右端不连续复杂网络的同步与控制

Synchronization is one of the most important collective behaviors of complex dynamical networks. Dynamical systems with discontinuous right-hand side （ induced by the discontinuity of functions on the right-hand side） have potential applications in many applied sciences and engineerings. Therefore, it is imperative to study the synchronization issue and its control problem for complex networks with discontinuous right-hand side. This project studies the problem according to different characteristics of discontinuous functions at discontinuity points.Based on theories and techniques such as Filippov's differential inclusion theory, stochastic theory, algebraic graph theory, matrices inequalities and Lyapunov functional method, this project aims to research the synchronization and its robustness of delayed complex networks with discontinuous right-hand side. Moreover, same problems shall also be considered under the conditions that this kind complex networks have reaction-diffusion terms and/or Markovian chains. Then synchronization criteria, which are easily verified, will be given for the considered systems. On the other hand, different kind of novel controllers guaranteeing the synchronization of the complex networks will be designed. Furthermore, efforts will be taken to find quantitative methods to describe the capacity and robustness of synchronization. By carrying out this project, we try to establish several novel theories and methods on studying the synchronization and corresponding control problem of complex networks with discontinuous right-hand side,which can make up for some of the shortcomings of the existing methods and accelerate the development of nonlinear science, information science and relative research and applied fields.

同步是动态复杂网络重要的群体行为。由函数的不连续性引起的右端不连续的动力系统在应用科学和工程中广泛存在而且有很多重要的用途。因此，有必要研究右端不连续复杂网络的同步及其控制问题。本项目拟根据不连续函数在间断点处的不同情况分类进行研究,结合Filippov微分包含理论、随机理论、矩阵不等式、Lyapunov函数方法等理论和技巧，研究有各种时滞的右端不连续复杂网络的同步及其鲁棒性等动力学行为，以及在具有反应扩散项和参数依赖于马尔科夫链等情况下的相关问题，给出这类复杂网络达到同步的容易验证的判据。设计各种不同的新的控制器使右端不连续的动态复杂网络在以上情况下达到同步，并寻找在这些情形下刻画其同步能力以及同步强壮性的定量方法；建立若干有创新性的研究右端不连续复杂网络同步与控制的理论与方法，以弥补某些现有方法的不足，促进非线性科学和信息科学等相关领域理论和应用研究的发展。

由函数的不连续性引起的右端不连续的动力系统在应用科学和工程中广泛存在而且有很多重要的用途，本项目研究右端不连续复杂网络的同步与控制问题。研究计划中的主要问题已经完成。在项目执行期间，以下问题得到完美解决：Filippov解的存在性、牵制控制下的同步、时滞脉冲控制下的指数同步、右端不连续混沌系统的有限时间同步、聚类同步、不确定外界干扰的不连续系统的滞后同步。给出了刻画右端不连续复杂网络同步能力以及同步强壮性的定量方法；建立了若干有创新性的研究右端不连续复杂网络同步与控制的理论与方法。另外，对连续的复杂网络也进行了深入的研究。发表的杂志包括信息科学、数学、控制科学等领域，研究结果丰富了非线性科学和信息科学等相关领域的理论框架。