Stability of non-autonomous difference equations with applications to transport and wave propagation on networks

TL;DR

This paper studies the stability of transport and wave systems on networks with changing parameters, providing new criteria for exponential stability that are robust to network variations and switching behaviors.

Contribution

It introduces a novel analysis of non-autonomous difference equations for network wave stability, extending classical criteria to arbitrary switching and variable edge lengths.

Findings

Exponential stability is robust under variations of edge lengths with rational dependence.

Wave equations on networks are stable if the network is a tree with sufficient damping.

The stability criterion applies to arbitrarily switching damping at external vertices.

Abstract

In this paper, we address the stability of transport systems and wave propagation on networks with time-varying parameters. We do so by reformulating these systems as non-autonomous difference equations and by providing a suitable representation of their solutions in terms of their initial conditions and some time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of such coefficients. In the case of difference equations with arbitrary switching, we obtain a delay-independent generalization of the well-known criterion for autonomous systems due to Hale and Silkowski. As a consequence, we show that exponential stability of transport systems and wave propagation on networks is robust with respect to variations of the lengths of the edges of the network preserving their rational dependence structure. This leads to our main result: the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but at most one.