Numerical solution to 3D bilinear Fokker–Planck control problem

A characterization and numerical scheme to control problem governed by a three-dimensional (3D) time-dependent Fokker–Planck (FP) equation is presented. We formulate a control formulation that controls the drift of the stochastic (FP) process. In this way, the probability density function attains a specific configuration. Moreover, a FP control strategy for collective motion is investigated and first-order optimality conditions are presented. On staggered grids, the Chang–Cooper discretization scheme that ensures the positivity, second-order accuracy, and conservativeness to the FP equation is employed to the discretized state (respectively adjoint) system. Furthermore, a line search strategy is applied to update the control variable. Results of numerical experiments show the efficiency of the proposed numerical scheme to stochastic (FP) control problems.