The Dirac--Bergmann approach to optimal control theory (2506.17610v1)

Abstract: We present a novel framework for optimal control in both classical and quantum systems. Our approach leverages the Dirac--Bergmann algorithm: a systematic method for formulating and solving constrained dynamical systems. In contrast to the standard Pontryagin principle, which is used in control theory, our approach bypasses the need to perform a variation to obtain the optimal solution. Instead, the Dirac--Bergmann algorithm generates the optimal solution dynamically. The efficacy of our framework is demonstrated through two quintessential examples: the classical and quantum brachistochrone problems, the latter relevant for quantum technological applications.

• The paper presents an alternative optimal control method by enforcing constraint evolution through the Dirac–Bergmann algorithm instead of traditional variational calculus.
• It demonstrates that optimal trajectories, such as the cycloid in classical and geodesic paths in quantum systems, are naturally obtained via systematic constraint stabilization.
• The framework is methodically detailed for high-dimensional, constrained systems and is amenable to automation in both engineering and quantum technology applications.

The Dirac–Bergmann Approach to Optimal Control Theory: A Rigorous Framework for Constrained Dynamics

This paper establishes an alternative framework for optimal control theory grounded in the Dirac–Bergmann algorithm, circumventing the traditional reliance on variational calculus intrinsic to the Pontryagin maximum principle (PMP). Rather than seeking extremal conditions of a cost functional via variation, the proposed method employs constraint evolution and stabilization principles to directly identify optimal controls and trajectories. The approach is systematically articulated for both classical and quantum systems and is applied to the canonical brachistochrone problems in both domains.

Theoretical and Methodological Overview

Contrast with Pontryagin Maximum Principle

PMP has long served as the mainstay for optimal control in both engineering and physics, using adjoint variables and Hamiltonian augmentation of the dynamical system to convert constrained optimization into a boundary value problem. Despite its efficacy, PMP requires a variational derivation and often leads to significant complications in cases with hard constraints or when controls lie on the boundary of admissible sets.

The Dirac–Bergmann algorithm, in contrast, arises from the Hamiltonian analysis of singular Lagrangians and constrained dynamical systems. Constraints—whether from physical laws or imposed optimality conditions—are enforced through their consistency with the time evolution. Instead of introducing variational equations, controls and adjoint variables are promoted to dynamical variables subject to primary and secondary constraints. This guarantees constraint satisfaction throughout the system's evolution and constructs a reduced phase space upon which dynamics are unambiguously defined via Dirac brackets.

Procedure for Classical Optimal Control

The key steps for applying the Dirac–Bergmann method to classical optimal control are as follows:

1. Formulate the augmented Lagrangian incorporating running and (potentially) terminal costs, dynamical states, and controls.
2. Identify canonical variables—including states, controls, adjoints—and determine their momenta by differentiation.
3. Enumerate primary constraints that arise because controls and adjoints lack independent velocities.
4. Construct the canonical Hamiltonian and augment it with all primary constraints using Lagrange multipliers.
5. Demand constraint preservation under Hamiltonian evolution. This yields relations that either fix the multipliers or generate secondary constraints.
6. Iterate the procedure for secondary (and higher) constraints until closure is achieved.
7. Classify constraints into first- and second-class, ultimately reducing the dynamics to the physical subspace via Dirac brackets.
8. Optimal control emerges not from extremization but as a necessary consistency condition for constraint stabilization.

Strong claims are made regarding automatic identification of optimal controls through this formalism, bypassing the need for variation and simplifying the derivation of trajectories (see Section 4 of the paper).

Quantum Systems and Hybridization of Methods

Quantum control is treated analogously. Despite the quantum system's evolution being governed by the Schrödinger equation, the method employs classical optimal control tools by introducing Lagrange multipliers (adjoint states) within the classical-analog Lagrangian. All variables are treated as canonically conjugate coordinates, resulting in a formal Poisson bracket structure suitable for subsequent Dirac–Bergmann analysis. Quantum optimal control problems are thus addressed with a unified methodology, and the quantum brachistochrone is solved entirely within this framework.

Exemplary Applications

Classical Brachistochrone

By explicitly reparametrizing the problem and systematizing the Dirac–Bergmann steps, the authors derive a sequence of constraints whose preservation determines the optimal control law dynamically. The culmination is the emergence of the well-known cycloid solution, including explicit parametric representations for the trajectory. This validates both the methodology and its equivalence (in this setting) to PMP-derived solutions.

Quantum Brachistochrone

For the quantum version, the approach formalizes the control Hamiltonian with energy constraints via Lagrange multipliers. Successive constraint consistency steps yield the optimal Hamiltonian and the expected geodesic evolution for quantum state transformation under bounded energy. The important result for the quantum case is the direct emergence of essential constraints such as $\langle \tilde{H} \rangle = 0$ and evolution along great circles in the projective Hilbert space, precisely matching previous results but now derived from the constraint dynamics viewpoint. The method also re-derives the time-optimal geodesic equation for quantum state transfer, showing congruence with earlier literature.

Notable Numerical and Methodological Claims

• The Dirac–Bergmann approach automatically selects optimal controls via constraint enforcement, without need for variational optimization.
• Recovery of classical (cycloid) and quantum (geodesic) brachistochrone trajectories, matching known analytical results.
• All constraints are second-class in both classical and quantum examples, allowing for a reduction to the physically meaningful subspace through explicit Dirac bracket computation.

Implementation Considerations

Applying this framework to practical optimal control problems involves several computational steps:

• Constraint Matrix Inversion: The Dirac–Bergmann algorithm requires inversion of the constraint Poisson bracket matrix; for the examples studied, these are $12 \times 12$ (classical) and $8 \times 8$ (quantum). For higher-dimensional and more complex systems, efficient symbolic or numerical algorithms (possibly leveraging computer algebra systems) will be required.
• Generalization to Practical Systems: The paper’s methodology is directly translatable to engineering and quantum technology scenarios, especially when optimality is tightly coupled to non-trivial dynamical constraints.
• Automatability: The systematic, algebraic structure of the Dirac–Bergmann procedure is amenable to automation, suggesting possible software tool development for control design workflows.
• Extension to Open Systems: The authors note ongoing work connecting their formalism to open quantum systems and Lindblad dynamics, hinting at future directions for algorithmic advances in non-Markovian control.

Implications and Future Developments

The formalism has both theoretical and practical implications:

• Practical Control Design: Elevating controls to dynamical variables could facilitate the design of controllers in settings where control channels are themselves subject to constraints and dynamics (e.g., quantum hardware, adaptive robotics).
• Algorithmic Innovation: The approach provides a foundational structure for algorithmic development in optimal control, particularly for systems with intricate constraints, gauge redundancies, or high-dimensional phase spaces.
• Generalization to Advanced Physics: The methodology shows promise for extension to field-theoretic and gauge-invariant optimal control problems, and possibly to emergent contexts such as fully quantum control theory.
• Bridging Classical and Quantum Control: By using a unified constraint-based approach, the work paves the way for handling hybrid classical–quantum control protocols and analyzing their limitations.

Conclusions

This work presents a rigorous re-framing of optimal control theory using the Dirac–Bergmann constrained dynamics framework. The method’s systematic enforcement of constraint evolution, together with the automatic selection of optimal controls and trajectories, provides a structured, algebraic alternative to variationally-based control analysis. The approach is validated through classical and quantum brachistochrone examples and offers explicit guidance for implementation in more complex, higher-dimensional, or physically constrained systems. Future research will likely address computational bottlenecks and extend applicability to open and fully quantum control scenarios, promising further practical innovations in control engineering and quantum technologies.