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.

提出了一个与时间无关的 Fokker-Planck (FP) 控制问题和一个两级数值方法。我们的目标是制定一个控制问题来控制随机过程的漂移，以便概率密度函数 (PDF) 获得特定的稳态配置。表征控制问题解决方案的一阶最优条件由 Chang-Cooper (CC) 方案离散化。对于固定 FP 控制公式和离散化中 PDF 的积极性和保守性，我们利用 CC 方案。我们研究了一种通过三因子策略进行粗化的双网格方法。发现通过三倍策略的粗化简化了网格间传输算子，从而简化了计算。