We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains 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 develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized, and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the 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。该问题涉及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，分析了一阶及必要充分二阶最优性条件，并研究了最优变量的正则性估计。我们发展了两种有限元离散策略：控制变量未被离散的半离散格式，以及控制变量采用分片常数函数离散的全离散格式。针对这两种格式，我们分析了离散化的收敛性质并推导了误差估计。