Spectral Galerkin Method for Optimal Control of Stochastic Fractional Laplacian Equations with White Noise on a Disk
Shengyue Li, Wanrong Cao · SIAM Journal on Control and Optimization · 2025
Abstract. We investigate a linear-quadratic optimal control problem governed by a two-dimensional stochastic fractional Laplacian equation, incorporating additive white noise over a disk. Regularity estimates are meticulously derived for both the white noise and the optimal control variables within a weighted Sobolev space. To discretize the state equation, we propose a spectral Galerkin method, leveraging a truncated spectral approximation of the white noise. Rigorous consistency analysis and error estimates are established for the numerical solution. Building on these insights, we further develop the spectral Galerkin approach for the optimal control problem and derive a priori error estimates for the approximated optimal variables. Numerical experiments are conducted to validate our theoretical findings.