# Critical cones for sufficient second order conditions in PDE constrained optimization

## TL;DR

This paper introduces a new extended cone for second order sufficient optimality conditions in PDE-constrained control problems, proving its effectiveness for ensuring strong local minima without Tikhonov regularization.

## Contribution

It proposes a novel, smaller extended cone for second order conditions in PDE control problems, enhancing the theoretical understanding of optimality criteria.

## Key findings

- New extended cone is smaller than previous ones.
- Second order condition based on this cone guarantees strong local minimum.
- Analysis applies to control problems with sparsity terms and no Tikhonov regularization.

## Abstract

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.