The Dirac--Bergmann approach to optimal control theory

TL;DR

This paper introduces a new optimal control framework using the Dirac--Bergmann algorithm, which automatically derives solutions for classical and quantum systems without variational methods, demonstrated through classical and quantum examples.

Contribution

It presents a novel control approach based on the Dirac--Bergmann algorithm, bypassing traditional variational techniques in favor of algebraic closure methods.

Findings

Successfully applied to classical brachistochrone problem

Extended to time-optimal control of quantum systems

Demonstrated effectiveness for both closed and open quantum systems

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.