Jet space extensions of infinite-dimensional Hamiltonian systems

Authors

Till Preuster, Bernhard Maschke, Manuel Schaller

Abstract

We analyze infinite-dimensional Hamiltonian systems corresponding to partial Differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and non-quadratic Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

Keywords

port-Hamiltonian systems; jet spaces; distributed parameter systems

Citation

• Journal: IFAC-PapersOnLine
• Year: 2024
• Volume: 58
• Issue: 6
• Pages: 298–303
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2024.08.297
• Note: 8th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2024- Besançon, France, June 10 – 12, 2024

BibTeX

@article{Preuster_2024,
  title={{Jet space extensions of infinite-dimensional Hamiltonian systems}},
  volume={58},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2024.08.297},
  number={6},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Preuster, Till and Maschke, Bernhard and Schaller, Manuel},
  year={2024},
  pages={298--303}
}

Download the bib file

References

• Bendimerad-Hohl, A., Haine, G., Matignon, D. & Maschke, B. Structure-preserving discretization of a coupled Allen-Cahn and heat equation system. IFAC-PapersOnLine vol. 55 99–104 (2022) – 10.1016/j.ifacol.2022.08.037
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Courant, (1988)
• Jacob, (2012)
• 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
• Macchelli, (2009)
• Maschke, (1993)
• Maschke, B. & van der Schaft, A. J. On alternative Poisson brackets for fluid dynamical systems and their extension to Stokes-Dirac structures. IFAC Proceedings Volumes vol. 46 109–114 (2013) – 10.3182/20130925-3-fr-4043.00083
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 10.1007/978-1-4612-4350-2
• 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. & Mehrmann, V. Linear Port-Hamiltonian DAE Systems Revisited. SSRN Electronic Journal (2022) doi:10.2139/ssrn.4292998 – 10.2139/ssrn.4292998
• 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
• 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
• Vincent, B., Couenne, F., Lefèvre, L. & Maschke, B. Port Hamiltonian systems with moving interface: a phase field approach. IFAC-PapersOnLine vol. 53 7569–7574 (2020) – 10.1016/j.ifacol.2020.12.1353
• 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
• Zwart, H., Le Gorrec, Y. & Maschke, B. Building systems from simple hyperbolic ones. Systems & Control Letters vol. 91 1–6 (2016) – 10.1016/j.sysconle.2016.02.002