Numerical solution to 3D bilinear Fokker–Planck control problem

ABSTRACT

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.

KEYWORDS:

• Fokker–Planck equation
• optimal control
• stochastic process
• Chang–Cooper scheme
• staggered grid

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q84
• 35Q93
• 49K20
• 65N06

Acknowledgments

The author would like to thank the anonymous referees for their useful comments and suggestions that improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.