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A Sound and Complete Proof Rule for Region Stability of Hybrid Systems

  • Andreas Podelski
  • Silke Wagner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4416)

Abstract

Region stability allows one to formalize hybrid systems whose trajectories may oscillate (within a given allowance) even after having ‘stabilized’. Unfortunately, until today no proof rule (giving necessary and sufficient conditions for the purpose of verifying region stability) has been available. This paper fills the gap. Our (sound and complete) proof rule connects region stability with the finiteness of specific state sequences and thus with the emerging set of verification methods for program termination.

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References

  1. 1.
    Branicky, M.S.: Stability of hybrid systems: State of the art. In: CDC’97 (1997)Google Scholar
  2. 2.
    Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. Trans. on Automatic Control (1998)Google Scholar
  3. 3.
    Bradley, A.R., Sipma, H., Manna, Z.: Termination of Polynomial Programs. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Cousot, P.: Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Relaxation and Semidefinite Programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Cook, B., Podelski, A., Rybalchenko, A.: Abstraction Refinement for Termination. In: Hankin, C., Siveroni, I. (eds.) SAS 2005. LNCS, vol. 3672, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Cook, B., Podelski, A., Rybalchenko, A.: Termination proofs for systems code. In: PLDI’06 (2006)Google Scholar
  7. 7.
    Colon, M., Sipma, H.: Synthesis of Linear Ranking Functions. In: Margaria, T., Yi, W. (eds.) ETAPS 2001 and TACAS 2001. LNCS, vol. 2031, Springer, Heidelberg (2001)Google Scholar
  8. 8.
    Colon, M., Sipma, H.: Practical Methods for Proving Program Termination. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, Springer, Heidelberg (2002)Google Scholar
  9. 9.
    Gotsman, A., et al.: Proving that software eventually does something good. In: POPL’07 (to appear)Google Scholar
  10. 10.
    Henzinger, T.A.: The Theory of Hybrid Automata. In: LICS’96 (1996)Google Scholar
  11. 11.
    Liberzon, D.: Switching in Systems and Control. Birkhäuser, Basel (2003)zbMATHGoogle Scholar
  12. 12.
    Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990)zbMATHGoogle Scholar
  13. 13.
    Pettersson, S.: Analysis and Design of Hybrid Systems. Ph.D. Thesis, Chalmers University of Technology, Göteborg, Sweden (1999)Google Scholar
  14. 14.
    Podelski, A., Rybalchenko, A.: Transition Invariants. In: LICS (2004)Google Scholar
  15. 15.
    Podelski, A., Rybalchenko, A.: A Complete Method for the Synthesis of Linear Ranking Functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, Springer, Heidelberg (2004)Google Scholar
  16. 16.
    Podelski, A., Rybalchenko, A.: Transition Predicate Abstraction and Fair Termination. In: POPL’05 (2005)Google Scholar
  17. 17.
    Podelski, A., Wagner, S.: Model Checking of Hybrid Systems: From Reachability towards Stability. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Tiwari, A.: Termination of linear programs. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, Springer, Heidelberg (2004)Google Scholar
  19. 19.
    Ye, H., Michel, A.N., Hou, L.: Stability Analysis of discontinuous Dynamical Systems with Applications. In: IFAC’96 (1996)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Andreas Podelski
    • 1
  • Silke Wagner
    • 2
  1. 1.Universität FreiburgGermany
  2. 2.Max-Planck-Institut für Informatik, SaarbrückenGermany

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