Mean-field optimal control by leaders

In this paper we deal with a social dynamics model, where one controls a small number of leaders in order to influence the behavior of the whole group (leaders and followers). We first provide a general mathematical framework to deal with optimal control of the microscopic problem, where the number of agents is finite, and its mean-field limit with an infinite number of followers. Then we focus on a migration-type model and develop optimal control strategies for the microscopic model. Such strategies are tested in their behavior both for the number of agents tending to infinity and for different initial conditions and initial locations of the leaders.