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Sufficient lyapunov-like conditions for stabilization

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Abstract

In this paper we study the stabilizability problem for nonlinear control systems. We provide sufficient Lyapunov-like conditions for the possibility of stabilizing a control system at an equilibrium point of its state space. The stabilizing feedback laws are assumed to be smooth except possibly at the equilibrium point of the system.

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References

  1. D. Aeyels, Stabilization of a class of nonlinear systems by a smooth feedback control,Systems Control Lett.,5 (1985), 289–294.

    Article  MathSciNet  Google Scholar 

  2. Z. Artstein, Stabilization with relaxed controls,Nonlinear Anal.,7 (1983), 1163–1173.

    Article  MathSciNet  Google Scholar 

  3. J. P. Aubin and A. Cellina,Differential Inclusions, Springer-Verlag, New York, 1984.

    MATH  Google Scholar 

  4. A. Bacciotti, The local stabilizability problem for nonlinear systems,IMA J. Math. Control Inform.,5 (1988), 27–39.

    Article  MathSciNet  Google Scholar 

  5. A. Bacciotti, Further remarks on potentially global stabilization, submitted.

  6. S. P. Banks, Stabilizability of finite- and infinite-dimensional bilinear systems,IMA J. Math. Control Inform.,3 (1986), 255–271.

    Article  Google Scholar 

  7. B. R. Barmish, M. J. Corless, and G. Leitmann, A new class of stabilizing controllers for uncertain dynamical systems,SIAM J. Control Optim.,21 (1983), 246–255.

    Article  MathSciNet  Google Scholar 

  8. N. P. Bhatia and G. P. Szegö,Stability Theory of Dynamical Systems, Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

  9. R. W. Brockett, Asymptotic stability and feedback stabilization, inDifferential Geometric Control Theory (R. W. Brockett, R. S. Millman, and H. J. Sussmann, eds.), pp. 181–191, Birkhauser, Boston, 1983.

    Google Scholar 

  10. M. J. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform boundedness for uncertain dynamical systems,IEEE Trans. Automat. Control,26 (1981), 1139–1144.

    Article  MathSciNet  Google Scholar 

  11. P. E. Crouch, Spacecraft attitude control and stabilization: applications of geometric control theory,IEEE Trans. Automat. Control,29 (1984), 321–333.

    Article  Google Scholar 

  12. P. O. Gutman, Stabilizing controllers for bilinear systems,IEEE Trans. Automat. Control,26 (1981), 917–922.

    Article  MathSciNet  Google Scholar 

  13. H. Hermes, On the synthesis of a stabilizing feedback control via Lie algebraic methods,SIAM J. Control Optim.,18 (1980), 352–361.

    Article  MathSciNet  Google Scholar 

  14. D. H. Jacobson,Extensions of Linear Quadratic Control, Optimization, and Matrix Theory, Academic Press, New York, 1977.

    Google Scholar 

  15. V. Jurdjevic and J. P. Quinn, Controllability and stability,J. Differential Equations,28 (1978), 381–389.

    Article  MathSciNet  Google Scholar 

  16. N. Kalouptsidis and J. Tsinias, Stability improvement of nonlinear systems by feedback,IEEE Trans. Automat. Control,29 (1984), 364–367.

    Article  MathSciNet  Google Scholar 

  17. K. K. Lee and A. Arapostathis, Remarks on smooth feedback stabilization of nonlinear systems,Systems Control Lett.,10 (1988), 41–44.

    Article  MathSciNet  Google Scholar 

  18. R. Longchamp, State-feedback control of bilinear systems,IEEE Trans. Automat. Control,25 (1980), 302–306.

    Article  MathSciNet  Google Scholar 

  19. J. L. Massera, Contributions to stability theory,Ann. of Math.,64 (1956), 182–206; erratum,Ann. of Math.,68 (1958), 202.

    Article  MathSciNet  Google Scholar 

  20. E. Roxin, Stability in general control systems,J. Differential Equations,19 (1965), 115–150.

    Article  MathSciNet  Google Scholar 

  21. E. P. Ryan and N. J. Buckingham, On asymptotically stabilizing feedback control of bilinear systems,IEEE Trans. Automat. Control,28 (1983), 863–864.

    Article  Google Scholar 

  22. A. J. van der Schaft, Stabilization of Hamiltonian systems,Nonlinear Anal.,10 (1986), 1021–1035.

    Article  MathSciNet  Google Scholar 

  23. M. Slemrod, Stabilization of bilinear control systems with applications to nonconservative problems in elasticity,SIAM J. Control Optim. 16 (1978), 131–141.

    Article  MathSciNet  Google Scholar 

  24. E. D. Sontag, Nonlinear regulation: the piecewise linear approach,IEEE Trans. Automat. Control,26 (1981), 346–358.

    Article  MathSciNet  Google Scholar 

  25. E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability,SIAM J. Control Optim.,21 (1983), 462–471.

    Article  MathSciNet  Google Scholar 

  26. E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback,Proceedings of the 19th IEEE Conference on Decision and Control, Albuquerque, NM, 1980, pp. 916–921.

  27. H. J. Sussmann, Subanalytic sets and feedback control,J. Differential Equations,31 (1979), 31–52.

    Article  MathSciNet  Google Scholar 

  28. J. Tsinias, Stabilization of affine in control nonlinear systems,Nonlinear Anal. 12 (1988), 1283–1296.

    Article  MathSciNet  Google Scholar 

  29. J. Tsinias and N. Kalouptsidis, Prolongations and stability analysis via Lyapunov functions of dynamical polysystems,Math. Systems Theory,20 (1987), 215–233.

    Article  MathSciNet  Google Scholar 

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  1. Department of Mathematics, National Technical University, Zografou Campus, 15773, Athens, Greece

    John Tsinias

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  1. John Tsinias
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Tsinias, J. Sufficient lyapunov-like conditions for stabilization. Math. Control Signal Systems 2, 343–357 (1989). https://doi.org/10.1007/BF02551276

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  • DOI: https://doi.org/10.1007/BF02551276

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