Gradient Projection Method and Stochastic Search for Some Optimal Control Models with Spin Chains. I

This article (I) considers optimal control models for both transferring an excitation from the first spin to the last spin in a chain of N spins and keeping an excitation initially given at the last spin. Each model employs an external controlled parabolic magnetic field represented by two control functions. For these models, the corresponding infinite-dimensional gradients and the projection-type linearized Pontryagin maximum principle are given along with their finite-dimensional analogues. The finite-dimensional gradient projection method in its one-, two-, and three-step forms is represented. A special class of controls for the transfer model is introduced for a genetic algorithm, and the corresponding transfer example for the case of N = 30 spins is given.