Optimal Control of Nonlocal Balance Equations

TL;DR

This paper develops a Pontryagin maximum principle for optimal control problems involving nonlocal balance laws, transforming them into a continuum of probability measures and Hilbert space ODEs, unifying existing mean field control approaches.

Contribution

It introduces a novel framework that generalizes and unifies various formulations of mean field control problems with nonlocal dynamics.

Findings

Established a Pontryagin maximum principle for nonlocal balance law control problems.

Unified existing formulations within a single theoretical framework.

Provided a method to transform balance laws into Hilbert space ODEs.

Abstract

The paper presents an approach to studying optimal control problems in the space of nonnegative measures with dynamics given by a nonlocal balance law. This approach relies on transforming the balance law into a continuity equation in the space of probabilities, and subsequently into an ODE in a Hilbert space. The main result is a version of Pontryagin's maximum principle for the addressed problem, which encompasses all known formulations of this type in mean field control theory.