Mean-field selective optimal control via transient leadership

TL;DR

This paper introduces a mean-field optimal control framework for multi-population systems with transient leadership, allowing selective influence based on population membership and demonstrating convergence from finite-particle models to the mean-field limit.

Contribution

It develops a novel mean-field control model incorporating transient leadership and proves convergence from finite-particle systems, with applications to opinion dynamics.

Findings

Finite-particle control problem analyzed and its mean-field limit established.

Convergence of optimal controls via $$-convergence proven.

Numerical experiments illustrate the model's application to opinion dynamics.

Abstract

A mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process. This way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via $\Gamma$-convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion. Specific applications in the context of opinion dynamics are discussed with some numerical experiments.