  In this article we analyze the optimal control strategy for rotating a
monitored qubit from an initial pure state to an orthogonal state in minimum
time. This strategy is described for two different cost functions of interest
which do not have the usual regularity properties. Hence, as classically smooth
cost functions may not exist, we interpret these functions as viscosity
solutions to the optimal control problem. Specifically we prove their existence
and uniqueness in this weak-solution setting. In addition, we also give bounds
on the time optimal control to prepare any pure state from a mixed state.
