Optimal rotation control for a qubit subject to continuous measurement

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.