Deterministic classical limit of the optimal control problem of quantum particles with spin

We study the optimal control problem applied to a gas of particles with spin confined in a material with Rashba spin-orbit coupling effect, in the presence of an external magnetic field. The evolution of the particle gas is described in the Wigner formalism. We investigate the classical limit of the optimal control problem, and we prove the convergence of the solution of the quantum problem toward the solution of a simplified optimal control model, based on an ODE description of the particle gas in terms of a single spin vector traveling along a classical trajectory in the phase-space.