TY - JOUR

T1 - Instantaneous control of interacting particle systems in the mean-field limit

AU - Burger, Martin

AU - Pinnau, René

AU - Totzeck, Claudia

AU - Tse, Oliver

AU - Roth, Andreas

PY - 2020/3/15

Y1 - 2020/3/15

N2 - 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.

AB - 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.

KW - Instantaneous control

KW - Interacting particle systems

KW - Mean-field limit

KW - Numerics

UR - http://www.scopus.com/inward/record.url?scp=85076922269&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2019.109181

DO - 10.1016/j.jcp.2019.109181

M3 - Article

AN - SCOPUS:85076922269

VL - 405

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

M1 - 109181

ER -