Stokes-Lagrange and Stokes-Dirac representations of $ N $-dimensional port-Hamiltonian systems for modeling and control

Antoine Bendimerad-Hohl; Ghislain Haine; Laurent Lefèvre; Denis Matignon; Antoine Bendimerad-Hohl; Ghislain Haine; Laurent Lefèvre; Denis Matignon

doi:10.3934/cam.2025020

Communications in Analysis and Mechanics

2025, Volume 17, Issue 2: 474-519. doi: 10.3934/cam.2025020

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Research article

Stokes-Lagrange and Stokes-Dirac representations of $ N $-dimensional port-Hamiltonian systems for modeling and control

1.
Fédération ENAC ISAE-SUPAERO ONERA, Université de Toulouse, Toulouse, France
2.
Université Grenoble Alpes, Grenoble INP, Laboratoire de Conception et d'Intégration des Systèmes, Valence, France

Received: 22 November 2024 Revised: 04 April 2025 Accepted: 16 April 2025 Published: 09 May 2025
93C20; 37K06; 74K20; 35Q61

In this paper, we extended the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an $ N $-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We proposed the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration-by-parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrated the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modeling and control of port-Hamiltonian systems.
- port-Hamiltonian systems,
- Stokes-Dirac structure,
- Stokes-Lagrange structure,
- Reissner-Mindlin model,
- Kirchhoff-Love model,
- Maxwell's equations,
- passivity-based control
Citation: Antoine Bendimerad-Hohl, Ghislain Haine, Laurent Lefèvre, Denis Matignon. Stokes-Lagrange and Stokes-Dirac representations of $ N $-dimensional port-Hamiltonian systems for modeling and control[J]. Communications in Analysis and Mechanics, 2025, 17(2): 474-519. doi: 10.3934/cam.2025020

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Abstract

In this paper, we extended the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an $ N $-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We proposed the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration-by-parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrated the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modeling and control of port-Hamiltonian systems.

References

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