几类随机动力系统的稳定性研究

Stability is one of the most basic and important characteristics of dynamical systems, which is the first consideration of any system analysis and control system design. The stability of dynamical systems is easily affected by many uncertain factors such as unavoidable delays, random noises and impulses, so it is very important to study the influence of these factors on the systems. In this project, we plan to study stability of several classes of stochastic dynamical systems, including (1) Stability of impulsive neutral stochastic functional differential equations; (2) Stability of stochastic neutral neural networks with impulses; (3) Stability of mild solutions to neutral stochastic functional partial differential equations driven by fractional Brownian motion. Due to the simultaneous occurrence of neutral terms, stochastic disturbance and impulses, we encountered some difficulties in dealing with it. This project plan to combine methods and techniques in stochastic analysis, dynamical systems, functional analysis, and stochastic differential equations to study it. This project is expected to establish stability criteria by using effective methods, and to analyze the effects of delays, random noises and impulses on the stability of the systems. By studying this project, we will perfect and deepen the study of stochastic dynamical systems, so it is a very meaningful work.

稳定性是动力系统最基本也是最重要的特性之一，是任何系统分析和控制系统设计都必须首先考虑的问题。动力系统的稳定性容易受到不可避免的时滞、随机噪声和脉冲等诸多不确定因素的影响，因此研究这些因素对系统的影响就显得十分重要。本项目计划研究几类随机动力系统的稳定性问题，主要包括：（1）脉冲中立型随机泛函微分方程的稳定性；（2）脉冲中立型随机神经网络系统的稳定性；（3）由分数布朗运动驱动的脉冲中立型随机泛函偏微分方程 mild解的稳定性。由于中立项、随机干扰和脉冲的同时出现，我们在研究方法上遇到了一些问题。本项目计划通过结合随机分析、动力系统、泛函分析和随机微分方程等领域的思想方法和工具对它们进行研究。期望通过选择有效的研究方法，建立系统稳定性的判定准则，进而分析时滞、随机噪声和脉冲对系统稳定性的影响。本课题的完成将完善和深化对随机动力系统的研究，是十分有意义的工作。

动力系统的稳定性容易受到不可避免的时滞、随机噪声和脉冲等诸多不确定因素的影响，因此研究这些因素对系统的影响就显得十分重要。本项目主要针对于n维空间中的脉冲中立型随机泛函微分方程零解的稳定性和Hilbert空间中由fBm驱动的脉冲中立型随机泛函偏微分方程mild解的稳定性进行了讨论，具体来讲有以下三部分：（1）脉冲中立型随机泛函微分方程的稳定性；（2）脉冲中立型随机神经网络系统的稳定性；（3）由分数布朗运动驱动的脉冲中立型随机泛函偏微分方程mild解的稳定性。本项目利用不动点定理和逐步逼近法建立了解的存在唯一性的充分条件；利用不动点定理和脉冲-积分不等式方法，建立了系统稳定性的判定准则。本课题的完成将完善和深化对随机动力系统的研究，是十分有意义的工作。