We adopt the integral definition of the fractional Laplace operator and study, on Lipschitz domains, an optimal control problem that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. Regularity estimates for optimal variables are also analyzed. We devise two strategies of finite element discretization: a semidiscrete scheme where the control variable is not discretized and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniques, we analyze convergence properties of discretizations and derive error estimates.

翻译：我们采用分数阶拉普拉斯算子的积分定义，研究Lipschitz域上的最优控制问题。该问题包含一个分数阶椭圆偏微分方程作为状态方程，控制变量以系数形式进入状态方程，同时考虑控制变量的逐点约束。我们证明了最优解的存在性，并分析了一阶以及必要与充分二阶最优性条件。此外，还分析了最优变量的正则性估计。我们设计了两种有限元离散策略：一种为半离散方案（控制变量不进行离散化），另一种为全离散方案（控制变量采用分段常数函数离散化）。针对这两种求解技术，我们分析了离散格式的收敛性质并推导了误差估计。