DOI: 10.1093/imanum/drad091 ISSN: 0272-4979

Semilinear optimal control with Dirac measures

Enrique Otárola
  • Applied Mathematics
  • Computational Mathematics
  • General Mathematics


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.

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