The quantum dark side of the optimal control theory
Contreras
M
author
Pena
J
P
author
2019
English
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 Schrodinger 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 h -> 0 of these three different quantum theories. Also, the exact relation of the phase S(x, t) of the wave function Psi(x, t) = e(i/hS(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. (C) 2018 Elsevier B.V. All rights reserved.
Optimal control theory
Pontryagin's equations
Hamilton-Jacobi-Bellman equation
Constrained systems
Dirac's method
Quantum mechanics
WOS:000452941100042
exported from refbase (http://ficpubs.uai.cl/show.php?record=952), last updated on Mon, 07 Jan 2019 13:25:11 +0000
text
http://ficpubs.uai.cl/files/903_G.+Pena2019.pdf
10.1016/j.physa.2018.09.134
Contreras+Pena2019
Physica A-Statistical Mechanics And Its Applications
Physica A
2019
Elsevier Science Bv
continuing
periodical
academic journal
515
450
473
0378-4371