The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable.

中文翻译：

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采用狄拉克测度的半线性最优控制

这项工作的目的是研究以狄拉克测度的线性组合作为强迫项的半线性椭圆偏微分方程的最优控制问题；控制变量对应于此类奇异源的幅度。我们分析最优解的存在性，并推导出一阶最优性条件和充分必要条件二阶最优性条件。我们开发了一种用连续分段线性有限元离散化状态和伴随方程的求解技术；控制变量已经是离散的。我们分析离散化的收敛特性，并在二维中获得最优控制变量的基本近似的先验误差估计。