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 consist of a collection of N 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 of 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 traveling spatial pattern. For a population of (N−1) identical agents and an atypic 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, via interactions with its fellows, is able to safely pilot a whole swarm to avoid an obstacle. A series of simulations experiments comfort our analytic findings. 1 Homogeneous versus heterogeneous mutually interacting stochastic agents The capability of a collection of interacting stochastic agents to exhibit an emergent collective behaviour (i.e flocking behaviour) even in random environments stimulates a strong research activity devoted to both experimental and theoretical modeling approaches. For suitable range of mutual interactions, flocking (phase) transitions, namely the self-organized capability to create finite and persistent spatio-temporal patterns, are observed (see [1, 3–5, 10, 13, 14]). Agents societies can be composed of either dynamically homogeneous or heterogeneous individuals requiring for each case drastically different approaches. For large and homogeneous population of agents, the classical statistical mechanics concepts and in particular, the mean-field description (MF) directly offers an appropriate tool to analytically discuss the global dynamics. In the MF description, one ∗Supported by the Swiss National Funds for Scientific Research