This paper is devoted to stochastic convergence theorems for stochastic impulsive systems (SISs) and their application to discrete-time stochastic feedback control (DTSFC). A general stochastic Barb_lat's lemma, that only requires that the concerned stochastic processes are almost surely integrable rather than absolutely integrable in the sense of expectation, for piecewise continuous adapted processes is first proposed, which is truly parallel to the deterministic one. As an extension of this lemma, a general stochastic convergence theorem is established for SISs, which can reveal and sufficiently apply the possible active contribution of the existing noise in the underlying system. To derive easy-to-check stability conditions, a series of LaSalle-type theorems and dwell-time-based conditions are established for stochastic stability/convergence of SISs. In contrast to preceding results, these stability criteria can not only characterize the stabilizing noise but also be applicable to SISs with both continuous and discrete unstable dynamics. Moreover, supported by the LaSalle-type theorems, the almost sure exponential stabilization problems by DTSFC in both time- and event-triggered control schemes are solved. Particularly, the proposed methods remove the global Lipschitz condition required in the literature and provide an explicit computation of the maximum allowable sampling period. Finally, four numerical examples with comparisons are used to illustrate the theoretical results.

中文翻译：

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随机脉冲系统收敛定理及其在离散时间随机反馈控制中的应用


本文致力于研究随机脉冲系统 (SIS) 的随机收敛定理及其在离散时间随机反馈控制 (DTSFC) 中的应用。一个一般的随机Barb_lat引理，只要求有关的随机过程几乎肯定可积，而不是期望意义上的绝对可积，因为分段连续适应过程是首次提出的，它与确定性过程真正平行。作为该引理的扩展，建立了 SIS 的一般随机收敛定理，该定理可以揭示并充分应用底层系统中现有噪声的可能主动贡献。为了导出易于检查的稳定性条件，为 SIS 的随机稳定性/收敛性建立了一系列拉萨尔型定理和基于驻留时间的条件。与之前的结果相比，这些稳定性准则不仅可以表征稳定噪声，而且适用于具有连续和离散不稳定动态的 SIS。此外，在拉萨尔型定理的支持下，DTSFC 在时间触发和事件触发控制方案中几乎肯定的指数稳定问题都得到了解决。特别是，所提出的方法消除了文献中所需的全局 Lipschitz 条件，并提供了最大允许采样周期的显式计算。最后通过四个数值算例并进行比较来说明理论结果。