On Distributed Control of Continuum Swarms: Local Controllers as Differential Operators

TL;DR

This paper introduces a PDE-based framework for the distributed control of robotic swarms modeled as continuum densities, highlighting the limitations of pointwise controllers and proposing a simple stabilizing control law.

Contribution

It formalizes distributed controllers as differential operators and demonstrates a simple control law that stabilizes swarm densities with strong stability properties.

Findings

Pointwise controllers are incompatible with natural symmetries and stability.

Mixing behavior is necessary for effective stabilization.

A first-order control law successfully stabilizes the swarm density.

Abstract

We study the problem of distributed control of large-scale robotic swarms which can be modeled as continuum densities evolving under the continuity equation. We propose a formalization of distributed controllers as (generally nonlinear) differential operators, in which control inputs depend only on local information about the state and environment. This perspective yields a fully local, PDE-based framework for analysis and design. We apply this framework to the problem of stabilizing a swarm density around an arbitrary target density, and investigate fundamental limitations of low-order distributed controllers in achieving this goal. In particular, we show that controllers which act in a purely pointwise manner are incompatible with natural system symmetries and strong forms of stability, and must rely on mixing-type behavior to achieve stabilization. In contrast, we present a simple first-order control law which achieves stabilization and enjoys substantially stronger properties.