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  and , 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.
Unable to display preview. Download preview PDF.
- Bacciotti, A., Local Stabilizability of Nonlinear Control Systems, World Scientific, London, 1991.Google Scholar
- 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
- 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
- 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
- Fleming, W.H. et. al., Future Directions in Control Theory: A Mathematical Perspective, SIAM Publications, Philadelphia, 1988.Google Scholar
- Krstic, M., I. Kanellakopoulos, and P. Kokotovic, Nonlinear and adaptive control design, John Wiley & Sons, New York, 1995.Google Scholar
- 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
- Ledyaev, Yu.S., and E.D. Sontag, “A Lyapunov characterization of robust stabilization,” Proc. Automatic Control Conference, Philadelphia, June 1998.Google Scholar
- Sontag, E.D., Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition, Springer, New York, 1998.Google Scholar
- 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