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

This article (I) considers the known optimal control model of a quantum information transfer along a spin chain with controlled external parabolic magnetic field, with an arbitrary length. The article adds certain lower and upper pointwise constraints on controls, adds the problem of keeping the signal at the last spin, considers various classes of controls. For these problems under piecewise continuous controls, the projection-type linearized Pontryagin maximum principle, gradient projection method’s constructions in its one- and two- and three-step forms were adapted by analogy with [Morzhin O.V., Pechen A.N. J. Phys. A: Math. Theor., 2025]. Moreover, an example with a genetic algorithm’s successful use under a special class of controls is given.

💡 Research Summary

This paper investigates optimal control strategies for quantum information transfer along a spin‑chain subjected to an externally controlled parabolic magnetic field. The authors extend the well‑known model by imposing pointwise lower and upper bounds on the two control functions—magnetic‑field intensity (u₁) and its shifted position (u₂)—and by adding a “signal‑keeping” objective that requires the quantum state to remain localized at the last spin throughout the evolution interval.

The dynamics are described by the Schrödinger equation dψ/dt = –i