### 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 original | English |
---|---|

Páginas (desde-hasta) | 450-473 |

Número de páginas | 24 |

Publicación | Physica A: Statistical Mechanics and its Applications |

Volumen | 515 |

DOI | |

Estado | Published - 1 feb 2019 |

Publicado de forma externa | Sí |

### Huella dactilar

### ASJC Scopus subject areas

- Statistics and Probability
- Condensed Matter Physics

### Citar esto

*Physica A: Statistical Mechanics and its Applications*,

*515*, 450-473. https://doi.org/10.1016/j.physa.2018.09.134

}

*Physica A: Statistical Mechanics and its Applications*, vol. 515, pp. 450-473. https://doi.org/10.1016/j.physa.2018.09.134

**The quantum dark side of the optimal control theory.** / G., Mauricio Contreras; Peña, Juan Pablo.

Resultado de la investigación: Article

TY - JOUR

T1 - The quantum dark side of the optimal control theory

AU - G., Mauricio Contreras

AU - Peña, Juan Pablo

PY - 2019/2/1

Y1 - 2019/2/1

N2 - 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.

AB - 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.

KW - Constrained systems

KW - Dirac's method

KW - Hamilton–Jacobi–Bellman equation

KW - Optimal control theory

KW - Pontryagin's equations

KW - Quantum mechanics

UR - http://www.scopus.com/inward/record.url?scp=85054429941&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2018.09.134

DO - 10.1016/j.physa.2018.09.134

M3 - Article

AN - SCOPUS:85054429941

VL - 515

SP - 450

EP - 473

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

ER -