  Recently two papers [K. Jacobs, Phys. Rev. A {\bf 67}, 030301(R) (2003); H.
M. Wiseman and J. F. Ralph, New J. Physics {\bf 8}, 90 (2006)] have derived
control strategies for rapid purification of qubits, optimized with respect to
various goals. In the former paper the proof of optimality was not
mathematically rigorous, while the latter gave only heuristic arguments for
optimality. In this paper we provide rigorous proofs of optimality in all
cases, by applying simple concepts from optimal control theory, including
Bellman equations and verification theorems.
