一致仿缩矩阵无穷乘积的收敛性及在Vicsek模型同步问题中的应用  

THE CONVERGENCE OF INFINITE PRODUCT OF UNIFORMLY PARACONTRACTING MATRICES AND ITS APPLICATION IN CONSENSUS OF THE VICSEK MODEL

作　　者：王兴平[1] 程兆林[2] 

机构地区：[1]海军航空工程学院数学与系统科学研究所,烟台264001 [2]山东大学数学学院,济南250100

出　　处：《系统科学与数学》2013年第6期724-731,共8页Journal of Systems Science and Mathematical Sciences

摘　　要：研究仿缩矩阵无穷乘积的收敛问题.将文献中的仿缩矩阵概念推广,提出一致仿缩集合概念,并推导出一致仿缩集的若干的有用的性质.基于这些性质,利用R^n空间的紧致性质证明了如下结论:取自一致仿缩集的任意无穷矩阵乘积都是收敛的,并给出了极限矩阵的构造.这一结果改进了现有的结论.应用得到的结论,研究了Vicsek模型的同步问题,用新的方法证明了一个基于最近邻居规则的粒子运动方位同步的结论.This paper addresses the convergence of infinite product of matrices. By extending the paracontracting matrix notion in the existing literature, we propose the concept of uniformly paracontracting set and develop some interesting properties of it. Based on these properties, we prove a result by the Bolzano-Weierstrass theorem that any infinite matrices product chosen from a uniformly paracontracting set is convergent. Moreover, the construction of the limit matrix is explicitly presented as well. This result includes and improves the existing results on the convergence of infinite product of finite paracontracting matrices. With the obtained result, we study the consensus problem of the Vicsek model with time-varying Laplacian matrix. In a new way, a consensus result based on a nearest neighbor rule is proved that all the particles in the Vicsek model will eventually move in the same direction provided that the topology is jointly connected over any infinite interval.

关 键 词：一致仿缩集合 矩阵无穷乘积 Vicsek模型 同步 

分 类 号：O151.21[理学—数学]