Real Eventual Exponential Positivity of Complex-valued Laplacians: Applications to Consensus in Multi-agent Systems

TL;DR

This paper investigates the property of eventual exponential positivity in complex matrices, linking spectral properties of complex Laplacians to stability in multi-agent consensus systems, with practical numerical demonstrations.

Contribution

It establishes conditions under which the real part of complex Laplacian matrices exhibits eventual exponential positivity, impacting stability analysis in multi-agent networks.

Findings

Real part of complex Laplacian matrices can be eventually exponentially positive.

Spectral properties of complex Laplacians determine system stability.

Numerical examples validate theoretical results.

Abstract

In this paper, we explore the property of eventual exponential positivity (EEP) in complex matrices. We show that this property holds for the real part of the matrix exponential for a certain class of complex matrices. Next, we present the relation between the spectral properties of the Laplacian matrix of an unsigned digraph with complex edge-weights and the property of real EEP. Finally, we show that the Laplacian flow system of a network is stable when the negated Laplacian admits real EEP. Numerical examples are presented to demonstrate the results.