# Optimal control of elliptic equations with positive measures

## TL;DR

This paper develops a framework for solving optimal control problems involving elliptic equations with positive measure controls, establishing existence, deriving optimality conditions, and proposing a numerical solution method.

## Contribution

It introduces a novel approach to ensure existence of solutions using Radon measures and Fenchel duality, and presents a discretization and semismooth Newton method for computation.

## Key findings

- Existence of optimal controls in Radon measure space under certain constraints.
- Derivation of optimality conditions via Fenchel duality.
- Numerical method combining discretization and semismooth Newton algorithm.

## Abstract

Optimal control problems without control costs in general do not possess solutions due to the lack of coercivity. However, unilateral constraints together with the assumption of existence of strictly positive solutions of a pre-adjoint state equation, are sufficient to obtain existence of optimal solutions in the space of Radon measures. Optimality conditions for these generalized minimizers can be obtained using Fenchel duality, which requires a non-standard perturbation approach if the control-to-observation mapping is not continuous (e.g., for Neumann boundary control in three dimensions). Combining a conforming discretization of the measure space with a semismooth Newton method allows the numerical solution of the optimal control problem.