# A convex penalty for switching control of partial differential equations

## TL;DR

This paper introduces a convex penalty method for switching controls in PDEs, enabling efficient numerical solutions and control structure analysis through a semismooth Newton approach.

## Contribution

It proposes a novel convex penalty for switching controls in PDEs and develops an efficient solution method using Moreau-Yosida approximation and semismooth Newton techniques.

## Key findings

- Efficient numerical solution demonstrated via examples.
- Controls exhibit the desired switching structure.
- The method is computationally effective for PDE control problems.

## Abstract

A convex penalty for promoting switching controls for partial differential equations is introduced; such controls consist of an arbitrary number of components of which at most one should be simultaneously active. Using a Moreau-Yosida approximation, a family of approximating problems is obtained that is amenable to solution by a semismooth Newton method. The efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples.