Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations

In this paper, we mainly show the following two statements. (1) A discrete-time Markovian jump linear system is uniformly exponentially stable if and only if it is robustly periodically stable, by using a Gel’fand-Berger-Wang formula proved here. (2) A random linear ODE driven by a semiflow with closing by periodic orbits property is uniformly exponentially stable if and only if it is robustly periodically stable, by using Shantao Liao’s perturbation technique and the semi-uniform ergodic theorems. Our proofs involve ergodic theory in both of the above two cases. In addition, counterexamples are constructed to the robustness condition and to spectral finiteness of linear cocycle.