Journal of Computational and Applied Mathematics ( IF 2.037 ) Pub Date : 2020-10-08 , DOI: 10.1016/j.cam.2020.113233 Fangyuan Wang; Zhongqiang Zhang; Zhaojie Zhou
A spectral Galerkin approximation of a optimal control problem governed by a fractional advection–diffusion–reaction equation with integral fractional Laplacian is investigated in 1D. We first derive a first-order optimality condition and analyze the regularity of the solution based on this optimality condition. We present a spectral Galerkin scheme for the control problem using weighted Jacobi polynomials and prove optimal error estimates of the spectral method for state, adjoint state and control variables. We also propose a fast projected gradient algorithm of quasilinear complexity and present two numerical examples verifying our theoretical findings.