  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 partial differential equation 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 the space of
configurations; "centralized" means that the strategy is computed by an
external agent knowing the configuration of all agents. We stress that we do
not only design a control function (in a sampled feedback form), but also a
time-varying control domain on which the action is applied. The sparsity
constraint reflects the fact that one cannot act on the whole crowd at every
instant of time.
  Our approach is based on geometric considerations on the velocity field of
the kinetic Cucker-Smale PDE, and in particular on the analysis of the particle
flow generated by this vector field. The control domain and the control
functions are designed to satisfy appropriate constraints, and such that, for
any initial configuration, the velocity part of the support of the measure
solution asymptotically shrinks to a singleton, which means flocking.
