Positive Systems pp 91-99

Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 389) | Cite as

Convergence to Consensus by General Averaging

  • Dirk A. Lorenz
  • Jan Lorenz

Abstract

We investigate sufficient conditions for a discrete nonlinear non-homogeneous dynamical system to converge to consensus. We formulate a theorem which is based on the notion of averaging maps. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angeli, D., Bliman, P.A.: Stability of leaderless multi-agent systems. Extension of a result by Moreau. Mathematics of Control, Signals & Systems 18(4), 293–322 (2006)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Hegselmann, R., Ulrich Krause, U.: Opinion Dynamics Driven by Various Ways of Averaging. Computational Economics 25(4), 381–405 (2004)CrossRefGoogle Scholar
  3. 3.
    Hendrickx, J.M., Blondel, V.D.: Convergence of Different Linear and Non-Linear Vicsek Models. CESAME research report 2005.57 (2005)Google Scholar
  4. 4.
    Krause, U.: Compromise, consensus, and the iteration of means. Elemente der Mathematik 63, 1–8 (2008)MathSciNetGoogle Scholar
  5. 5.
    Lorenz, D.A., Lorenz, J.: On conditions for convergence to consensus. arXiv.org/abs/0803.2211 (March 2008)Google Scholar
  6. 6.
    Lorenz, J.: A Stabilization Theorem for Dynamics of Continuous Opinions. Physica A 355(1), 217–223 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Lorenz, J.: Repeated Averaging and Bounded Confidence – Modeling, Analysis and Simulation of Continuous Opinion Dynamics. PhD thesis, Universität Bremen (March 2007)Google Scholar
  8. 8.
    Moreau, L.: Stability of Multiagent Systems with Time-Dependent Communication Links. IEEE Transactions on Automatic Control 50(2) (2005)Google Scholar
  9. 9.
    Seneta, E.: Non-Negative Matrices and Markov Chains, 2nd edn. Springer, Heidelberg (1981)MATHGoogle Scholar
  10. 10.
    Wolfowitz, J.: Products of Indecomposable, Aperiodic, Stochastic Matrices. In: Proceedings of the American Mathematical Society Eugene, vol. 15, pp. 733–737 (1963)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Dirk A. Lorenz
    • 1
  • Jan Lorenz
    • 2
  1. 1.Institute for Analysis and AlgebraTU BraunschweigBraunschweigGermany
  2. 2.Chair of Systems Design, Department of Management, Technology, and EconomicsETH ZürichZürichSwitzerland

Personalised recommendations