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:我们采用了分片拉普特操作员的整体定义,并分析了对利普西茨聚顶顶顶部产生的分片、半线性和椭圆性最佳控制问题的解决方案技术。我们考虑了两种分解战略:半分解计划,其中可允许的控件组不离散;完全分解计划,其中这种组件与小片不变功能分解。作为一个工具步骤,我们为半单形和分级的半线性椭圆性部分方程式(PDEs)的有限元素分解得出误差估计。我们掌握这些估计,我们得出半分解计划误差的界限,并改进文献中为完全分解方案提供的分流性半线性半线性极异方程式(PDEs)的误差估计值。</s>