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

arXiv: https://arxiv.org/abs/2301.05140 · HTML: https://mlsanity.com/paper/2301.05140/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jan 2023.

## Built on by (most cited first, 1 of 1)

- [Robust Online Learning over Networks](https://mlsanity.com/paper/2309.00520/) (2023)

## Abstract (arXiv)

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.