Semilinear optimal control with Dirac measures

12/14/2021

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by   Enrique Otarola, et al.

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The purpose of this work is to analyze an optimal control problem for a semilinear elliptic partial differential equation (PDE) involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. We analyze the existence of optimal solutions and derive first and, necessary and sufficient, second order optimality conditions. We devise a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze convergence properties of discretizations and obtain an a priori error estimate for the underlying approximation of an optimal control variable.