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.

翻译：本文旨在研究一类以狄拉克测度线性组合作为源项的半线性椭圆型偏微分方程的最优控制问题；控制变量对应此类奇异源的幅值。我们分析了最优解的存在性，并推导了一阶及必要且充分的二阶最优性条件。我们开发了一种求解技术，采用连续分片线性有限元对状态方程和伴随方程进行离散化；控制变量本身已是离散形式。我们分析了离散化的收敛性质，并在二维情形下获得了最优控制变量近似解的先验误差估计。