SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1140-1165, January 2021.
We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $L^2(0,T;L^2(\Omega))$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.

中文翻译：

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抛物线形分数阶偏微分方程最优控制的误差估计

SIAM数值分析学报，第59卷，第2期，第1140-1165页，2021年1月。
我们考虑分数拉普拉斯算子的积分定义，并针对所谓的分数热方程分析线性二次最优控制问题。还考虑了控制约束。我们得出存在性和唯一性结果，一阶最优性条件以及最优变量的正则性估计。为了离散化状态方程，我们提出了一种完全离散的方案，该方案依赖于时间上的隐式有限差分离散化和空间中的分段线性有限元离散化。我们导出稳定性结果，并得出一个新颖的$ L ^ 2（0，T; L ^ 2（\ Omega））$先验误差估计。基于上述解决方案技术，我们针对最优控制问题提出了一种完全离散的方案，该方案将控制变量与分段常数函数离散化，然后我们得出它的先验误差估计。我们通过一维和二维数值实验来说明该理论。