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

Xiongping Dai

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

机译：鲁棒的周期稳定性意味着Markovian跳跃线性系统和随机线性常微分方程的一致指数稳定性

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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.

机译：在本文中，我们主要显示以下两个语句。（1）通过使用此处证明的Gel'fand-Berger-Wang公式，当且仅当鲁棒周期稳定时，离散时间拓扑马尔可夫跳跃线性系统才是一致指数稳定的。（2）通过使用廖善涛摄动法和半均匀遍历定理，由且具有周期轨道性的半流驱动的随机线性ODE在且仅当其具有鲁棒周期性时，才是均匀指数稳定的。在以上两种情况下，我们的证明都涉及遍历理论。另外，针对鲁棒性条件和线性cocycle的光谱有限性构造了反例。

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《Journal of the Franklin Institute》 |2014年第5期|2910-2937|共28页
作者
Xiongping Dai;
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Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China;

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5. Stability analysis of jump-linear systems driven by finite-state machines with Markovian inputs. [D] . Patilkulkarni, Sudarshan S. 2004

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6. Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations [O] . Ling Zhang -1

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7. Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations [O] . Xiongping Dai 2014

机译：坚固的周期稳定性意味着马尔可夫跳跃线性系统和随机线性常微分方程的均匀指数稳定性
8. On a Criterion of Instability in the Sense of Lyapunov of Solutions of a Linear System of Ordinary Differential Equations [R] . Vinograd, R. E. 1964

机译：关于常微分方程线性系统解的Lyapunov定理的不稳定判据

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

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