Two-level method for a time-independent Fokker–Planck control problem

A time-independent Fokker–Planck (FP) control problem and a two-level numerical method are presented. We aim to formulate a control problem that controls the drift of the stochastic process so that the probability density function (PDF) attains a specific steady-state configuration. First-order optimality conditions, which characterize the solution of the control problem, are discretized by the Chang-Cooper (CC) scheme. For positivity and conservativeness of a PDF in the stationary FP control formulation and discretization, we take advantage of CC-scheme. We investigate a two-grid method with coarsening by a factor-of-three strategy. It is found that the coarsening by a factor-of-three strategy simplifies the inter-grid transfer operators and hence the computations. We present several numerical experiments to show the effectiveness of the proposed two-level framework to solve Fokker–Planck or stochastic models control problems with and without control-constrained.