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

Oleg V. Morzhin

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.

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

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 -dimensional spin chain, and correspondingly adapts a projection-type condition for optimality, gradient projection method (GPM). For the case , 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 and the final time is not assigned.

Paper Structure

This paper contains 6 sections, 14 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction-II
  2. Finite-Dimensional Gradients, Projection-Type Condition for Optimality, and GPM Forms
  3. GPM, GA for the Transfer and Keeping Problems, $N=3$
  4. Non-Assigned $T$ for a Transfer. The Case $N=20$ and GA
  5. Some Methodological Comments
  6. Conclusion-II

Figures (3)

  • Figure 1: For Example \ref{['example1']}: (a), (b), (g)--(i) in Case 1, the provided by GPM-2S excitement's exchange in the terms of the resulting $\{\psi_m\}_{m=1}^3$ and controls; (c), (e), (d) in Cases 1, 2, 3, the lg-values of $\Phi_1,~I_1$ at the GPM iterations; (j)--(l) in Case 4, the resulting $\{\psi_m\}_{m=1}^3$ obtained via GPM-2S; (f) in Cases 1--6, the signal is (approximately) concentrated at the last spin site.
  • Figure 2: For Example 2: (a)--(c) the results obtained via GA for the approximated (as PConst.) class of controls (2.11); (d), (e) the resulting controls and the iterative behavior of GPM-3S; (f) the iterative behavior of GPM-2S.
  • Figure 3: For Example 3 with $N=20$, the GA results (before adding normally distributed noises): (a), (b) transfer in the terms of $|\psi_m^u(t)|^2$; (c) resulting controls.