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.

一维研究了由分数对流-扩散-反应方程控制的带有分数分数拉普拉斯积分的最优控制问题的谱Galerkin近似。我们首先导出一阶最优条件，并基于该最优条件分析解的正则性。我们提出了使用加权Jacobi多项式控制问题的频谱Galerkin方案，并证明了状态，伴随状态和控制变量的频谱方法的最佳误差估计。我们还提出了一种拟线性复杂度的快速投影梯度算法，并提供了两个数值示例来验证我们的理论发现。