SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 924-952, April 2024.
Abstract. This paper provides some sufficient conditions for the existence and uniqueness and the stochastic stability of the global solution for nonlinear neutral stochastic functional differential equations. When the drift term and the diffusion term satisfy a locally Lipschitz condition, and the Lyapunov monotonicity condition has a sign-changed time-varying coefficient, the existence and uniqueness of the global solution for such equations will be studied by using the Lyapunov–Krasovskii function approach and the theory of stochastic analysis. The stability in [math]th-moment, the asymptotical stability in [math]th-moment, and the exponential stability in [math]th-moment will be investigated. Different characterizations for these three kinds of stochastic stability in moment will be established, which are presented with respect to integration conditions. These results have seldom been reported in the existing literature. The almost surely exponential stability for the global solution of such equations is also discussed. Some discussions and comparisons are provided. Two examples are given to check the effectiveness of the theoretical results obtained.

中文翻译：

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非线性中性随机泛函微分方程的稳定性分析

SIAM 控制与优化杂志，第 62 卷，第 2 期，第 924-952 页，2024 年 4 月。
摘要。本文为非线性中性随机泛函微分方程全局解的存在唯一性和随机稳定性提供了一些充分条件。当漂移项和扩散项满足局部Lipschitz条件，且Lyapunov单调性条件具有变号时变系数时，利用Lyapunov-Krasovskii函数研究此类方程组全局解的存在唯一性方法和随机分析理论。将研究第 矩的稳定性、第 矩的渐近稳定性以及第 矩的指数稳定性。将建立这三种随机力矩稳定性的不同表征，并根据积分条件给出。这些结果在现有文献中很少报道。还讨论了此类方程的全局解的几乎肯定的指数稳定性。提供了一些讨论和比较。给出了两个例子来检验所获得的理论结果的有效性。