Global minima for semilinear optimal control problems

TL;DR

This paper establishes a verifiable condition to identify global minima in semilinear elliptic PDE control problems, ensuring convergence of discrete solutions to the global solution of the continuous problem, supported by numerical examples.

Contribution

It introduces an explicit, discretization-level condition to determine global optimality in semilinear PDE control problems and proves convergence of discrete solutions under this condition.

Findings

The condition can be explicitly evaluated at the discrete level.

Discrete solutions converge to the global solution of the limit problem.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

We consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization. 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. Numerical examples with unique global solutions are presented.