In this work, we use the integral definition of the fractional Laplace operator and study a sparse optimal control problem involving a fractional, semilinear, and elliptic partial differential equation as state equation; control constraints are also considered. We establish the existence of optimal solutions and first and second order optimality conditions. We also analyze regularity properties for optimal variables. We propose and analyze two finite element strategies of discretization: a fully discrete scheme, where the control variable is discretized with piecewise constant functions, and a semidiscrete scheme, where the control variable is not discretized. For both discretization schemes, we analyze convergence properties and a priori error bounds.

翻译：本文利用分数阶拉普拉斯算子的积分定义，研究了一类涉及分数阶半线性椭圆型偏微分方程作为状态方程的稀疏最优控制问题，同时考虑了控制约束条件。我们建立了最优解的存在性以及一阶和二阶最优性条件，并分析了最优变量的正则性性质。我们提出并分析了两种有限元离散策略：一种是完全离散格式（其中控制变量采用分片常数函数离散），另一种是半离散格式（其中控制变量不进行离散）。针对这两种离散格式，我们分析了收敛性性质并给出了先验误差界。