Calculus of Variations and Geometric Measure Theory

G. Albi - S. Almi - M. Morandotti - F. Solombrino

Mean-field selective optimal control via transient leadership

created by morandott on 14 Jun 2021
modified by solombrino on 20 Apr 2022


Published Paper

Inserted: 14 jun 2021
Last Updated: 20 apr 2022

Journal: Applied Mathematics and Optimization
Volume: 85
Pages: Art. n. 9
Year: 2022
Doi: 10.1007/s00245-022-09837-4

ArXiv: 2106.07254 PDF


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.

Keywords: Gamma-convergence, superposition principle, mean-field optimal control, selective control, population dynamics, leader-follower dynamics