分数阶时滞和退化动力网络的同步与结构研究

Synchronization and structure of complex networks have become a very important hot issue, ability and speed of synchronization of networks are closely related to stability and transmission efficiency of networks, and the structure of networks has important effect on its synchronization. Since time delay、fractional-order derivatives and Singular phenomenon widely exist in many practical systems and have important effection on property of the systems, this project will deeply consider fractional-order、time delay、singular phenomenon and structure of networks. First, combined with theory and method of fractional-order、functional differential equations and singular differential systems, we study stability of fractional-order delayed and singular differential systems by using Lyapunov direct method; Then, we investigate synchronization and control of fractional-order delayed and singular dynamical networks based on our obtained results of stability; Finally, the relationship between the structure and ability、speed of syncronization of networks will be erected. The scientific problems that will be resolved in this project have a profound theoretical significance and very important value of the applications, not only enrich the stability theory of fractional differential equations, but make fractional-order delayed and singular differential equations describe the concrete complex systems more precisely and reveal the intrinsical effect of fractional-order、time delay and singular factors.

复杂网络的同步和结构研究是复杂系统领域的热点课题，网络的同步能力与同步速度关系到网络的稳定性和传输效率等实际问题，而网络的结构性能对其同步能力和同步速度有着重要的，甚至是本质的影响。由于时滞、分数阶导数和退化现象广泛存在于各种实际系统中，且对系统的性态具有重要的影响，本项目将充分考虑分数阶、时滞、退化和网络结构等因素，首先结合分数阶、泛函微分方程和退化系统的最新理论和方法，利用李雅普诺夫直接法研究分数阶时滞和退化系统的稳定性；其次利用稳定性理论研究分数阶时滞和退化动力网络的同步控制问题；最后从网络结构着手，利用谱图理论研究网络的结构参数与同步能力和同步速度之间的联系。本项目拟解决的科学问题具有重要的理论意义和应用价值，不仅能丰富和发展分数阶微分方程的稳定性理论，而且能促使分数阶时滞和退化微分方程更精确地描述具体的复杂系统，揭示分数阶导数、时滞和退化因素对系统内在和本质的影响。

复杂网络的同步和结构研究是复杂系统领域中的热点课题。由于时滞、分数阶导数和退化现象广泛存在于各种实际系统中，且对系统的性态具有重要的影响，本项目主要研究：分数阶时滞、退化系统解的存在性、稳定性和可控性; 分数阶时滞网络的同步控制；分数阶时滞耦合多智能体系统的一致性控制等问题。取得的主要研究成果有：(1)提出了一种简便有效的新方法研究Riemann-Liouville意义下分数阶时滞和退化系统的稳定性，这种新方法允许我们计算李雅普诺夫函数的整数阶导数，避免了计算其分数阶导数所带来的复杂性；(2)克服了分数阶微积分算子不具有半群性质这一困难，提出了新型的分数阶Razumikhind方法，并给出了Caputo意义下分数阶时滞系统的渐近稳定的若干新结果，这种方法不仅适用于分数阶有界时滞系统，还适用于分数阶时变时滞和分布时滞系统；(3)利用所获得的稳定性结果研究了几类分数阶时滞耦合网络的同步控制和多智能体系统的一致性控制问题；(4)研究了几类分数阶时滞系统解的存在性和可控性问题, 并给出了一些简单的充分条件。本项目所解决的科学问题不仅丰富了分数阶微分系统理论知识，还能为其在物理、工程学等众多领域的广泛应用提供理论指导和技术支持。