论文标题
分数半线性最佳控制：最佳条件，收敛和误差分析
Fractional semilinear optimal control: optimality conditions, convergence, and error analysis
论文作者
论文摘要
我们采用了分数拉普拉斯操作员的整体定义，并分析了分数半椭圆形偏微分方程（PDE）的最佳控制问题；还考虑了控制约束。我们建立了分数半线性椭圆PDE的适当性，并分析了规则性特性和合适的有限元离散性。在我们的最佳控制问题的设置中，我们得出了最佳解决方案的存在以及一阶和二阶最佳条件；还分析了最佳变量的规律性估计。我们设计了一个完全离散的方案，该方案用分段常数函数近似控制变量。通过连续的分段线性有限元元素离散状态和伴随方程。我们分析离散化的收敛性能并得出先验误差估计。
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.
