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Coupling between hyperbolic and diffusive systems: A port-Hamiltonian formulation

Le Gorrec, Yann and Matignon, Denis Coupling between hyperbolic and diffusive systems: A port-Hamiltonian formulation. (2013) European Journal of control, 19 (6). 505-512. ISSN 0947-3580

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Official URL: http://dx.doi.org/10.1016/j.ejcon.2013.09.003


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 detailled.

Item Type:Article
Additional Information:Thanks to Elsevier editor. The definitive version is available at http://www.sciencedirect.com The original PDF of the article can be found at European Journal of control website: http://www.sciencedirect.com/science/journal/09473580/19/6
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:Other partners > Ecole Nationale Supérieure de Mécanique et des Microtechniques - ENSMM (FRANCE)
Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
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Deposited On:25 Apr 2013 12:11

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