  In this paper, we investigate a spectral Petrov-Galerkin method for an
optimal control problem governed by a two-sided space-fractional
diffusion-advection-reaction equation. Taking into account the effect of
singularities near the boundary generated by the weak singular kernel of the
fractional operator, we establish the regularity of the problem in weighted
Sobolev space. Error estimates are provided for the presented spectral
Petrov-Galerkin method and the convergence orders of the state and control
variables are determined. Furthermore, a fast projected gradient algorithm with
a quasi-linear complexity is presented to solve the resulting discrete system.
Numerical experiments show the validity of theoretical findings and efficiency
of the proposed fast algorithm.
