# Multigoal-oriented optimal control problems with nonlinear PDE constraints

## TL;DR

This paper develops an adaptive solution strategy for multigoal-oriented optimal control problems constrained by nonlinear PDEs, using a dual-weighted residual method to balance errors and improve accuracy.

## Contribution

It introduces a combined a posteriori error estimator for multiple quantities of interest in nonlinear PDE-constrained control problems, enabling adaptive mesh refinement.

## Key findings

- Effective error balancing between discretization and nonlinear iteration.
- Enhanced accuracy in control solutions through adaptive mesh refinement.
- Numerical examples demonstrate the method's efficiency and robustness.

## Abstract

In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized $p$-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated via several numerical examples.