  Optimal control theory, also known as Pontryagin's Maximum Principle, is
applied to the quantum parameter estimation in the presence of decoherence. An
efficient procedure is devised to compute the gradient of quantum Fisher
information with respect to the control parameters and is used to construct the
optimal control protocol. The proposed procedure keeps the control problem in
the time-invariant form so that both first-order and second-order optimality
conditions derived from Pontryagin's Maximum Principle apply; the second-order
condition turns out to be crucial when the optimal control contains singular
arcs. Concretely we look for the optimal control that maximizes quantum Fisher
information for "twist and turn" problem. We find that the optimal control is
singular without dissipation but can become unbounded once the quantum
decoherence is introduced. An amplitude constraint is needed to guarantee a
bounded solution. With quantum decoherence, the maximum quantum Fisher
information happens at a finite time due to the decoherence, and the asymptotic
value depends on the specific decoherence channel and the control of
consideration.
