Overview of complexity and decidability results for three classes of elementary nonlinear systems

  • Vincent D. Blondel
  • John N. Tsitsiklis
Part A Learning And Computational Issues
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 241)


It has become increasingly apparent this last decade that many problems in systems and control are NP-hard and, in some cases, undecidable. The inherent complexity of some of the most elementary problems in systems and control points to the necessity of using alternative approximate techniques to deal with problems that are unsolvable or intractable when exact solutions are sought.

We survey some of the decidability and complexity results available for three classes of discrete time nonlinear systems. In each case, we draw the line between the problems that are unsolvable, those that are NP-hard, and those for which polynomial time algorithms are known.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Asarin et al., 1995]
    [Asarin et al., 1995] Asarin, A., O. Maler and A. Pnueli (1995). Reachability analysis of dynamical systems having piecewise-constant derivatives, Theoretical Computer Science, 138, 35–66.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Blondel and Tsitsiklis, 1997a]
    Blondel, V. D. and J. N. Tsitsiklis (1997). Complexity of elementary hybrid systems, Proc. of the 4th European Control Conference, Brussels.Google Scholar
  3. [Blondel and Tsitsiklis, 1997b]
    Blondel, V. D. and J. N. Tsitsiklis (1997). When is a pair of matrices mortal?, Information Processing Letters, 63, 283–286.CrossRefMathSciNetGoogle Scholar
  4. [Blondel and Tsitsiklis, 1997c]
    Blondel, V. D. and J. N. Tsitsiklis (1997). Survey of complexity results for systems and control problems, (in preparation).Google Scholar
  5. [Blondel and Tsitsiklis, 1997d]
    Blondel, V. D. and J. N. Tsitsiklis (1997). Decidability limits for low-dimensional piecewise linear systems, (submitted).Google Scholar
  6. [Davis, 1982]
    Davis, M. (1982). Computability and Unsolvability, New York, Dover.zbMATHGoogle Scholar
  7. [Garey and Johnson, 1979]
    Garey, M. R. and D. S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman and Co., New York.zbMATHGoogle Scholar
  8. [Hopcroft and Ullman, 1969]
    Hopcroft, J. E. and J. D. Ullman (1969). Formal languages and their relation to automata, Addison-Wesley.Google Scholar
  9. [Hyotyniemu, 1997]
    Hyotyniemu, H. (1997). On unsolvability of nonlinear system stability, Proc. ECC conference, to appear.Google Scholar
  10. [Kilian and Siegelmann, 1996]
    Kilian, J. and H. Siegelmann (1996). The dynamic universality of sigmoidal neural networks, Information and Computation, 128, 48–56.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Koiran, 1996]
    Koiran, P. (1996). A family of universal recurrent networks, Theor. Comp. Sciences, 168, 473–480.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Koiran et al., 1994]
    [Koiran et al., 1994] Koiran, P., M. Cosnard and M. Garzon (1994). Computability properties of low-dimensional dynamical systems, Theoretical Computer Science, 132, 113–128.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Liu and Michel, 1994]
    Liu, D. and A. Michel (1994). Dynamical systems with saturation nonlinearities: analysis and design, Springer-Verlag, London, 1994.zbMATHGoogle Scholar
  14. [Matiyasevich and Sénizergues, 1996]
    Matiyasevich, Y. and G. Sénizergues (1996). Decision problem for semi-Thue systems with a few rules, preprint.Google Scholar
  15. [Papadimitriou, 1994]
    Papadimitriou, C. H. (1994). Computational complexity, Addison-Wesley, Reading.zbMATHGoogle Scholar
  16. [Siegelmann and Sontag, 1991]
    Siegelmann, H. T. and E. D. Sontag (1991). Turing computability with neural nets, Applied Mathematics Letters, 4, 77–80.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Siegelmann and Sontag, 1995]
    Siegelmann, H. and E. Sontag (1995). On the computational power of neural nets, J. Comp. Syst. Sci., 132–150.Google Scholar
  18. [Sontag, 1981]
    Sontag, E. (1981). Nonlinear regulation: the piecewise linear approach, IEEE Trans. Automat. Control, 26, 346–358.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [Sontag, 1990]
    Sontag, E. (1990). Mathematical control theory, Springer, New York.zbMATHGoogle Scholar
  20. [Sontag, 1993]
    Sontag, E. (1993). Neural networks for control in Essays on Control: Perspectives in the Theory and its Applications (H.L. Trentelman and J.C. Willems, eds.), Birkhauser, Boston, pp. 339–380.Google Scholar
  21. [Sontag, 1995]
    Sontag, E. (1995). From linear to nonlinear: some complexity comparisons, Proc. IEEE Conference Decision and Control, New Orleans, 2916–2920.Google Scholar
  22. [Sontag, 1996]
    Sontag, E. (1996). Interconnected automata and linear systems: A theoretical framework in discrete-time, in Hybrid Systems III: Verification and Control (R. Alur, T. Henzinger, and E.D. Sontag, eds.), Springer, 436–448.Google Scholar
  23. [Tsitsiklis, 1994]
    Tsitsiklis, J. N. (1994). Complexity theoretic aspects of problems in control theory, Transactions of the eleventh Army, ARO Report.Google Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Vincent D. Blondel
    • 1
  • John N. Tsitsiklis
    • 2
  1. 1.Institute of MathematicsUniversity of Liège B37LiègeBelgium
  2. 2.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations