Optimal Control for Steady Circulation of a Diffusion Process via Spectral Decomposition of Fokker-Planck Equation

TL;DR

This paper develops a low-cost spectral method for optimal control of 2D diffusion processes, enabling desired circulation and faster convergence to steady states, demonstrated through numerical simulations.

Contribution

It introduces a spectral decomposition approach to formulate an efficient optimal control problem for Fokker-Planck equations, achieving targeted circulation and accelerated convergence.

Findings

Successfully achieves desired circulation in diffusion processes.

Accelerates convergence to the stationary distribution.

Demonstrates effectiveness through numerical simulations.

Abstract

We present a formulation of an optimal control problem for a two-dimensional diffusion process governed by a Fokker-Planck equation to achieve a nonequilibrium steady state with a desired circulation while accelerating convergence toward the stationary distribution. To achieve the control objective, we introduce costs for both the probability density function and flux rotation to the objective functional. We formulate the optimal control problem through dimensionality reduction of the Fokker-Planck equation via eigenfunction expansion, which requires a low-computational cost. We demonstrate that the proposed optimal control achieves the desired circulation while accelerating convergence to the stationary distribution through numerical simulations.