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

## TL;DR

This paper develops an instantaneous control method for large interacting particle systems, using mean-field approximations and adjoint-based optimization to effectively steer particles into desired regions, with demonstrated convergence as particle number grows.

## Contribution

It introduces a novel control approach for mean-field particle systems using adjoint optimization, applicable to high-dimensional collective dynamics.

## Key findings

- Optimal controls converge in the mean-field limit
- Numerical results validate the control strategy
- Effective steering of particles demonstrated

## Abstract

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.