Sparse control to prevent Black Swan clustering in collective dynamics

In this paper, we elaborate control strategies to prevent clustering effects, as opposed to numerous works in the literature seeking to control multi-agent systems to achieve consensus. We consider general controlled collective dynamics and show how the group variance should be replaced by an entropy-type functional to measure clustering. Then we focus on Hegselmann-Krause type models and propose sparse declustering controls for the discrete system as well as for its mean-field limit. The behavior or the interaction function at zero and at infinity characterizes whether clustering can be avoided by controlling the system. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety regions (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering manifold) and collapse prevention.