SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1344-1371, June 2024.
Abstract. We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first- and necessary and sufficient second-order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.

中文翻译：

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分数拉普拉斯算子的双线性最优控制：分析和离散化


《SIAM 数值分析杂志》，第 62 卷，第 3 期，第 1344-1371 页，2024 年 6 月。

抽象的。采用分数拉普拉斯算子的积分定义，研究了Lipschitz域上的最优控制问题，该问题涉及分数椭圆偏微分方程作为状态方程，控制变量作为系数进入状态方程；还考虑了控制变量的逐点约束。我们建立最优解的存在性并分析一阶和充分二阶最优性条件。还分析了最佳变量的规律性估计。我们开发了两种有限元离散化策略：半离散方案（其中控制变量不离散）和全离散方案（其中控制变量用分段常数函数离散）。对于这两种方案，我们分析了离散化的收敛特性并得出误差估计。