  The problem of optimal control of non-Markovian open quantum system via weak
measurement is presented. Based on the non-Markovian master equation, we
evaluate exactly the non-Markovian effect on the dynamics of the system of
interest interacting with a dissipative reservoir. We find that the
non-Markovian reservoir has dual effects on the system: dissipation and
backaction. The dissipation exhausts the coherence of the quantum system,
whereas the backaction revives it. Moreover, we design the control Hamiltonian
with the control laws attained by the stochastic optimal control and the
corresponding optimal principle. At last, we considered the exact decoherence
dynamics of a qubit in a dissipative reservoir composed of harmonic
oscillators, and demonstrated the effectiveness of our optimal control
strategy. Simulation results showed that the coherence will completely lost in
the absence of control neither in non-Markovian nor Markovian system. However,
the optimal feedback control steers it to a stationary stochastic process which
fluctuates around the target. In this case the decoherence can be controlled
effectively, which indicates that the engineered artificial reservoirs with
optimal feedback control may be designed to protect the quantum coherence in
quantum information and quantum computation.
