Authors 
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Document type 
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Communication à un colloque (Conference Paper) – Présentation orale avec comité de sélection

Abstract 
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The Vicsek model describes the evolution of a system composed by different agents moving in the plane. Each agent has a constant speed and updates its heading using a local rule depending on the headings of its “neighbors”. Although the original model by Vicsek is nonlinear, most of the convergence results obtained so far deal with linearized versions. In this paper, we introduce a new linear model in which the relative importance of each neighbor can vary with the distance and we prove the convergence of all the agents headings. For this purpose, we derive a theorem on the convergence of long products of stochastic products that applies to infinite set of matrices. Using this result we also prove convergence properties for the original nonlinear Vicsek model.
Moreau [7] obtains similar results to ours but using a proof technique based on convexity and system theory. We present here proofs that are based on elementary linear algebra tools. The results we obtain are somewhat weaker than those of Moreau but have the advantage of being indistinctly applicable to continuous and discontinuous systems. 
Access type 
: 
Accès restreint 
Publication date 
: 
2006 
Language 
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Anglais 
Conference 
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"MTNS 2006", Kyoto, Japan 
Peer reviewed 
: 
yes 
Host document 
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"Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems" 
Publication status 
: 
Publié 
Affiliation 
: 
UCL
 FSA/INMA  Département d'ingénierie mathématique

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