# Sufficient conditions for unique global solutions in optimal control of semilinear equations with $C^1-$nonlinearity

## TL;DR

This paper establishes sufficient conditions ensuring that solutions to certain semilinear elliptic optimal control problems are globally optimal, extending previous results and providing explicit criteria at both continuous and discrete levels.

## Contribution

It generalizes prior work by deriving explicit global optimality conditions for semilinear elliptic control problems with $C^1$ nonlinearities, including the case of $ ext{sign}(s)$ nonlinearity.

## Key findings

- Derived explicit global optimality conditions for continuous problems.
- Extended conditions to discrete problem settings.
- Numerical tests demonstrate practical applicability.

## Abstract

We consider a $C^1-$semilinear elliptic optimal control problem possibly subject to control and/or state constraints. Generalizing previous work we provide a condition which guarantees that a solution of the necessary first order conditions is a global minimum. A similiar result also holds at the discrete level where the corresponding condition can be evaluated explicitly. Our investigations are motivated by G\"unter Leugering, who raised the question whether our previous results can be extended to the nonlinearity $\phi(s)=s|s|$. We develop a corresponding analysis and present several numerical test examples demonstrating its usefulness in practice.