  This paper presents a novel methodology to tackle feedback optimal control
problems in scenarios where the exact state of the controlled process is
unknown. It integrates data assimilation techniques and optimal control solvers
to manage partial observation of the state process, a common occurrence in
practical scenarios. Traditional stochastic optimal control methods assume full
state observation, which is often not feasible in real-world applications. Our
approach underscores the significance of utilizing observational data to inform
control policy design. Specifically, we introduce a kernel learning backward
stochastic differential equation (SDE) filter to enhance data assimilation
efficiency and propose a sample-wise stochastic optimization method within the
stochastic maximum principle framework. Numerical experiments validate the
efficacy and accuracy of our algorithm, showcasing its high efficiency in
solving feedback optimal control problems with partial observation.
