Authors

Y. Le Gorrec, D. Matignon

Abstract

The aim of this paper is to study a conservative wave equation coupled to a diffusion equation : this coupled system naturally arises in musical acoustics when viscous and thermal effects at the wall of the duct of a wind instrument are taken into account. The resulting equation, known as Webster-Lokshin model, has variable coefficients in space, and a fractional derivative in time. The port-Hamiltonian formalism proves adequate to reformulate this coupled system, and could enable another well-posedness analysis, using classical results from port-Hamiltonian systems theory. First, an equivalent formulation of fractional derivatives is obtained thanks to so-called diffusive representations: this is the reason why we first concentrate on rewriting these diffusive representations into the port-Hamiltonian formalism; two cases must be studied separately, the fractional integral operator as a low-pass filter, and the fractional derivative operator as a high-pass filter. Second, a standard finite-dimensional mechanical oscillator coupled to both types of dampings, either low-pass or high-pass, is studied as a coupled pHs. The more general PDE system of a wave equation coupled with the diffusion equation is then found to have the same structure as before, but in an appropriate infinite-dimensional setting, which is fully detailed.

Keywords

energy storage; port-Hamiltonian systems; partial differential equations; fractional derivatives; diffusive representation

Citation

  • Journal: IFAC Proceedings Volumes
  • Year: 2012
  • Volume: 45
  • Issue: 19
  • Pages: 254–259
  • Publisher: Elsevier BV
  • DOI: 10.3182/20120829-3-it-4022.00037
  • Note: 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control

BibTeX

@article{Le_Gorrec_2012,
  title={{Diffusive systems coupled to an oscillator: a Hamiltonian formulation}},
  volume={45},
  ISSN={1474-6670},
  DOI={10.3182/20120829-3-it-4022.00037},
  number={19},
  journal={IFAC Proceedings Volumes},
  publisher={Elsevier BV},
  author={Le Gorrec, Y. and Matignon, D.},
  year={2012},
  pages={254--259}
}

Download the bib file

References

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