# A Nonlinear Perron-Frobenius Approach for Stability and Consensus of Discrete-Time Multi-Agent Systems

## TL;DR

This paper introduces a nonlinear Perron-Frobenius method to analyze stability and consensus in discrete-time multi-agent systems with heterogeneous nonlinear interactions, extending linear results to a broader nonlinear class.

## Contribution

It generalizes linear stability and consensus results to nonlinear, sub-homogeneous, order-preserving maps using Perron-Frobenius theory, without relying on Lyapunov functions.

## Key findings

- Established convergence conditions for nonlinear multi-agent systems.
- Demonstrated the method's effectiveness through examples.
- Extended linear results to nonlinear, heterogeneous interactions.

## Abstract

In this paper we propose a novel method to establish stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) evolving according to nonlinear heterogeneous local interaction rules which is not based on Lyapunov function arguments. In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by sub-homogeneous and order-preserving nonlinear maps. This paper directly generalizes results for sub-homogeneous and order-preserving linear maps which are shown to be the counterpart to stochastic matrices thanks to nonlinear Perron-Frobenius theory. We provide sufficient conditions on the structure of local interaction rules among agents to establish convergence to a fixed point and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the method are provided to corroborate the theoretical analysis.