Abstract

In this article, we establish the LaSalle-type theorem for stochastic functional differential equations with Markovian switching (SFDEwMSs) under much weaken conditions. We would like to emphasize that we do not require the linear growth condition and the bounded moment condition on the solutions. Indeed, we allow the Lyapunov function operator could be dependent of time to cover a much wider class of SFDEwMSs. As a bonus, we obtain the criterion on the asymptotical stability and asymptotical boundedness for SFDEwMSs. For comparison and verification, we also present a specific example with much general coefficients.

中文翻译：

---

具有马尔可夫转换的随机泛函微分方程的拉萨尔型定理

摘要

在本文中，我们建立了具有马尔可夫切换的随机泛函微分方程 (SFDEwMS) 在弱弱条件下的 LaSalle 型定理。我们想强调的是，我们不需要解的线性增长条件和有界矩条件。事实上，我们允许 Lyapunov 函数算子可以依赖于时间来覆盖更广泛的 SFDEwMS 类。作为奖励，我们获得了 SFDEwMS 的渐近稳定性和渐近有界的标准。为了比较和验证，我们还提供了一个具有很多通用系数的特定示例。