Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint

Andrea Brugnoli; Ghislain Haine; Denis Matignon; Andrea Brugnoli; Ghislain Haine; Denis Matignon

doi:10.3934/cam.2023018

Communications in Analysis and Mechanics

2023, Volume 15, Issue 3: 362-387. doi: 10.3934/cam.2023018

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Research article

Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint

1.
Technische Universität Berlin, Berlin, Germany
2.
ISAE-SUPAERO, Université de Toulouse, Toulouse, France

Received: 25 November 2022 Revised: 21 April 2023 Accepted: 14 June 2023 Published: 05 July 2023
93A30, 35Q74, 35Q61

In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by van der Schaft and Maschke in 2002. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.
- port-Hamiltonian system,
- Stokes-Dirac structure,
- boundary control,
- partial differential equation,
- wave equation,
- Kirchhoff-Love thin plate,
- 3D elasticity,
- Maxwell equations
Citation: Andrea Brugnoli, Ghislain Haine, Denis Matignon. Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint[J]. Communications in Analysis and Mechanics, 2023, 15(3): 362-387. doi: 10.3934/cam.2023018

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Abstract

In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by van der Schaft and Maschke in 2002. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.

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