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

Yacine Chitour (1, 2) , Guilherme Mazanti (3, 4) , Mario Sigalotti (4, 3)

1 UP11 - Université Paris-Sud - Paris 11
2 L2S - Laboratoire des signaux et systèmes
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 GECO - Geometric Control Design

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.

Mots clés: difference equations; switching systems; transport equation; wave propagation; networks; exponential stability

Domaines: Equations aux dérivées partielles [math.AP]

Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. 2015. ⟨hal-01139814v1⟩