The Dirac--Bergmann approach to optimal control theory
Davit Aghamalyan, Aleek Maity, Varun Narasimhachar, V V Sreedhar
TL;DR
This paper introduces a Dirac–Bergmann constrained-dynamics framework for time-optimal control in both classical and quantum settings, offering an alternative to Pontryagin's maximum principle by deriving optimal controls from the closure of the Poisson-bracket algebra of all constraints. By promoting controls and adjoint variables to dynamical phase-space coordinates and using Dirac brackets to enforce second-class constraints, the method automatically identifies optimal controls without variational extremization and provides reduced equations of motion on the physical subspace. The authors demonstrate the approach on two archetypal problems: the classical brachistochrone and the quantum brachistochrone, including open-system generalizations via Lindblad dynamics, showing that the framework reproduces known minimal-time solutions and yields tractable equations for dissipative control. The work highlights potential advantages for constrained and open quantum control and points to computational and theoretical expansions, such as fully quantum control and non-local control schemes, aided by automated constraint-handling tools.
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 automatically, through the closure of the Poission Bracket algebra of the full set of constraints and the Hamiltonian. The efficacy of our framework is demonstrated through two quintessential examples: (1) the classical brachistochrone problem and (2) the time-optimal control of a generic quantum system, relevant for quantum technological applications. In the latter example, both closed and open quantum systems are discussed.