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.
Highlights
In this paper, we address the stability of transport systems and wave propagation on networks with time-varying parameters
We address stability issues first for transport systems with timedependent transmission conditions and for wave propagation on networks with time-dependent damping terms
When the time-dependent coefficients switch arbitrarily in a given bounded set, we prove that the stability is robust with respect to variations of the lengths of the edges of the network preserving their rational dependence structure
Summary
For p ∈ [1, +∞], the maximal Lyapunov exponent of Σδ(L, A) in Xδp is defined as λp(L, A) = lim sup sup sup t→+∞ A∈A u0∈Xδp ln ut p , t u0 p=1 where u denotes the solution of Σδ(L, A) with initial condition u0. The following result, which is a generalization of [8, Proposition 4.1], uses the representation formula (15) for the solutions of Σδ(L, A) in order to provide upper bounds on their growth.
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