The main objective of control is to modify the behavior of a dynamical system, typically with the purpose of regulating certain variables or of tracking desired signals. Often, either stability of the closed-loop system is an explicit requirement, or else the problem can be recast in a form that involves stabilization (e.g., of an error signal). For linear systems, the associated problems can now be treated fairly satisfactorily, but in the nonlinear case the area is still far from being settled. Both of the late 1980s reports [9] and [18], with dealt with challenges and future directions for research in control theory, identified the problem of stabilization of finite-dimensional deterministic systems as one of the most important open problems in nonlinear control. We discuss some questions in this area.


Nonlinear Control System Feedback Stabilization Universal Formula Feedback Design Uniform Asymptotic Stability 
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  1. [1]
    Artstein, Z., “Stabilization with relaxed controls,” Nonlinear Analysis, Theory, Methods & Applications 7 (1983): 1163–1173.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Bacciotti, A., Local Stabilizability of Nonlinear Control Systems, World Scientific, London, 1991.Google Scholar
  3. [3]
    Brockett, R.W., “Asymptotic stability and feedback stabilization,” in Differential Geometric Control theory ( R.W. Brockett, R.S. Millman, and H.J. Sussmann, eds.), Birkhauser, Boston, 1983, pp. 181–191.Google Scholar
  4. [4]
    Clarke, F.H., Methods of Dynamic and Nonsmooth Optimization. Volume 57 of CBMS-NSF Regional Conference Series in Applied Mathematics, S.I.A.M., Philadelphia, 1989.Google Scholar
  5. [5]
    Clarke, F.H., Yu.S. Ledyaev, R.J. Stern, and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.MATHGoogle Scholar
  6. [6]
    Clarke, F.H., Yu.S. Ledyaev, E.D. Sontag, and A.I. Subbotin, “Asymptotic stabilization implies feedback stabilization,” IEEE Trans. Automat. Control 42 (1997): 1394–1407MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Coron, J.-M., and L. Rosier, “A relation between continuous time-varying and discontinuous feedback stabilization,” J. Math. Systems Estim. Control 4 (1994), 67–84.MathSciNetMATHGoogle Scholar
  8. [8]
    Coron, J.M., L. Praly, and A. Teel, “Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input-output techniques,” in Trends in Control: A European Perspective ( A. Isidori, Ed.), Springer, London, 1995 (pp. 293–348 ).Google Scholar
  9. [9]
    Fleming, W.H. et. al., Future Directions in Control Theory: A Mathematical Perspective, SIAM Publications, Philadelphia, 1988.Google Scholar
  10. [10]
    Krstic, M. and H. Deng, Stabilization of Uncertain Nonlinear Systems, Springer-Verlag, London, 1998.MATHGoogle Scholar
  11. [11]
    Krstic, M., I. Kanellakopoulos, and P. Kokotovic, Nonlinear and adaptive control design, John Wiley & Sons, New York, 1995.Google Scholar
  12. [12]
    Kurzweil, J., “On the reversibility of the first theorem of Lyapunov concerning the stability of motion,” Czechoslovak Math. J. 5(80) (1955),382–398 (in Russian). English translation in: Amer. Math. Soc. Translations, Series 2, 24 (1956), 19–77.Google Scholar
  13. [13]
    Ledyaev, Yu.S., and E.D. Sontag, “A Lyapunov characterization of robust stabilization,” Proc. Automatic Control Conference, Philadelphia, June 1998.Google Scholar
  14. [14]
    Lin, Y., and E.D. Sontag, “Control-Lyapunov universal formulae for restricted inputs,” Control: Theory and Advanced Technology 10 (1995): 1981–2004.MathSciNetGoogle Scholar
  15. [15]
    Massera, J.L., “Contributions to stability theory,” Annals of Math 64 (1956): 182–206. Erratum in Annals of Math 68 (1958): 202.MATHCrossRefGoogle Scholar
  16. [16]
    Isidori, A., Nonlinear Control Systems: An Introduction, Springer-Verlag, Berlin, third ed., 1995.MATHGoogle Scholar
  17. [17]
    Jurdjevic, V. and J.P. Quinn, “Controllability and stability,” J.of Diff.Eqs. 28 (1978): 381–389.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Levis, A.H., et. al., “Challenges to control: A collective view. Report from the workshop held at the University of Santa Clara on September 18–19, 1986,” IEEE Trans. Autom. Control 32 (1987): 275–285.CrossRefGoogle Scholar
  19. [19]
    Ryan, E.P., “On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback,” SIAM J. Control Optim. 32 (1994) 1597–1604.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Sontag, E.D., Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition, Springer, New York, 1998.Google Scholar
  21. [21]
    Sontag, E.D., and H.J. Sussmann, “Remarks on continuous feedback,” Proc. IEEE Conf Decision and Control, Albuquerque, Dec. 1980, pp. 916–921.Google Scholar
  22. [22]
    Sontag, E.D., “A Lyapunov-like characterization of asymptotic controllability,” SIAM J. Control & Opt 21 (1983): 462–471.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Sontag, E.D., “A ‘universal’ construction of Artstein’s theorem on non-linear stabilization,” Systems and Control Letters, 13 (1989): 117–123.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Sussmann, H.J., “Subanalytic sets and feedback control,” J. Diff. Eqs. 31 (1979): 31–52.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Eduardo D. Sontag
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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