Semilinear optimal control with Dirac measures

TL;DR

This paper investigates an optimal control problem involving semilinear elliptic PDEs with Dirac measures, establishing existence, optimality conditions, and convergence analysis for discretized solutions.

Contribution

It introduces a novel analysis of control problems with Dirac measures, deriving optimality conditions and providing convergence and error estimates for discretized solutions.

Findings

Existence of optimal solutions established

First and second order optimality conditions derived

Convergence and error estimates for discretized controls obtained

Abstract

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.