Reachable sets of an open quantum system driven by coherent control

We will discuss the structure of optimal control in the time minimization problem for a two-level open quantum system (qubit) using coherent control. This system is remarkable in that the optimal control must essentially be impulsive (in the class of measurable functions, there is usually no optimal control). However, we have managed to prove that the impulses can only occur at the first and last moments in time. The proof relies on a surprising fact: it turns out that there exist values of the adjoint multiplier at the initial time for which the Pontryagin maximum principle has no solution! This allowed us to obtain both upper and lower bounds for the optimal time of motion. Furthermore, we propose an explicit formula for a control with 4 impulses that is close to optimal and yields practically optimal time of motion.