# Optimal control of a non-smooth semilinear elliptic equation

## TL;DR

This paper investigates optimal control problems involving non-smooth semilinear elliptic equations, establishing differentiability properties, deriving optimality conditions, and demonstrating numerical solution methods.

## Contribution

It characterizes the directional differentiability and Bouligand subdifferential of the control-to-state map, and develops numerical methods for solving the resulting optimality conditions.

## Key findings

- Control-to-state map is directionally differentiable.
- First-order optimality conditions are derived and interpreted.
- Numerical examples show semi-smooth Newton method effectiveness.

## Abstract

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.