Optimal control problems of nonlocal interaction equations

TL;DR

This paper investigates the existence of solutions for optimal control problems involving nonlocal transport equations that model populations influenced by control agents, focusing on mildly singular potentials in a gradient flow framework.

Contribution

It extends previous work by considering mildly singular potentials within a gradient flow formulation for nonlocal transport equations in optimal control.

Findings

Established existence of solutions for the control problem.

Extended the model to include mildly singular potentials.

Provided a mathematical framework for controlling population dynamics.

Abstract

In the present work we deal with the existence of solutions for optimal control problems associated to transport equations. The behaviour of a population of individuals will be influenced by the presence of a population of control agents whose role is to lead the dynamics of the individuals towards a specific goal. The dynamics of the population of individuals is described by a suitable nonlocal transport equation, while the role of the population of agents is designed by the optimal control problem. This model has been first studied in [12] for a class of continuous nonlocal potentials, while in the present project we consider the case of mildly singular potentials in a gradient flow formulation of the target transport equation.