The pth moment input-to-state stability and the pth moment integral input-to-state stability of stochastic systems with Markovian switching are investigated in this article. Sufficient conditions are provided by indefinite multiple Lyapunov functions (iMLFs) with a time-varying sojourn time condition. These criteria quantify the stability of each individual subsystem with time-varying infinitesimal generator of Lyapunov function, which allows infinitesimal generators to be positive in some intervals. An integral form of the sojourn time is introduced to replace the switch count process, which proposes a relaxed switching constraint. Uniformly asymptotic stability and uniformly exponential stability conditions with iMLFs are also presented. Some comparisons are given through theoretical derivations to show the advantages of the proposed results. Finally, two numerical examples verify the effectiveness and the advantages of the presented results.

中文翻译：

---

具有马尔可夫切换的非线性时变随机系统的p阶矩输入状态稳定性和p阶矩积分输入状态稳定性的不定多重Lyapunov函数

第p时刻输入状态稳定性和p本文研究了具有马尔可夫切换的随机系统的第 时刻积分输入到状态稳定性。具有时变逗留时间条件的不定多李雅普诺夫函数 (iMLF) 提供了充分条件。这些标准用 Lyapunov 函数的时变无穷小生成器量化每个单独子系统的稳定性，这允许无穷小生成器在某些区间为正。引入逗留时间的积分形式来代替切换计数过程，这提出了宽松的切换约束。还介绍了 iMLF 的均匀渐近稳定性和均匀指数稳定性条件。通过理论推导给出了一些比较，以显示所提出结果的优点。最后，