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Dirac Structures and Reduction of Optimal Control Problems with Symmetries

This paper establishes a symmetry reduction framework for optimal control problems using Dirac structures, demonstrating that the reduced first-order conditions derived from the Pontryagin Maximum Principle correspond to Hamiltonian equations on the dual of an Atiyah algebroid while proving the commutativity of reduction with constraint algorithms and analyzing the geometric nature of abnormal extremals.

Original authors: A. Ibort, M. Leok, R. Salmoni, C. Sardón

Published 2026-10-23

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CC BY 4.0

Original authors: A. Ibort, M. Leok, R. Salmoni, C. Sardón

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where every movement, from the spin of a planet to the flight of a drone, is governed by a set of invisible rules that balance energy, momentum, and the path taken. In physics and engineering, finding the most efficient way to move an object—whether it is a satellite adjusting its orbit or a robot arm reaching for a tool—is known as an optimal control problem. These problems are notoriously difficult because they involve not just where an object is, but how it got there, how fast it is going, and what forces are acting upon it at every single moment. For decades, mathematicians have developed powerful tools to solve these puzzles, often relying on the idea of symmetry. If a system looks the same no matter how you rotate it or shift it in space, that symmetry can be used to simplify the math, stripping away unnecessary complexity to reveal the core motion underneath. However, when these systems become irregular or "singular"—meaning the usual rules for finding the best path break down and the controls cannot be easily solved for—the standard methods often fail, leaving a gap in our understanding of how these complex machines truly behave.

A team of researchers has now filled that gap by applying a sophisticated geometric framework known as Dirac structures to the problem of symmetry reduction in optimal control. Think of a Dirac structure not as a physical object, but as a comprehensive map that records every possible way a system can move while obeying its physical laws, including the constraints that might stop it from moving in certain directions. The researchers took this map and showed how to fold it down, or "reduce" it, when a system possesses symmetry, much like how a spinning top can be described by its center of mass and spin rate without tracking every atom in its body. What makes this work significant is that they proved two different ways of simplifying the problem—one based on the energy of the system and the other based on the geometry of its possible movements—lead to exactly the same result. This agreement is crucial because it confirms that the simplified equations they derived are not just a mathematical trick, but a faithful representation of the real-world physics, preserving both the motion and the underlying reasons why that motion is optimal.

The researchers then tackled the most difficult part of the puzzle: the singular cases where the usual rules do not apply. In these scenarios, the controls that steer the system cannot be eliminated from the equations, and the path to the solution requires a step-by-step algorithm to filter out impossible movements. The team demonstrated that this filtering process works just as well on the simplified, reduced map as it does on the full, complex one. This means that engineers and mathematicians can now solve these stubborn problems on a smaller, more manageable version of the system and then reconstruct the full solution with confidence, knowing that no critical information was lost in the process. They also discovered that in these singular cases, the "abnormal" paths—those that seem to ignore the usual cost of movement—are strictly tied to the curvature of the space in which the system moves. If the space is flat, these paths might not exist; if the space curves in a specific way, these paths appear exactly where the curvature vanishes.

To prove their theory, the team tested it on three very different real-world scenarios. First, they looked at a rigid spacecraft equipped with internal spinning wheels. In this case, the symmetry group was complex, but the geometric connection was flat, meaning the curvature played no role. The reduction successfully isolated the rotational dynamics, showing how the spacecraft's attitude could be controlled without tracking the full orientation at every step. Second, they examined the movement of a particle in a sub-Riemannian space, specifically on the Heisenberg group, which models a system where movement is restricted to certain directions. Here, the symmetry group was simple, but the space was curved everywhere. The reduction revealed that the particle's motion was governed entirely by this curvature, behaving like a charged particle moving through a magnetic field. Finally, they studied the Martinet problem, a scenario where the curvature vanishes on a specific surface. This case was the most revealing, as it showed that the "abnormal" paths, which usually cause trouble for standard solvers, were exactly the curves lying on that surface of zero curvature.

The paper concludes by providing a clear set of tests to determine the nature of these paths. The researchers showed that one can check if a path is abnormal, if it is rigid (meaning it cannot be slightly altered to improve the outcome), and if it is strictly abnormal (meaning it cannot be explained by the standard rules of optimization) by looking at specific geometric properties of the system. These tests rely on the curvature of the space and the way the system's connection changes, offering a complete toolkit for analyzing even the most irregular control problems. By unifying the geometric reduction of symmetries with the constraint algorithms needed for singular systems, this work provides a robust foundation for understanding and controlling complex mechanical systems, from spacecraft to autonomous robots, ensuring that the simplified models used for design and control are as accurate and reliable as the full, complicated reality.

Technical Summary: Dirac Structures and Reduction of Optimal Control Problems with Symmetries

Problem Statement
This paper addresses the geometric formulation and symmetry reduction of optimal control problems (OCPs). Specifically, it seeks to extend the reduction theory for Dirac structures, developed by Yoshimura and Marsden for implicit Lagrangian and Hamiltonian systems, to the realm of optimal control. The central challenge is to reduce OCPs admitting a Lie group of symmetries while simultaneously preserving both the geometric dynamics and the underlying variational character of the problem. Standard presymplectic reduction of the Pontryagin Maximum Principle (PMP) often reduces the degrees of freedom but loses the variational information, failing to guarantee that reduced solutions correspond to extremals of the original cost functional. Furthermore, the paper addresses the "singular" case where the control cannot be eliminated due to degeneracy, requiring a constraint algorithm.

Methodology
The authors formulate the first-order necessary conditions of the PMP as an implicit Dirac system on the Pontryagin bundle $M = T^*P \times_P C$ . The methodology proceeds through several key steps:

Dirac Formulation: The PMP is cast as a Dirac system $(X_P, dH_P) \in D_\Omega$ on the Pontryagin bundle, where $D_\Omega$ is the canonical Dirac structure associated with the presymplectic form $\Omega = pr_P^* \omega_P$ .
Symmetry Reduction: For a $G$ -invariant OCP, the authors perform a Lie–Dirac reduction. They construct the reduced Dirac structure $[D_\Omega]_G$ as a subbundle of the quotient Courant algebroid $(TM \oplus T^*M)/G$ . Unlike standard reductions that project to a Dirac structure on the quotient manifold, this approach retains the necessary data (specifically the coadjoint and adjoint bundle components) to reconstruct the original extremals.
Variational vs. Geometric Reduction: The reduced equations are derived via two independent paths:
- Reducing the Lagrange–Pontryagin variational principle directly.
- Reducing the Dirac structure and Hamiltonian directly.
  The paper proves these two procedures yield identical reduced equations.
Constraint Algorithm: For singular problems (where the Hessian of the Hamiltonian with respect to controls is degenerate), the authors adapt the Gotay–Nester–Hinds constraint algorithm into a Dirac-geometric form. They prove that this algorithm commutes with symmetry reduction, allowing the algorithm to be run on the reduced space.
Abnormal Extremals: By extending the state space to include the cost variable (Mayer form), the authors treat normal and abnormal extremals uniformly. They identify abnormal extremals as solutions lying on the leaf where the base costate vanishes ( $p_z = 0$ ).

Key Contributions and Results

Equivalence of Reductions: The primary theoretical result (Theorem 4) establishes that the reduction of the variational principle and the reduction of the Dirac structure lead to the same reduced Pontryagin Maximum Principle equations. These equations are identified as Hamiltonian equations on the dual of the Atiyah algebroid $TP/G$ , recovering formulations by Martínez and Grabowski–Jóźwikowski, but derived here via Dirac geometry.
Commutation of Reduction and Constraints: Theorem 5 is the paper's main new result regarding singular systems. It proves that symmetry reduction commutes with the constraint algorithm. Consequently, for singular problems, the constraint algorithm can be executed on the reduced space, and the unreduced constraint submanifolds can be recovered by pullback.
Geometric Structure of Reduced Equations: The reduced equations (26) explicitly display two terms distinguishing them from the unreduced PMP:
1. A curvature term involving the curvature $F_A$ of the principal connection.
2. A coadjoint term involving the Lie algebra structure (specifically $ad^*_\xi \mu$ ).
  The paper demonstrates that these terms arise naturally from the Dirac reduction process.
Characterization of Abnormal Extremals:
- Location: For control-linear systems, abnormal extremals occupy the leaf where the base costate vanishes.
- Obstruction: The first secondary constraint in the algorithm is the degeneracy of the curvature pairing $\langle \mu, F_A \rangle$ (Theorem 6). This links the obstruction to abnormality directly to the curvature of the connection.
- Strictness: The paper provides a criterion (Theorem 8) to determine if an abnormal extremal is strictly abnormal (i.e., admits no normal lift). This depends on whether a parallel coadjoint multiplier satisfies Wong's equation.
Second-Order Theory: For abelian structure groups, the authors derive a reduced index form (Theorem 9) for abnormal extremals. This form is metric-free and depends only on the curvature and its covariant derivative. This leads to a trichotomy (Theorem 10) for analyzing admissible curves: testing for abnormality (via curvature), rigidity (via the reduced Legendre form), and strictness (via the parallelism obstruction).

Illustrative Examples
The theory is validated through three distinct examples that exercise different aspects of the reduced equations:

Rigid Spacecraft with Internal Rotors: A system with a non-abelian symmetry group ( $SO(3)$ ) but a flat connection. Here, the curvature term vanishes, and the dynamics are governed by the coadjoint term.
Heisenberg Group Geodesics: A system with an abelian symmetry group (center of the Heisenberg group) but non-flat curvature. Here, the coadjoint term vanishes, and the dynamics are governed by the curvature term (acting as a magnetic field).
Martinet Problem: A system where the curvature degenerates on a hypersurface. This example illustrates the interplay between normal and abnormal extremals, showing how the curvature degeneracy locus contains the abnormal curves, and how strictness can be determined by the metric-dependent second-order conditions.

Significance and Claims
The paper claims that its primary contribution lies not in the reduced equations themselves (which were previously known in the context of Lie algebroids) but in the Dirac-geometric framework that unifies them. Specifically:

It provides a framework where the constraint algorithm and symmetry reduction interact naturally.
It proves that reduction commutes with the constraint algorithm, a result that was not previously established in this generality.
It offers a geometric interpretation of the obstruction to abnormality as the degeneracy of the curvature pairing.
It supplies a second-order theory for abnormal extremals in reduced variables, separating the geometric (curvature-based) constraints from the metric-based optimality conditions.

The authors explicitly state that they do not claim to solve the general regularity problem for abnormal minimizers or to provide sufficient conditions for minimality in all cases, but rather to clarify the geometric structure and the relationship between the reduced and unreduced systems. The work is presented as a completion of the reduction picture for symmetric optimal control problems, bridging the gap between variational principles, Dirac geometry, and the analysis of singular controls.

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