On O-Minimal Hybrid Systems

  • Thomas Brihaye
  • Christian Michaux
  • Cédric Rivière
  • Christophe Troestler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2993)


This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures. In particular, we recover the main result of a paper by Lafferriere G., Pappas G.J. and Sastry S. on o-minimal hybrid systems.

Mathematics Subject Classification: 68Q60, 03C64.


Hybrid System Transition System Transition Relation Hybrid Automaton Continuous Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ACH+]
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theoretical Computer Science 138, 3–34 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [AD]
    Alur, R., Dill, D.L.: A theory of timed automata. Theoretical Computer Science 126, 183–235 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [AHLP]
    Alur, R., Henzinger, T.A., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. Proc. IEEE 88, 971–984 (2000)CrossRefGoogle Scholar
  4. [ASY]
    Asarin, E., Schneider, G., Yovine, S.: On the Decidability of the Reachability Problem for Planar Differential Inclusions. In: Di Benedetto, M., Sangiovanni-Vincentelli, A. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 89–104. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. [BFH]
    Bouajjani, A., Fernandez, J.-C., Halbwachs, N.: Minimal model generation. In: Kurshan, R.P., Clarke, E.M. (eds.) CAV 1990. LNCS, vol. 531, pp. 197–203. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  6. [Cau]
    Caucal, D.: Bisimulation of context-free grammars and push-down automata. In: Modal logic and process algebra, Stanford. CSLI Lecture Notes, vol. 53, pp. 85–106 (1995)Google Scholar
  7. [vdD96]
    van den Dries, L.: o-Minimal Structures. In: Logic: from Foundations to Applications, European Logic Colloquium, pp. 137–185. Oxford University Press, Oxford (1996)Google Scholar
  8. [vdD98]
    van den Dries, L.: Tame Topology and O-minimal Structures. Cambridge University Press, Cambridge (1998)zbMATHCrossRefGoogle Scholar
  9. [Hen95]
    Henzinger, T.A.: Hybrid automata with finite bisimulations. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 324–335. Springer, Heidelberg (1995)Google Scholar
  10. [Hen96]
    Henzinger, T.A.: The Theory of Hybrid Automata. In: Proceedings of the 11th Annual Symposium on Logic in Computer Science, pp. 278–292. IEEE Computer Society Press, Los Alamitos (1996)CrossRefGoogle Scholar
  11. [HKPV]
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? Journal of Computer and System Sciences 57, 94–124 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Ho]
    Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  13. [LGS]
    Lygeros, J., Godbole, D.N., Sastry, S.: Verified hybrid controllers for automated vehicles. IEEE Transactions on Automatic Control 43(4), 522–539 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [LPS]
    Lafferriere, G., Pappas, G.J., Sastry, S.: O-Minimal Hybrid Systems. Mathematics of control, signals and systems 13, 1–21 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [TPS]
    Tomlin, C., Pappas, G.J., Sastry, S.: Conflict resolution for air traffic management: A study in multi-agent hybrid systems. IEEE Transactions on Automatic Control 43(4), 509–521 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Tr]
    Truss, J.: Infinite permutation groups II - subgroups of small index. J. Algebra 120, 494–515 (1989)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Thomas Brihaye
    • 1
  • Christian Michaux
    • 1
  • Cédric Rivière
    • 1
  • Christophe Troestler
    • 1
  1. 1.Institut de MathématiqueUniversité de Mons-HainautMons

Personalised recommendations