On the Stability of Zeno Equilibria

  • Aaron D. Ames
  • Paulo Tabuada
  • Shankar Sastry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3927)


Zeno behaviors are one of the (perhaps unintended) features of many hybrid models of physical systems. They have no counterpart in traditional dynamical systems or automata theory and yet they have remained relatively unexplored over the years. In this paper we address the stability properties of a class of Zeno equilibria, and we introduce a necessary paradigm shift in the study of hybrid stability. Motivated by the peculiarities of Zeno equilibria, we consider a form of asymptotic stability that is global in the continuous state, but local in the discrete state. We provide sufficient conditions for stability of these equilibria, resulting in sufficient conditions for the existence of Zeno behavior.


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  1. 1.
    Ames, A.D., Zheng, H., Gregg, R.D., Sastry, S.: Is there life after Zeno? Taking executions past the breaking (Zeno) point (Submitted to the 2006 American Control Conference)Google Scholar
  2. 2.
    Brogliato, B.: Nonsmooth Mechanics. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  3. 3.
    Ames, A.D., Abate, A., Sastry, S.: Sufficient conditions for the existence of Zeno behavior. In: 44th IEEE Conference on Decision and Control and European Control Conference ECC (2005)Google Scholar
  4. 4.
    Branicky, M.S.: Stability of hybrid systems: State of the art. In: Proceedings of the 36th IEEE Conference on Decision and Control, San Diego (1997)Google Scholar
  5. 5.
    Heymann, M., Lin, F., Meyer, G., Resmerita, S.: Analysis of Zeno behaviors in hybrid systems. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vagas (2002)Google Scholar
  6. 6.
    Johansson, K.H., Lygeros, J., Sastry, S., Egerstedt, M.: Simulation of Zeno hybrid automata. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix (1999)Google Scholar
  7. 7.
    Zhang, J., Johansson, K.H., Lygeros, J., Sastry, S.: Zeno hybrid systems. Int. J. Robust and Nonlinear Control 11(2), 435–451 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Zheng, H., Lee, E.A., Ames, A.D.: Beyond Zeno: Get on with it (To appear in Hybrid Systems: Computation and Control, 2006)Google Scholar
  9. 9.
    Ames, A.D., Sastry, S.: A homology theory for hybrid systems: Hybrid homology. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 86–102. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Ames, A.D., Sangiovanni-Vincentelli, A., Sastry., S.: Homogenous semantic preserving deployments of heterogenous networks of embedded systems. In: Workshop on Networked Embedded Sensing and Control, Notre Dame (2005)Google Scholar
  11. 11.
    Lane, S.M.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5. Springer, Heidelberg (1998)MATHGoogle Scholar
  12. 12.
    Ames, A.D., Tabuada, P., Sastry, S.: H-categories and graphs (Technical Note)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Aaron D. Ames
    • 1
  • Paulo Tabuada
    • 2
  • Shankar Sastry
    • 1
  1. 1.Department of Electrical Engineering and Computer SciencesUniversity of California at BerkeleyBerkeleyUSA
  2. 2.Department of Electrical EngineeringUniversity of Notre DameNotre DameUSA

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