Soft control of swarms: Analytical approach
We analytically study the collective dynamics of mutually interacting heterogeneous agents evolving in a random environment. Our formal framework consists of a collection of JV scalar drifted Brownian motions (BM) diffusing on R. The mutual interactions are introduced via a ranked-based, real-time mechanism always endowing the laggard (i.e the agent with the leftmost position) with an extra positive drift. The extra drift generates a net tendency for any agents not to remain the laggard of the society. For well chosen individual and extra laggard's drifts, the agents organize with time to flock towards a tight and stable travelling spatial pattern. For a population of (N - 1) identical agents and an atypical fellow (called hereafter the shill), we are able to analytically discuss the dynamics. In particular we exhibit how a single turbulent shill, stylized here by a ballistic diffusion process, can destroy the cohesion of a swarm. Conversely, we also analytically show how a single shill is able to safely pilot a whole swarm to avoid an obstacle, via interactions with its fellows. A series of simulations experiments comfort our analytic findings.