  In this paper, we address the so-called general Fokker-Planck control problem
for discrete-time first-order linear systems. Unlike conventional treatments,
we don't assume the distributions of the system states to be Gaussian. Instead,
we only assume the existence and finiteness of the first several order power
moments of the distributions. It is proved in the literature that there doesn't
exist a solution, which has a form of conventional feedback control, to this
problem. We propose a moment representation of the system to turn the original
problem into a finite-dimensional one. Then a novel feedback control term,
which is a mixture of a feedback term and a Markovian transition kernel term is
proposed to serve as the control input of the moment system. The states of the
moment system are obtained by maximizing the smoothness of the state
transition. The power moments of the transition kernels are obtained by a
convex optimization problem, of which the solution is proved to exist and be
unique. Then they are mapped back to the probability distributions. The control
inputs to the original system are then obtained by sampling from the realized
distributions. Simulation results are provided to validate our algorithm in
treating the general discrete-time Fokker-Planck control problem.
