Instantaneous control of interacting particle systems in the mean-field limit.

• Tailored optimization methods for interacting particle systems in the mean-field limit. • Derivation of the adjoint equation for a nonlocal Vlasov equation. • The " first optimize, then discretize " approach allows for performant discretizations of the forward and adjoint equations. • Numerical solution of a large scale optimization problem with 4D phase space. • Numerical verification of the mean-field limit in optimization problems. Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We introduce an appropriate mathematical model and the corresponding optimization problem. In particular, we are interested in the interaction of numerous particles, which can be approximated by a mean-field equation. Due to the high-dimensional phase space this will require a tailored optimization strategy. The arising control problems are solved using adjoint information to compute the descent directions. Numerical results on the microscopic and the macroscopic level indicate the convergence of optimal controls and optimal states in the mean-field limit, i.e., for an increasing number of particles. [ABSTRACT FROM AUTHOR]