Implicit Lagrangians, partial Dirac structures and extremals of the maximum principle in infinite dimensions

Fernand Pelletier; Patrick Cabau; Fernand Pelletier; Patrick Cabau

doi:10.3934/cam.2026033

Communications in Analysis and Mechanics

2026, Volume 18, Issue 4: 806-876. doi: 10.3934/cam.2026033

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Research article

Implicit Lagrangians, partial Dirac structures and extremals of the maximum principle in infinite dimensions

Fernand Pelletier ¹,
Patrick Cabau ^{2
,
,}

1.
UMR 5127 CNRS, Université de Savoie Mont Blanc, LaMa, Campus Scientifique, 73370 Le Bourget-du-Lac, France
2.
Independent Researcher

Received: 04 December 2025 Revised: 10 August 2026 Accepted: 16 September 2026 Published: 19 October 2026
46T05, 58B20, 70G45, 70H45

In a previous work, the notion of a Dirac structure was extended from finite-dimensional manifolds to the convenient setting of Frölicher, Kriegl, and Michor through the introduction of partial Dirac structures on convenient manifolds. This framework encompasses a wide range of infinite-dimensional settings, including Hilbert, Banach, and Fréchet manifolds, as well as direct limits of Banach manifolds. It is also particularly well-suited to variational methods, since the category of convenient vector spaces is Cartesian closed, unlike the category of locally convex vector spaces. In this paper, we develop a variational framework for the study of constrained Lagrangian systems on convenient manifolds and for optimal control problems in infinite dimensions. The variational tools introduced by Yoshimura and Marsden for constrained dynamics on Banach manifolds were extended to singular distributions arising in the context of the Pontryagin maximum principle. As a consequence, a characterization of normal extremals was obtained in this infinite-dimensional setting. The theory was then applied to conical Finsler geometry on Banach manifolds, yielding a characterization of normal geodesics.
- partial Dirac structures,
- convenient Lie algebroids,
- convenient manifolds,
- implicit Lagrangians,
- direct limits,
- extremals,
- geodesics,
- conical Finsler metrics
Citation: Fernand Pelletier, Patrick Cabau. Implicit Lagrangians, partial Dirac structures and extremals of the maximum principle in infinite dimensions[J]. Communications in Analysis and Mechanics, 2026, 18(4): 806-876. doi: 10.3934/cam.2026033

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Abstract

In a previous work, the notion of a Dirac structure was extended from finite-dimensional manifolds to the convenient setting of Frölicher, Kriegl, and Michor through the introduction of partial Dirac structures on convenient manifolds. This framework encompasses a wide range of infinite-dimensional settings, including Hilbert, Banach, and Fréchet manifolds, as well as direct limits of Banach manifolds. It is also particularly well-suited to variational methods, since the category of convenient vector spaces is Cartesian closed, unlike the category of locally convex vector spaces. In this paper, we develop a variational framework for the study of constrained Lagrangian systems on convenient manifolds and for optimal control problems in infinite dimensions. The variational tools introduced by Yoshimura and Marsden for constrained dynamics on Banach manifolds were extended to singular distributions arising in the context of the Pontryagin maximum principle. As a consequence, a characterization of normal extremals was obtained in this infinite-dimensional setting. The theory was then applied to conical Finsler geometry on Banach manifolds, yielding a characterization of normal geodesics.

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