Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equation

TL;DR

This paper studies error estimates for pointwise tracking optimal control problems governed by semilinear elliptic equations, analyzing discretization strategies and convergence properties.

Contribution

It introduces two discretization methods for the control problem and provides convergence analysis and error estimates for both.

Findings

Both discretization schemes converge under certain conditions.

Error estimates are derived for the semidiscrete and fully discrete solutions.

Optimality conditions are established for the control problem.

Abstract

We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. We devise two strategies of discretization to approximate a solution of the optimal control problem: a semidiscrete scheme where the control variable is not discretized -- the so-called variational discretization approach -- and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniques, we analyze convergence properties of discretizations and derive error estimates.