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

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.

Figures (5)

• Figure 1: Stationary distribution $\rho_{\mathrm{s}}$ of the uncontrolled FPE.
• Figure 2: Desired flux rotation $\omega_{\mathrm{d}}$.
• Figure 3: Optimal control inputs. (a) Optimal control input $u_{1}$. The horizontal line represents $u_{1}(t) = 0$. (b) Optimal control input $u_{2}$. The horizontal line represents $u_{2}(t) = 1$.
• Figure 4: Comparison of the $L^{2}\left(\Omega;\rho_{\mathrm{s}}^{-1}\right)$ norm between the uncontrolled case and optimal control case. (a) $L^{2}\left(\Omega;\rho_{\mathrm{s}}^{-1}\right)$ norm between the stationary and current distribution. (b) $L^{2}\left(\Omega;\rho_{\mathrm{s}}^{-1}\right)$ norm between the desired and current flux rotation.
• Figure 5: The behavior of the flux obtained from the particles under control at the final time on the $xy$ plane. The length of the arrows represents the flux speed.