We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of discretization: a semidiscrete scheme where the admissible control set is not discretized and a fully discrete scheme where such a set is discretized with piecewise constant functions. As an instrumental step, we derive error estimates for finite element discretizations of fractional semilinear elliptic partial differential equations (PDEs) on quasi-uniform and graded meshes. With these estimates at hand, we derive error bounds for the semidiscrete scheme and improve the ones that are available in the literature for the fully discrete scheme.

翻译：本文采用分数阶拉普拉斯算子的积分定义，分析了Lipschitz多面体上分数阶半线性椭圆最优控制问题的求解技术。我们考虑了两种离散化策略：一种为半离散格式（控制集未被离散化），另一种为全离散格式（控制集采用分片常数函数离散化）。作为关键步骤，我们推导了拟均匀网格和渐变网格上分数阶半线性椭圆偏微分方程（PDE）有限元离散化的误差估计。基于这些估计，我们进一步推导了半离散格式的误差界，并改进了文献中已有的全离散格式误差界。