Reachability in One-Dimensional Controlled Polynomial Dynamical Systems

  • Margarita Korovina
  • Nicolai Vorobjov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7162)


In this paper we investigate a case of the reachability problem in controlled o-minimal dynamical systems. This problem can be formulated as follows. Given a controlled o-minimal dynamical system initial and target sets, find a finite choice of time points and control parameters applied at these points such that the target set is reachable from the initial set. We prove that the existence of a finite control strategy is decidable and construct a polynomial complexity algorithm which generates finite control strategies for one-dimensional controlled polynomial dynamical systems. For this algorithm we also show an upper bound on the numbers of switches in finite control strategies.


Integral Curve Switching Point Integral Curf Reachability Problem Local Minimum Point 
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. 1.
    Asarin, E., Maler, O., Pnueli, A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoret. Comput. Sci. 138(1), 35–65 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Asarin, E., Mysore, V., Pnueli, A., Schneider, G.: Low dimensional Hybrid Systems - Decidable, Undecidable, Don’t Know. Accepted for publication in Information and Computation (2010)Google Scholar
  3. 3.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Results in Mathematics and Related Areas (3), vol. 36. Springer, Berlin (1998)zbMATHGoogle Scholar
  4. 4.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry, 2nd edn. Algorithms and Computation in Mathematics, vol. 10, p. x+662. Springer, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Brihaye, T., Michaux, C.: On the expressiveness and decidability of o-minimal hybrid systems. Journal of Complexity 21(4), 447–478 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brihaye, T.: A note on the undecidability of the reachability problem for o-minimal dynamical systems. Math. Log. Q. 52(2), 165–170 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bouyer, P., Brihaye, T., Chevalier, F.: Control in o-minimal Hybrid Systems. In: Proc. LICS 2006, pp. 367–378 (2006)Google Scholar
  8. 8.
    Korovina, M., Vorobjov, N.: Pfaffian Hybrid Systems. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 430–441. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Korovina, M., Vorobjov, N.: Computing combinatorial types of trajectories in Pfaffian Dynamics. Journal of Logic and Algebraic Programming 79(1), 32–37 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lafferriere, G., Pappas, G.J., Sastry, S.: O-minimal hybrid systems. Math. Control Signals Systems 13(1), 1–21 (2000)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Margarita Korovina
    • 1
    • 2
  • Nicolai Vorobjov
    • 3
  1. 1.The University of ManchesterUK
  2. 2.IISSB RASNovosibirskRussia
  3. 3.University of BathUK

Personalised recommendations