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

Martin Burger, René Pinnau (Corresponding author), Claudia Totzeck, Oliver Tse, Andreas Roth

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Article number109181
Number of pages20
JournalJournal of Computational Physics
Publication statusPublished - 15 Mar 2020


  • Instantaneous control
  • Interacting particle systems
  • Mean-field limit
  • Numerics

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