A spectral Galerkin approximation of optimal control problem governed by fractional advection–diffusion–reaction equations

Abstract

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.