On the computation of optimal state transfers with application to the control of quantum spin systems

摘要

We study the problem of finding a trajectory from an initial state to a desired final state while minimizing an integral cost. We use an unconstrained optimization approach and obtain the desired terminal constraint through the use of a novel combination of terminal penalty and root finding. This approach is developed in detail for the linear quadratic optimal transfer problem, where the availability of closed form solutions provides key insights. An important use is made of the notion of positive definiteness of a quadratic functional - a significant concept for second order sufficiency conditions. The development continuous with a number of important results for the nonlinear optimal transfer problem. We briefly discuss the use of this approach for the computation of trajectories and propagators for quantum mechanical systems. The fact that the system evolves in a compact manifold alleviates many (stability related) boundedness difficulties that commonly affect trajectory optimization computations. On the other hand, we find that it is essential to respect the state manifold constraint when computing, for example, the second derivative of the terminal cost for use in a Newton descent method.