On Conditions for Convergence to Consensus

TL;DR

This paper introduces a new theorem based on averaging maps that provides conditions for convergence to consensus in multiagent systems, expanding upon and differing from previous Lyapunov-based approaches.

Contribution

The paper presents a novel theorem for consensus convergence using averaging maps, offering an alternative to existing set-valued Lyapunov methods and addressing their limitations.

Findings

The new theorem applies to cases where Moreau's theorem does not.

Examples demonstrate the differences and advantages of the new approach.

The theory of convergence to consensus remains incomplete, as shown by counterexamples.

Abstract

A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to results by Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005) about set-valued Lyapunov theory and convergence under switching communication topologies. We give examples that point out differences of approaches including examples where Moreau's theorem is not applicable but ours is. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete.