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Pré-Publication, Document De Travail Année : 2015

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

Résumé

In this paper, we address the stability of non-autonomous difference equations by providing an explicit formula expressing the solution at time t in terms of the initial condition and time-dependent matrix coefficients. We then relate the asymptotic behavior of such coefficients to that of solutions. As a consequence, we obtain necessary and sufficient stability criteria for non-autonomous linear difference equations. In the case of difference equations with arbitrary switching, we obtain a generalization of the well-known criterion for autonomous systems due to Hale and Silkowski. These results are applied to transport and wave propagation on networks. In particular, we show that 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 one.
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Dates et versions

hal-01139814 , version 1 (07-04-2015)
hal-01139814 , version 2 (16-04-2015)
hal-01139814 , version 3 (26-11-2015)
hal-01139814 , version 4 (04-03-2016)

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Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. 2015. ⟨hal-01139814v2⟩
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