Departmental Papers (ESE)


In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent’s heading is updated using a local rule based on the average of its own heading plus the headings of its “neighbors.” In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent’s set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.

Document Type

Journal Article

Subject Area


Date of this Version

June 2003


Copyright 2003 IEEE. Reprinted from IEEE Transactions on Automatic Control, Volume 48, Issue 6, June 2003, pages 988-1001.
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Cooperative control, graph theory, infinite products, multiagent systems, switched systems

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Date Posted: 15 November 2004

This document has been peer reviewed.