The control of a two-level open quantum system subject to dissipation due to environment interaction is considered. The evolution of this system is governed by a Lindblad master equation which is augmented by a stochastic term to model the effect of time-continuous measurements. In order to control this stochastic master equation model, a Fokker-Planck control framework is investigated. Within this strategy, the control objectives are defined based on the probability density functions of the two-level stochastic process and the controls are computed as minimizers of these objectives subject to the constraints represented by the Fokker-Planck equation. This minimization problem is characterized by an optimality system including the Fokker-Planck equation and its adjoint. This optimality system is approximated by a second-order accurate, stable, conservative, and positive preserving discretization scheme. The implementation of the resulting open-loop controls is realized with a receding-horizon algorithm over a sequence of time windows. Results of numerical experiments demonstrate the effectiveness of the proposed approach.
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