Gradient projection method and stochastic search for some optimal control models with spin chains. II

TL;DR

This paper extends previous work on optimal control of spin chains by deriving finite-dimensional gradients, adapting a gradient projection method, and demonstrating the effectiveness of GPM and genetic algorithms on problems with up to 20 spins.

Contribution

It introduces finite-dimensional gradients for infinite-dimensional problems and adapts the gradient projection method for spin chain control models.

Findings

GPM and genetic algorithms successfully solved transfer and keeping problems.

Multi-step GPM outperforms one-step GPM.

Algorithms effectively handled large spin chains with N=20.

Abstract

This article (II) continues the research described in [Morzhin O.V. Gradient projection method and stochastic search for some optimal control models with spin chains. I (submitted)] (Article I), derives the needed finite-dimensional gradients corresponding to the infinite-dimensional gradients obtained in Article I, both for transfer and keeping problems at a certain $N$-dimensional spin chain, and correspondingly adapts a projection-type condition for optimality, gradient projection method (GPM). For the case $N=3$, the given in this article examples together with Example 3 in Article I show that: a) the adapted GPM and genetic algorithm (GA) successfully solved numerically the considered transfer and keeping problems; b) the two- and three-step GPM forms significantly surpass the one-step GPM. Moreover, GA and a special class of controls were successfully used in such the transfer problem that $N=20$ and the final time is not assigned.