# Bilevel optimal control: existence results and stationarity conditions

## TL;DR

This paper reviews bilevel optimal control problems, focusing on existence results and stationarity conditions, especially for PDE-based models with parameter reconstruction, providing theoretical insights and solution approaches.

## Contribution

It offers a comprehensive review of bilevel optimal control, introduces new existence results, and derives necessary optimality conditions for PDE-based problems with parameter reconstruction.

## Key findings

- Existence of solutions for specific bilevel control problems.
- Derivation of necessary optimality conditions using the optimal value function.
- Application of relaxation approaches to establish stationarity conditions.

## Abstract

The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.