Controlling a self-organizing system of individuals guided by a few external agents -- particle description and mean-field limit

Abstract: Optimal control of large particle systems with collective dynamics by few agents is a subject of high practical importance (e.g. in evacuation dynamics), but still limited mathematical basis. In particular the transition from discrete optimal control to a continuum setting as the number of particles tends to infinity is by far not fully understood. In this paper we contribute to this issue by studying a canonical model of controlling an interacting particle system into a certain spatial region by repulsive forces from few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We discuss the appropriate modelling of such a problem and the associated optimality systems, providing some connections between the Lagrange multipliers in the discrete and continuum setting. As control strategies we investigate an Instantaneous Control and a global Optimal Control approach. The solutions of a family of control problems for the particle system with external agents are numerically compared to the mean-field controls as the number of particles tends to infinity. In both cases, this leads to a high dimensional phase space requiring tailored optimization strategies. All control problems arising are solved using adjoint information to compute the descent directions. The numerical results indicate the convergence of controls for both optimization strategies.