随机时滞耦合控制系统的数学建模与动力学性质分析

项目名称： 随机时滞耦合控制系统的数学建模与动力学性质分析

项目编号： No.11301112

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 李文学

作者单位： 哈尔滨工业大学

项目金额： 23万元

中文摘要： 随机时滞耦合控制系统（简称SRCCS）是复杂系统理论的研究对象之一。由于考虑到结点系统之间的耦合拓扑、随机扰动与时滞，使SRCCS可以更好地刻画科学与生产实践中的许多复杂现象。本项目拟建立SRCCS的数学模型并分析模型的动力学性质。内容包括：（1）基于图论建立SRCCS模型与状态反馈SRCCS模型。（2）结合Kirchhoff矩阵树定理与Lyapunov方法研究SRCCS的输入状态稳定性、积分输入状态稳定性与指数输入状态稳定性。研究状态反馈SRCCS的随机稳定性和有界性。揭示随机扰动、时滞与耦合拓扑对SRCCS动力学性质的影响。(3) 针对SRCCS构造高效的数值仿真方法，并进行数值仿真。本项目期望，综合随机微分方程理论、动力系统理论和图论等，发展出一套框架性的方法，并由此对SRCCS提供一系列简单有效的稳定性与有界性判定准则。在丰富SRCCS理论的同时，也为其在工程上的应用提供理论基础。

中文关键词： 耦合系统；图论；稳定性；同步性；有界性

英文摘要： Stochastic retarded coupled control system (SRCCS) is one of the research objects in complex system theory. In science and production practice, many complex phenomena could be described by SRCCSs better, since the coupled topology between node systems, the stochastic perturbations and the retard are taken into account. In this project some mathematical models of SRCCS will be established and their dynamic properties will be analyzed. The content includes: (1) Some models of SRCCSs and state feedback SRCCSs will be established based on graph theory. (2) Input-to-state stability, integral input-to-state stability and exponential input-to-state stability for SRCCSs will be studied, by combining Kirchhoff's matrix tree theorem and Lyapunov method. The stochastic stability and boundedness of the state feedback SRCCSs will be considered. The impacts of stochastic disturbations, retard and coupled topology on the dynamics of SRCCSs will be revealed. (3)Some highly effective numerical methods of SRCCSs will be constructed and the numerical simulation will be performed. The aims of this project are that, a framework approach can be developed, by combining stochastic differential equations theory, dynamical systems theory, graph theory, etc. and then a series of easy and effective criteria of stability and boundedness for

英文关键词： coupled systems；graph theory；stability；synchronization；boundedness