The quantum dark side of the optimal control theory

Mauricio Contreras G., Juan Pablo Peña

Resultado de la investigación: Contribución a una revistaArtículo

Resumen

In a recent article, a generic optimal control problem was studied from a physicist's point of view (Contreras et al. 2017). Through this optic, the Pontryagin equations are equivalent to the Hamilton equations of a classical constrained system. By quantizing this constrained system, using the right ordering of the operators, the corresponding quantum dynamics given by the Schrödinger equation is equivalent to that given by the Hamilton–Jacobi–Bellman equation of Bellman's theory. The conclusion drawn there were based on certain analogies between the equations of motion of both theories. In this paper, a closer and more detailed examination of the quantization problem is carried out, by considering three possible quantization procedures: right quantization, left quantization, and Feynman's path integral approach. The Bellman theory turns out to be the classical limit ħ→0 of these three different quantum theories. Also, the exact relation of the phase S(x,t) of the wave function Ψ(x,t)=e[Formula presented]S(x,t) of the quantum theory with Bellman's cost function J+(x,t) is obtained. In fact, S(x,t) satisfies a ‘conjugate’ form of the Hamilton–Jacobi–Bellman equation, which implies that the cost functional J+(x,t) must necessarily satisfy the usual Hamilton–Jacobi–Bellman equation. Thus, the Bellman theory effectively corresponds to a quantum view of the optimal control problem.

Idioma originalInglés
Páginas (desde-hasta)450-473
Número de páginas24
PublicaciónPhysica A: Statistical Mechanics and its Applications
Volumen515
DOI
EstadoPublicada - 1 feb 2019
Publicado de forma externa

Áreas temáticas de ASJC Scopus

  • Estadística y probabilidad
  • Física de la materia condensada

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