This article investigates the delay-dependent stability and stabilization problems for stochastic time-delay systems (STDSs) governed by the Poisson process and Brownian motion. First, this article exploits the decomposition of semimartingale to transform STDSs governed by the Poisson process and Brownian motion into STDSs governed by martingales, and then one can use effective properties and tools of martingale theory to investigate the stability and stabilization problems. Second, the expectations of stochastic crossterms involving stochastic integrals with respect to (w.r.t.) càdlàg martingale have been studied by martingale theory, and this article proves that the expectations of some particular stochastic crossterms are zero. Third, on the basis of above results on the expectations of stochastic crossterms and the equivalent Itô formula derived, this article uses the free weighting matrix method to give a delay-dependent stability condition without using bounding techniques. Thus, the conservatism induced by bounding techniques is eliminated, and this method leads to a less conservative condition by a linear matrix inequality. Then, this article designs a state-feedback controller which can stabilize the STDS governed by the Poisson process and Brownian motion. Finally, the effectiveness of derived results are illustrated by numerical examples.

中文翻译：

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泊松过程和布朗运动控制的随机时滞系统的时滞稳定性分析和镇定

本文研究了由泊松过程和布朗运动控制的随机时滞系统 (STDS) 的时滞相关稳定性和镇定问题。首先，本文利用半鞅的分解，将受泊松过程和布朗运动控制的 STDS 转化为受鞅控制的 STDS，然后可以利用鞅理论的有效性质和工具来研究稳定性和镇定问题。其次，鞅理论研究了涉及(wrt) càdlàg martingale的随机积分的随机交叉项的期望，本文证明了某些特定的随机交叉项的期望为零。第三，在上述随机交叉项的期望结果和推导出的等效Itô公式的基础上，本文采用自由加权矩阵方法，在不使用边界技术的情况下，给出了时滞相关的稳定性条件。因此，消除了由边界技术引起的保守性，并且该方法通过线性矩阵不等式导致不太保守的条件。然后，本文设计了一种状态反馈控制器，它可以稳定由泊松过程和布朗运动控制的 STDS。最后，通过数值例子说明了导出结果的有效性。并且该方法通过线性矩阵不等式导致不太保守的条件。然后，本文设计了一种状态反馈控制器，它可以稳定由泊松过程和布朗运动控制的 STDS。最后，通过数值例子说明了导出结果的有效性。并且这种方法通过线性矩阵不等式导致不太保守的条件。然后，本文设计了一种状态反馈控制器，它可以稳定由泊松过程和布朗运动控制的 STDS。最后，通过数值例子说明了导出结果的有效性。