Nonuniform Stability of Arbitrary Difference Equations

Abstract

We describe two approaches to study the robustness of the exponential stability of an arbitrary difference equation with finite delay on a Banach space. The first approach is based on a characterization of Perron-type of the exponential stability in terms of the invertibility of a certain linear operator between spaces of bounded sequences. The second approach is based on looking at the dynamics on a higher-dimensional space without delay and so for which one has the cocycle property. We emphasize that none of the results obtained with the two approaches implies the other. More precisely, the second approach allows to obtain the robustness of a weaker notion although also with a stronger hypothesis. We consider the general case of nonuniform exponential stability.