Novel Stability Conditions for Nonlinear Monotone Systems and Consensus in Multi-Agent Networks

TL;DR

This paper introduces a new class of nonlinear monotone systems called K-topical systems, proves their stability, and applies these results to achieve consensus in multi-agent networks with nonlinear interactions.

Contribution

It defines K-topical systems, establishes their stability, and demonstrates their application to consensus problems in nonlinear multi-agent systems.

Findings

K-topical systems are asymptotically stable.

Consensus is achieved under certain graph conditions.

Applicable to both continuous-time and discrete-time systems.

Abstract

In this work, we characterize a class of nonlinear monotone dynamical systems that have a certain translation invariance property which goes by the name of plus-homogeneity; usually called "topical" systems. Such systems need not be asymptotically stable, since they are merely nonexpansive but not contractive. Thus, we introduce a stricter version of monotonicity, termed "type-K" in honor of Kamke, and we prove the asymptotic stability of the equilibrium points, as well as the convergence of all trajectories to such equilibria for type-K monotone and plus-homogeneous systems: we call them "K-topical". Since topical maps are the natural nonlinear counterpart of linear maps defined by row-stochastic matrices, which are a cornerstone in the convergence analysis of linear multi-agent systems (MASs), we exploit our results for solving the consensus problem over nonlinear K-topical MASs. We first provide necessary and sufficient conditions on the local interaction rules of the agents ensuring the K-topicality of a MAS. Then, we prove that the agents achieve consensus asymptotically if the graph describing their interactions contains a globally reachable node. Finally, several examples for continuous-time and discrete-time systems are discussed to corroborate the enforceability of our results in different applications.