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FLOCKING OF A THERMODYNAMIC CUCKER-SMALE MODEL WITH LOCAL VELOCITY INTERACTIONS

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Acta Mathematica Scientia 2024年第2期44卷 632-649页

作者：金春银李双智 College of Science China Agricultural UniversityBeijing100083China

In this paper, we study the flocking behavior of a thermodynamic Cucker–Smale model with local velocity interactions. Using the spectral gap of a connected stochastic matrix, together with an elaborate estimate on perturbations of a linearized system, we provide a sufficient framework in terms of initial data and model parameters to guarantee flocking. Moreover, it is shown that the system achieves a consensus at an exponential rate.

关键词： flocking local interaction thermodynamical Cucker-Smale model stochastic matrix neighbor graph

Estimates for Eigenvalues of stochastic Matrices

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Acta Mathematicae Applicatae Sinica 2011年第3期27卷 503-508页

作者： Jia-li Zhou Min Wu Shu-you Zhang Guo-ping Yang Department of Mathematics Zhejiang University of Technology Zhejiang 31(}023 China State Key Lab of CAD & CG Zhejiang University Zhejiang 310027 China Department of Mathematics Zhejiang University of Science &= Technology Zhejiang 310023 China College of Information Technology Zhejiang Chinese Medical University Zhejiang 310053 China

It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r 2 , ··· , r N } be the eigenvalues of stochastic matrix X of size N × N . We will present in this paper a simple necessary and sufficient condition for X such that ｜r j ｜〈 1, j = 2, ··· , N . Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.

关键词： Directed graph eigenvalues spectral radius stochastic matrix

Robust consensus of multi-agent systems with noise

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Science in China(Series F) 2009年第5期52卷 824-834页

作者： WANG Lin LIU ZhiXin Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

The consensus problem of multi-agent systems has attracted wide attention from researchers in recent years, following the initial work of Jadbabaie et al. on the analysis of a simplified Vicsek model. While the original Vicsek model contains noise effects, almost all the existing theoretical results on consensus problem, however, do not take the noise effects into account. The purpose of this paper is to initiate a study of the consensus problems under noise disturbances. First, the class of multi-agent systems under study is transformed into a general time-varying system with noise. Then, for such a system, the equivalent relationships are established among （i） robust consensus, （ii） the positivity of the second smallest eigenvalue of a weighted Laplacian matrix, and （iii） the joint connectivity of the associated dynamical neighbor graphs. Finally, this basic equivalence result is shown to be applicable to several classes of concrete multi-agent models with noise.

关键词： time-varying systems stochastic matrix robust consensus connectivity exponential stability Vicsek model

Connectivity and synchronization of Vicsek model

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Science in China(Series F) 2008年第7期51卷 848-858页

作者： LIU ZhiXin GUO Lei Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190 China

The collective behavior of multi-agent systems is an important studying point for the investigation of complex systems, and a basic model of multi-agent systems is the so called Vicsek model, which possesses some key features of complex systems, such as dynamic behavior, local interaction, changing neighborhood, etc. This model looks simple, but the nonlinearly coupled relationship makes the theoretical analysis quite complicated. Jadbabaie et al. analyzed the linearized heading equations in this model and showed that all agents will synchronize eventually, provided that the neighbor graphs associated with the agents＇ positions satisfy a certain connectivity condition. Much subsequent research effort has been devoted to the analysis of the Vicsek model since the publication of Jadbabaie＇s work. However, an unresolved key problem is when such a connectivity is satisfied. This paper given a sufficient condition to guarantee the synchronization of the Vicsek model, which is imposed on the model parameters only. Moreover, some counterexamples are given to show that the connectivity of the neighbor graphs is not sufficient for synchronization of the Vicsek model if the initial headings are allowed to be in [0,2π）, which reveals some fundamental differences between the Vicsek model and its linearized version.

关键词： Vicsek model synchronization stochastic matrix connectivity neighbor graph

A NOTE ON THE stochastic ROOTS OF stochastic MATRICES

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Systems Science and Systems Engineering 2003年第2期12卷 210-223页

作者： Eldon GUNN Department of Industrial Engineering Dalhousie University HalifaxCanada B3J 2X4

In this paper, we study the stochastic root matrices of stochastic matrices. All stochastic roots of2×2 stochastic matrices are found explicitly.A method based on characteristic polynomial of matrix isdeveloped to find all real root matrices that are functions of the original 3×3 matrix, including allpossible(function)stochastic root matrices. In addition, we comment on some numerical methods forcomputing stochastic root matrices of stochastic matrices.

关键词： Markov chain stochastic matrix matrix root linear algebra

Robust Consensus of Multi-agent Systems with Noise

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Robust Consensus of Multi-agent Systems with Noise

第二十六届中国控制会议

作者： Wang Lin,Liu Zhixin,Guo Lei Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100080,P.R.China

The consensus problem of multi-agent systems has attracted wide attention from researchers in recent years,following the initial work of Jadbabaie et al [1] on the analysis of a simplified Vicsek *** the original Vicsek model contains noise effects,almost all the existing theoretical results on consensus problem,however,do not take the noise effects into *** purpose of this paper is to initiate a study of the consensus problems under noise ***,the class of multi-agent systems under study is transformed into the following general time-varying systems with noises: x（t+1） = P（t）x（t）+w（t+1）,where {P（t）} is a sequence of nonnegative stochastic ***,for such a general time-varying systems,the equivalent relationships are established among （i） robust consensus,（ii） the positivity of the second smallest eigenvalue of a weighted Laplacian matrix,and （iii） the joint connectivity of the associated dynamical neighbor ***,this basic equivalence result is shown to be applicable to several class of concrete multi-agent models with noises.

关键词： Time-varying systems stochastic matrix graph theory robust consensus Laplacian matrix connectivity exponen- tial stability Vicsek model

Consensus of Second Order Discrete-Time Multi-Agent Systems with Switching Topology

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Consensus of Second Order Discrete-Time Multi-Agent Systems ...

第三十届中国控制会议

作者： HO Daniel W.C. Department of Mathematics City University of Hong KongHong KongP.R.China

＜正＞Over the last few years,the second order multi-agent systems（MAS） have been intensively *** is well known that most of known results on the consensus of second order MAS depend on an important assumption that all system matrices have a common left *** is because that a common quadratic Lyapunov function can be constructed based on the above *** overcome the nonexistence of a common left eigenvector,a convex analysis approach is introduced in this *** particular,several consensus criteria are obtained under the condition of nonexistence of a common left *** indicates some potential engineering applications in the near future.

关键词： Convex Analysis Strongly Connected Graph stochastic matrix Consensus Second Order Multi-Agent Systems

Robust Consensus of Multi-Agent Systems under Directed Information Exchanges

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Robust Consensus of Multi-Agent Systems under Directed Infor...

第二十七届中国控制会议

作者： Wang Lin,Guo Lei Key Laboratory of Systems and Control,Chinese Academy of Sciences,Beijing 100190,P.R.China

This paper considers the problem of robust consensus for a basic class of multi-agent systems with bounded disturbances and with directed information flow.A necessary and sufficient condition on the robust consensus is first presented,which is then applied to the analysis,control and decision making problems in the noise *** particular,we will show how a soft control technique will synchronize a group of autonomous mobile agents without changing the existing local rule of interactions,and without assuming any kind of connectivity conditions on the system trajectories.

关键词： Time-varying systems stochastic matrix Robust consensus Directed graph Spanning tree Vicsek model Soft control Decision-making process

Some spectral properties of positive matrices

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Some spectral properties of positive matrices

中国数学力学物理学高新技术交叉研究会第8届学术研讨会

作者： Jiang Niansheng (Dept.of Computer Science and Mathematics,Sichuan Three Gorges College Chongqing 404000) Chen Zili (Dept.of Applied Mathematics,Southwest Jiaotong University Chengdu 610031)

＜正＞There have been a great deal of positive results concerning the positive matrices in many *** spectral properties of positive matrices we will survey some of what is known in this area,

关键词： Positive matrix Spectrum Ergodic stochastic matrix