A mathematical approach to crowd surveillance with drones

We formulate a surveillance problem, which consists in optimally monitoring a moving pedestrian crowd via a drone fleet. The crowd dynamics is given by a nonlinear transport equation for measures with finite-mass. The measure-theoretic framework allows multiscale representation of the crowd. The controlled drone dynamics is formulated in classical sense, but the optimal control problems are formulated in terms of the generalized Wasserstein distance between the crowd measure and the empirical measure of the drones’ positions. Accurate estimates allow to prove the existence of an optimal control under suitable assumptions. Then we compare the proposed surveillance problem with coverage ones, showing that the main difference is in the way the mass is assigned to the empirical measure.