This paper studies stochastic boundedness of trajectories of a non-vanishing stochastically perturbed stable linear time-invariant system. First, two definitions on stochastic boundedness are presented, then, the boundedness is analyzed via Lyapunov theory. A theorem is proposed, which shows that under a condition on the Lipchitz constant of the perturbation kernel, the trajectories remain stochastically bounded, and the bounds are calculated. Also, the limiting behaviour of the trajectories is studied. At the end, an illustrative example is presented, which shows the effectiveness of the proposed theory.

中文翻译：

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随机扰动不消失时稳定LTI系统状态轨迹的随机有界性

本文研究了一个不消失的随机扰动的稳定线性时不变系统的轨迹的随机有界性。首先，给出了关于随机有界性的两个定义，然后通过李雅普诺夫理论对有界性进行了分析。提出了一个定理，该定理表明，在摄动核的Lipchitz常数的条件下，轨迹保持随机有界，并计算了界。此外，研究了轨迹的极限行为。最后，给出了一个说明性的例子，说明了所提出理论的有效性。