, A., Haine, G., Lefèvre, L. & Matignon, D. Implicit port-Hamiltonian systems: structure-preserving discretization for the nonlocal vibrations in a viscoelastic nanorod, and for a seepage model. IFAC-PapersOnLine vol. 56 6789–6795 (2023) – 10.1016/j.ifacol.2023.10.387
• Ducceschi, M. & Bilbao, S. Conservative finite difference time domain schemes for the prestressed Timoshenko, shear and Euler–Bernoulli beam equations. Wave Motion vol. 89 142–165 (2019) – 10.1016/j.wavemoti.2019.03.006
• Dzektser, Generalization of the equation of motion of ground waters with free surface. Dokl. Akad. Nauk SSSR (1972)
• Heidari, H. & Zwart, H. Port-Hamiltonian modelling of nonlocal longitudinal vibrations in a viscoelastic nanorod. Mathematical and Computer Modelling of Dynamical Systems vol. 25 447–462 (2019) – 10.1080/13873954.2019.1659374
• Jacob, B. & Morris, K. On Solvability of Dissipative Partial Differential-Algebraic Equations. IEEE Control Systems Letters vol. 6 3188–3193 (2022) – 10.1109/lcsys.2022.3183479
• Maschke, Linear boundary port-Hamiltonian systems with implicitly defined energy. arXiv preprint (2023)
• Mehrmann, Differential–algebraic systems with dissipative Hamiltonian structure. Mathematics of Control, Signals, and Systems (2023)
• Philipp, Infinite-dimensional port-Hamiltonian systems–a system node approach. arXiv preprint (2023)
• Preuster, Jet space extensions of infinite-dimensional Hamiltonian systems. (2024)
• Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
• Schöberl, M. & Schlacher, K. Lagrangian and Port-Hamiltonian formulation for Distributed-parameter systems. IFAC-PapersOnLine vol. 48 610–615 (2015) – 10.1016/j.ifacol.2015.05.025
• Schöberl, M. & Siuka, A. Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica vol. 50 607–613 (2014) – 10.1016/j.automatica.2013.11.035
• Skrepek, N. Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains. Evolution Equations & Control Theory vol. 10 965 (2021) – 10.3934/eect.2020098
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• van der Schaft, A. & Maschke, B. Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam Journal of Mathematics vol. 48 929–939 (2020) – 10.1007/s10013-020-00419-x
• 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
• Yaghi, M., Couenne, F., Galfré, A., Lefèvre, L. & Maschke, B. Port Hamiltonian formulation of the solidification process for a pure substance: A phase field approach*. IFAC-PapersOnLine vol. 55 93–98 (2022) – 10.1016/j.ifacol.2022.08.036