  We analyze the dynamics of multi-agent collective behavior models and their
control theoretical properties. We first derive a large population limit to
parabolic diffusive equations. We also show that the non-local transport
equations commonly derived as the mean-field limit, are subordinated to the
first one. In other words, the solution of the non-local transport model can be
obtained by a suitable averaging of the diffusive one.
  We then address the control problem in the linear setting, linking the
multi-agent model with the spatial semi-discretization of parabolic equations.
This allows us to use the existing techniques for parabolic control problems in
the present setting and derive explicit estimates on the cost of controlling
these systems as the number of agents tends to infinity. We obtain precise
estimates on the time of control and the size of the controls needed to drive
the system to consensus, depending on the size of the population considered.
  Our approach, inspired on the existing results for parabolic equations,
possibly of fractional type, and in several space dimensions, shows that the
formation of consensus may be understood in terms of the underlying diffusion
process described by the heat semi-group. In this way, we are able to give
precise estimates on the cost of controllability for these systems as the
number of agents increases, both in what concerns the needed control
time-horizon and the size of the controls.
