Exponential convergence towards consensus for non-symmetric linear first-order systems in finite and infinite dimensions

TL;DR

This paper studies how non-symmetric linear first-order systems reach consensus exponentially fast, extending classical symmetric results by identifying a weighted mean and calculating the optimal decay rate in both finite and infinite dimensions.

Contribution

It introduces a method to prove exponential convergence for non-symmetric systems by defining a weighted mean and determines the sharp decay rate.

Findings

Exponential convergence towards consensus in non-symmetric systems.

Identification of a positive weight for defining the weighted mean.

Calculation of the sharp exponential decay rate.

Abstract

We consider finite and infinite-dimensional first-order consensus systems with timeconstant interaction coefficients. For symmetric coefficients, convergence to consensus is classically established by proving, for instance, that the usual variance is an exponentially decreasing Lyapunov function. We investigate here the convergence to consensus in the non-symmetric case: we identify a positive weight which allows to define a weighted mean corresponding to the consensus, and obtain exponential convergence towards consensus. Moreover, we compute the sharp exponential decay rate.