We study the numerical approximation of linear–quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed. Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.

中文翻译：

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分数阶拉普拉斯方程的最优控制问题的有限元法

我们研究分数阶拉普拉斯方程及其谱定义下的线性二次最优控制问题的数值逼近。我们使用Balakrishnan公式的离散化来计算状态方程的近似值，该方程式是基于空间中的有限元离散化和另外涉及的积分的Sinc正交逼近。针对所得线性系统的数值解，提出了一种量身定制的方法。关于最优控制问题的离散化，我们考虑两种方案。第一个是变分方法，其中控制集不离散化，第二个是完全离散方案，其中控制是通过分段常数函数离散化的。