奇异摄动随机泛函微分方程解的基本性质的研究

Singularly perturbed differential equations have a wide range of applications in the vibration theory, fluid mechanics, modern physics, automation, marine engineering and many other fields, and are important research directions of differential equations theories and research hotspots. On the other hand, delay phenomena and stochastic phenomena are common and inevitable in nature, and often make the system property decline or even causes instability. Therefore, it is necessary to investigate singularly perturbed stochastic functional differential equations in order to provide a rigorous mathematical theory for its practical applications. However, at present there are few results on singularly perturbed stochastic functional differential equations because its research is difficult. Based on this, we plan to study singularly perturbed stochastic functional differential equations, discuss the fundamental properties of its solutions, such as existence, stability, boundeness, attractivity and so on.

奇异摄动微分方程在振动理论、流体力学、近代物理、自动控制、海洋工程等诸多领域有着广泛的应用，是微分方程理论研究中的重要方向，也是研究热点之一。另一方面，时滞现象和随机现象在自然界中是普遍存在且不可避免的，往往是系统性能下降甚至不稳定的主要根源之一。因此，很有必要对奇异摄动随机泛函微分方程进行系统研究, 为其实际应用提供严格的数学理论基础。然而，由于其研究难度大，目前关于奇异摄动随机泛函微分方程的研究还很匮乏。基于此，我们计划研究奇异摄动随机泛函微分方程解的基本性质，如解的存在性、稳定性、有界性、吸引性等。

奇异摄动微分方程在振动理论、流体力学、近代物理、自动控制、海洋工程等诸多领域有着广泛的应用，是微分方程理论研究中的重要方向，也是研究热点之一。另一方面，时滞现象和随机现象在自然界中是普遍存在且不可避免的，往往是系统性能下降甚至不稳定的主要根源之一。因此，很有必要对奇异摄动随机泛函微分方程进行系统研究, 为其实际应用提供严格的数学理论基础。项目组按照原计划研究了奇异摄动随机泛函微分方程解的基本性质，如解的存在性、稳定性、有界性、吸引性等。