Control to Flocking of the Kinetic Cucker--Smale Model

Abstract

The well-known Cucker--Smale model is a macroscopic system reflecting flocking, i.e., the alignment of velocities in a group of autonomous agents having mutual interactions. In the present paper, we consider the mean-field limit of that model, called the kinetic Cucker--Smale model, which is a transport PDE involving nonlocal terms. It is known that flocking is reached asymptotically whenever the initial conditions of the group of agents are in a favorable configuration. For other initial configurations, it is natural to investigate whether flocking can be enforced by means of an appropriate external force, applied to an adequate time-varying subdomain. In this paper we prove that we can drive to flocking any group of agents governed by the kinetic Cucker--Smale model, by means of a sparse centralized control strategy, and this, for any initial configuration of the crowd. Here, “sparse control” means that the action at each time is limited over an arbitrary proportion of the crowd, or, as a variant, of th...