Mean-field selective optimal control via transient leadership

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 Γ -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. Contents 1. Introduction 1 1.1. Technical aspects 3 2. Mathematical setting 4 3. The finite particle control problem 6 4. Mean-field control problem 8 5. Numerical experiments 22 5.1. A leader-follower dynamics 22 5.2. Two-leader game 25 References 31