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

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.

An anisotropic interaction model with collision avoidance

“…We recall an isotropic interaction model to clarify the starting point of the modifications. Following Newton's Second Law, isotropic interaction models are second order ODE systems [2,13,17,47,58], given by…”

“…Up to the author's knowledge the community widely agrees that isotropic interaction models that are used for the modelling of swarms of birds, schools of fish, sheep-dog interactions or opinion dynamics [2,13,15,50,47] are inappropriate for a detailed and realistic description of pedestrian dynamics [4]. This lack of details is unfortunate since these models allow for formulations on the microscopic, mesoscopic and macroscopic scale and even the limiting procedures are well understood, see [2,14,17,26,27,35,43,58] for an overview.…”

Space mapping-based receding horizon control for stochastic interacting particle systems: dogs herding sheep

“…To do so we define a time dependent reference stateZ : [0, T] → R D . Similar to the approach in [8], we measure the spread of the crowd aroundZ. In particular, due to the stochastic behaviour of the state system we use the expected paths E[X].…”

“…from f 0 (x, v), it is well-known that f (t, x, v) assigns the probability of finding a particle at time t at position x with velocity v. Hence, one could use the deterministic mean-field problem for f as coarse model for a space-mapping in order to control the stochastic limit for many particles. Note that a similar deterministic optimization problem was solved in [8].…”

“…Based on this knowledge, the investigation of the interaction of crowds and external agents became of interest [3,21]. In particular, the idea of controlling crowds with the help of the external agents [7,8]. The corresponding optimal control problem (OCP) is then constrained by the dynamics of the respective ODE or SDE system.…”