# Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations

## TL;DR

This paper analyzes Tikhonov regularization for semilinear PDE-constrained optimal control problems, providing error estimates and necessary conditions for convergence, with numerical validation.

## Contribution

It introduces new a-priori error estimates and necessary conditions for regularization convergence in semilinear PDE control problems, including sparse controls.

## Key findings

- Derived a-priori regularization error estimates.
- Established necessary conditions for convergence rates.
- Numerical experiments confirm theoretical results.

## Abstract

In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under suitable conditions. These conditions comprise second-order sufficient optimality conditions as well as regularity conditions on the control, which consists of a source condition and a condition on the active sets. In addition, we show that these conditions are necessary for convergence rates under certain conditions. We also consider sparse optimal control problems and derive regularization error estimates for them. Numerical experiments underline the theoretical findings.