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.

翻译：在本文中,我们调查了一种光谱Petrov-Galerkin 方法,以最佳控制问题为最佳控制,该方法由双向空间折射扩散-反射-反应方程式管理。考虑到分数操作员的微弱单核内核在边界附近产生的奇点效应,我们确定了在加权Sobolev空间中问题的规律性。提供了Petrov-Galerkin光谱方法的误差估计,确定了州和控制变量的趋同顺序。此外,还提出了一种具有准线性复杂性的快速预测梯度算法,以解决由此产生的离散系统。数字实验显示了拟议的快速算法的理论结果和效率的有效性。