# A shape optimal control problem and its probabilistic counterpart

## TL;DR

This paper investigates a shape optimization problem with sign-changing data, establishing the existence of optimal domains and deriving necessary optimality conditions, including a probabilistic extension where data is uncertain.

## Contribution

It introduces the first existence and optimality conditions for shape optimization problems with sign-changing data and extends the analysis to probabilistic data uncertainty.

## Key findings

- Existence of optimal domains despite sign-changing data
- Necessary conditions for optimality in shape problems
- Extension to probabilistic data in the state equation

## Abstract

In this paper we consider a shape optimization problem in which the data in the cost functional and in the state equation may change sign, and so no monotonicity assumption is satisfied. Nevertheless, we are able to prove that an optimal domain exists. We also deduce some necessary conditions of optimality for the optimal domain. The results are applied to show the existence of an optimal domain in the case where the cost functional is completely identified, while the right-hand side in the state equation is only known up to a probability P in the space $L^2$.