# A convex analysis approach to optimal controls with switching structure for partial differential equations

## TL;DR

This paper introduces a convex analysis framework for optimal control problems with switching structures in PDEs, enabling explicit characterization and efficient computation of solutions with zero optimality gap.

## Contribution

It develops a convex relaxation and a primal-dual system for hybrid control costs, facilitating semismooth Newton methods for PDE control problems with switching controls.

## Key findings

- Explicit pointwise characterization of optimal controls.
- Zero optimality gap for controls with switching structure.
- Numerical examples demonstrate method effectiveness.

## Abstract

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.