Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations. (English) Zbl 1372.93209
Summary: In this paper, we mainly show the following two statements.
(1) A discrete-time topological 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 a perturbation technique of Shantao Liao and the semi-uniform ergodic theorems.
Our proofs involve the ergodic theory in both of the above two cases. In addition, counterexamples are constructed to the robustness condition and to the spectral finiteness of linear cocycle.
(1) A discrete-time topological 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 a perturbation technique of Shantao Liao and the semi-uniform ergodic theorems.
Our proofs involve the ergodic theory in both of the above two cases. In addition, counterexamples are constructed to the robustness condition and to the spectral finiteness of linear cocycle.
MSC:
| 93E15 | Stochastic stability in control theory |
| 93D20 | Asymptotic stability in control theory |
| 60J75 | Jump processes (MSC2010) |
| 93C05 | Linear systems in control theory |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
robust periodic stability; uniform exponential stability; Markovian jump linear systems; random linear ordinary differential equationsReferences:
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