具有耦合时滞的分数阶复杂网络的稳定性和同步控制

With the development of fractional calculus theory, the study on dynamic behavior of fractional-order complex networks has become the current research hotspot. We carry out research on fractional-order complex networks with and without coupling delays, focus on the stability and synchronization of the considered networks. The research contents of this project are as follows: 1. We construct Lyapunov function/functional of the considered networks based on Kirchhoff matrix tree theorem and Lyapunov function/functional of each node system. By using the fractional Lyapunov method, we study stability of fractional-order delayed complex networks with/without impulsive effect, and some novel stability criteria of the considered networks are derived. The effects of the order of fractional-order systems, coupling delays, impulse and topology structure on the stability of fractional-order complex networks are further discussed. 2. Different types of synchronization of fractional-order complex networks with and without coupling delays can be realized by adaptive control, impulsive control, adaptive impulsive control, impulsive pinning control, adaptive impulsive pinning control and other control strategies, and some novel synchronization criteria of the considered networks are derived. The effects of the order of fractional-order systems, coupling delays, feedback gain, and topology structure on the stability of fractional-order complex networks are further discussed.

随着分数阶微积分理论的发展，分数阶复杂网络的动力学行为研究已成为当前研究的热点。本项目拟对同时具有无时滞耦合和常时滞耦合的分数阶复杂网络开展研究，重点研究网络的稳定性和同步控制问题。本项目研究内容如下：1. 拟用Kirchhoff矩阵树定理和节点系统的Lyapunov函数/泛函构造分数阶网络系统的Lyapunov函数/泛函，利用分数阶Lyapunov 方法研究具有/不具有脉冲效应的分数阶耦合时滞网络的稳定性，得到网络稳定的判别准则，并探讨分数阶系统的阶数、耦合时滞、脉冲、网络的拓扑结构等对分数阶网络系统稳定性的影响。2. 拟采用自适应控制、脉冲控制、自适应-牵制控制、脉冲-牵制控制、自适应脉冲-牵制控制等控制策略，实现同时具有无时滞耦合和常时滞耦合的分数阶复杂网络的不同类型同步，得到网络同步的判别准则，并讨论分数阶的阶数、耦合时滞、反馈增益、网络的拓扑结构对复杂网络同步效果的影响。

随着分数阶微积分理论的发展，分数阶复杂网络的动力学行为研究已成为当前研究的热点.本项目主要对无时滞耦合和常时滞耦合的分数阶复杂动力网络的稳定性和同步控制开展研究。涉及的稳定性有鲁棒指数稳定性；涉及的控制策略包括线性控制策略，自适应控制策略，混合连续控制策略，混杂脉冲控制策略，有限时间控制策略；涉及的同步类型有完全同步、拟同步、有限时间同步。给出网络鲁棒指数稳定判据和不同类型同步的判别准则，并讨论了分数阶的阶数、脉冲、耦合时滞、反馈增益、网络的拓扑结构对复杂动力网络稳定和同步效果的影响。此外，还对整数阶复杂动力网络、斑块环境下的SI传染病模型及HIV病毒-细胞动力学模型进行了研究，建立了系统稳定及同步判据。这些研究不仅丰富和完善了微分方程稳定性理论、网络科学理论、控制论等相关学科内容，还为控制科学、信息工程、计算机科学等诸多工程领域提供了理论支持和新的研究工具，引起了国内外学者的广泛关注和跟踪。