  We model, simulate and control the guiding problem for a herd of evaders
under the action of repulsive drivers. The problem is formulated in an optimal
control framework, where the drivers (controls) aim to guide the evaders
(states) to a desired region of the Euclidean space. The numerical simulation
of such models quickly becomes unfeasible for a large number of interacting
agents. To reduce the computational cost, we use the Random Batch Method (RBM),
which provides a computationally feasible approximation of the dynamics. At
each time step, the RBM randomly divides the set of particles into small
subsets (batches), considering only the interactions inside each batch. Due to
the averaging effect, the RBM approximation converges to the exact dynamics as
the time discretization gets finer. We propose an algorithm that leads to the
optimal control of a fixed RBM approximated trajectory using a classical
gradient descent. The resulting control is not optimal for the original
complete system, but rather for the reduced RBM model. We then adopt a Model
Predictive Control (MPC) strategy to handle the error in the dynamics. While
the system evolves in time, the MPC strategy consists in periodically updating
the state and computing the optimal control over a long-time horizon, which is
implemented recursively in a shorter time-horizon. This leads to a
semi-feedback control strategy. Through numerical experiments we show that the
combination of RBM and MPC leads to a significant reduction of the
computational cost, preserving the capacity of controlling the overall
dynamics.
