# Error analysis for global minima of semilinear optimal control problems

## TL;DR

This paper develops an error estimate for discrete global solutions in semilinear elliptic PDE optimal control problems, providing conditions to verify global optimality and demonstrating convergence of solutions.

## Contribution

It introduces an explicit, verifiable condition for global optimality at the discrete level and proves convergence of discrete solutions to the continuous global minimum.

## Key findings

- Explicit condition for global optimality can be evaluated at the discrete level
- Discrete solutions converge to the continuous global solution under certain conditions
- Numerical experiments confirm the theoretical error estimates

## Abstract

In [1] we consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization, where we provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. With the present work we complement our investigations of [1] in that we prove an error estimate for those discrete global solutions. Numerical experiments confirm our analytical findings.