We adopt the integral definition of the fractional Laplace operator and analyze an optimal control problem for a fractional semilinear elliptic partial differential equation (PDE); control constraints are also considered. We establish the well-posedness of fractional semilinear elliptic PDEs and analyze regularity properties and suitable finite element discretizations. Within the setting of our optimal control problem, we derive the existence of optimal solutions as well as first and second order optimality conditions; regularity estimates for the optimal variables are also analyzed. We devise a fully discrete scheme that approximates the control variable with piecewise constant functions; the state and adjoint equations are discretized with continuous piecewise linear finite elements. We analyze convergence properties of discretizations and derive a priori error estimates.

翻译：我们采用分解拉普特操作员的整体定义,并分析分解半线椭圆偏差部分方程(PDE)的最佳控制问题; 也考虑控制限制; 我们确定分解半线性椭圆极半线性PDE的稳妥性, 分析常规性特性和适当的有限元素分解。 在确定最佳控制问题时, 我们得出最佳解决方案以及第一和第二级最佳条件; 对最佳变量的定期性估计也进行了分析。 我们设计了一个完全离散的方案, 将控制变量与片断常数功能相近; 状态和连接方程与连续的片断线性线性限定元素分离。 我们分析离散的趋同特性, 并得出先验误估计值。