AI 中文总结
针对带有谱分数阶拉普拉斯算子和Dirac测度线性组合的椭圆方程，研究其偏微分方程约束优化问题，证明最优解的存在唯一性，推导一阶最优性条件，提出有限元离散化方法并推导先验误差界。
AI 中文摘要
我们研究了一个偏微分方程约束优化问题，该问题针对带有谱分数阶拉普拉斯算子的椭圆方程，其强迫项为Dirac测度的线性组合；控制量是这些奇异源的振幅。我们证明了最优解的存在性与唯一性，并推导了一阶最优性条件。随后，我们提出了一种基于有限元的离散化方法。由于容许控制集是有限维的，控制变量本身无需离散化。最后，我们推导了先验误差界。
英文摘要
We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds
