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In this paper, we investigate a spectral Petrov-Galerkin method for anoptimal control problem governed by a two-sided space-fractionaldiffusion-advection-reaction equation. Taking into account the effect ofsingularities near the boundary generated by the weak singular kernel of thefractional operator, we establish the regularity of the problem in weightedSobolev space. Error estimates are provided for the presented spectralPetrov-Galerkin method and the convergence orders of the state and controlvariables are determined. Furthermore, a fast projected gradient algorithm witha quasi-linear complexity is presented to solve the resulting discrete system.Numerical experiments show the validity of theoretical findings and efficiencyof the proposed fast algorithm.

On spectral Petrov-Galerkin method for solving optimal control problem governed by a two-sided fractional diffusion equation