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

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