Hamiltonian formulation of distributed-parameter systems with boundary energy flow

Authors

Abstract

A Hamiltonian formulation of classes of distributed-parameter systems is presented, which incorporates the energy flow through the boundary of the spatial domain of the system, and which allows to represent the system as a boundary control Hamiltonian system. The system is Hamiltonian with respect to an infinite-dimensional Dirac structure associated with the exterior derivative and based on Stokes’ theorem. The theory is applied to the telegraph equations for an ideal transmission line, Maxwell’s equations on a bounded domain with non-zero Poynting vector at its boundary, and a vibrating string with traction forces at its ends. Furthermore, the framework is extended to cover Euler’s equations for an ideal fluid on a domain with permeable boundary. Finally, some properties of the Stokes–Dirac structure are investigated, including the analysis of conservation laws.

Keywords

Distributed-parameter systems; Hamiltonian systems; Boundary variables; Dirac structures; Stokes’ theorem; Conservation laws

Citation

Journal: Journal of Geometry and Physics
Year: 2002
Volume: 42
Issue: 1-2
Pages: 166–194
Publisher: Elsevier BV
DOI: 10.1016/s0393-0440(01)00083-3

BibTeX

@article{van_der_Schaft_2002,
  title={{Hamiltonian formulation of distributed-parameter systems with boundary energy flow}},
  volume={42},
  ISSN={0393-0440},
  DOI={10.1016/s0393-0440(01)00083-3},
  number={1–2},
  journal={Journal of Geometry and Physics},
  publisher={Elsevier BV},
  author={van der Schaft, A.J. and Maschke, B.M.},
  year={2002},
  pages={166--194}
}

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References

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