Dynamic optimization and its relation to classical and quantum constrained systems

Abstract

We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two second-class constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closed-loop λ-strategy, the optimality condition for the action gives a consistency relation, which is associated to the Hamilton-Jacobi-Bellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Ψ(x, t) = e iS(x,t) in the quantum Schrödinger equation, a non-linear partial equation is obtained for the S function. For the right-hand side quantization, this is the Hamilton-Jacobi-Bellman equation, when S(x, t) is identified with the optimal value function. Thus, the Hamilton-Jacobi-Bellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem.

Behavioral Capital Theory via Canonical Quantization

“…An important precedent to our work is that of Contreras et al [ 36 , 37 ] who made an extensive study of the relationship between optimal control theory and quantum theory. In this work, they built on their observation of the equivalence between the Pontryagin equations and those of classical constrained systems and showed that the Schröedinger equation that follows from the quantization of the classical system is dynamically equivalent to the Hamilton–Bellman–Jacobi equation.…”

Loop quantum gravity with optimal control path integral, and application to black hole tunneling

“…To illustrate the subject, an application to the harmonic oscillator is presented. For further technical details, concerning the relations between standard physics and optimal control, we refer to [27].…”

“…Remark that this is unrelated to stochastic optimal control where path integrals are also used. The path integral of the optimal control problem is formally given by [27]:…”

“…Since, Optimal Control Theory (OCT) has been one of the most successful theories of mathematics, with applications in engineering, aerospace [24], robotics, finance, quantum technologies [25,26],... Despite its close relation to the well-known variational principles used in classical physics, optimal control has a few small differences that allows us to tackle more more general situations [27]. In particular it can handle dynamical problems without natural canonical adjoint state of the generalized coordinates.…”

“…In particular it can handle dynamical problems without natural canonical adjoint state of the generalized coordinates. More recently, several papers have outlined the precise relationship between OCT and classical/quantum physics [27][28][29][30][31]. As an example, one can explain how quantum mechanics can be understood through stochastic optimization on space-times [32,33].…”