Optimal Control for Steady Circulation of a Diffusion Process via Spectral Decomposition of Fokker-Planck Equation
This work addresses the control of diffusion processes for applications in fields like physics or engineering, but it appears incremental as it builds on existing spectral methods for Fokker-Planck equations.
The authors tackled the problem of controlling a two-dimensional diffusion process to achieve a desired circulation and accelerate convergence to a stationary distribution, using an optimal control formulation based on spectral decomposition of the Fokker-Planck equation, with numerical simulations demonstrating successful achievement of these goals.
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.