Turnpike solutions in optimal control problems for quantum-mechanical systems

We consider the problem of optimal impulse control for a quantum-mechanical spin sequence as a system of oscillators. We study this problem with a double transformation to a derivative problem in which phase variables are oscillator amplitudes. As a result, we get the turnpike solutions. For special (turnpike) sets of boundary conditions they correspond exactly to generalized solutions of the original problem, while for other conditions they can be used as initial approximations for iterative procedures. This problem is a generalization of the special case of two oscillators which we study exhaustively and use as an illustrative example.