A bilinear pointwise tracking optimal control problem for a semilinear elliptic PDE

TL;DR

This paper studies a bilinear optimal control problem for a semilinear elliptic PDE with pointwise state tracking, establishing existence, regularity, and optimality conditions for solutions in two and three dimensions.

Contribution

It introduces a comprehensive analysis of the control problem, including existence, regularity, and optimality conditions, specifically addressing pointwise state tracking with Dirac measures.

Findings

Existence of optimal solutions in Lipschitz domains.

Optimal controls belong to H^1(Ω).

In convex polygons, controls are Lipschitz continuous.

Abstract

We consider a bilinear optimal control problem with pointwise tracking for a semilinear elliptic PDE in two and three dimensions. The control variable enters the PDE as a (reaction) coefficient and the cost functional contains point evaluations of the state variable. These point evaluations lead to an adjoint problem with a linear combination of Dirac measures as a forcing term. In Lipschitz domains, we derive the existence of optimal solutions and analyze first and necessary and sufficient second order optimality conditions. We also prove that every locally optimal control $\bar u$ belongs to $H^1(\Omega)$. Finally, assuming that the domain $\Omega \subset \mathbb{R}^2$ is a convex polygon, we prove that $\bar u \in C^{0,1}(\bar \Omega)$.