Asymptotic controllability implies feedback stabilization

IEEE Transactions on Automatic Control

Volumn 42, Issue 10, 1997, Pages 1394-1407

  Clarke, Francis H ^b,c   Ledyaev, Yuri S ^d   Sontag, Eduardo D ^a,e   Subbotin, Andrei I ^f  

^a IEEE   (United States)

^b UNIVERSITÉ LYON 1   (France)

^c UNIVERSITÉ DE MONTRÉAL   (Canada)

^d STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

^e RUTGERS UNIVERSITY   (United States)

^f INSTITUTE OF MATHEMATICS AND MECHANICS   (Russian Federation)

Author keywords

Control lyapunov functions; Feedback; Nonsmooth analysis; Stabilization

Indexed keywords

ASYMPTOTIC STABILITY; CONTROL SYSTEM ANALYSIS; FEEDBACK CONTROL; ITERATIVE METHODS; LYAPUNOV METHODS; SYSTEM STABILITY;

ASYMPTOTIC CONTROLLABILITY; NON SMOOTH ANALYSIS;

CONTROLLABILITY;

EID: 0031257967     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/9.633828     Document Type: Article

References (33)

1. J.-P. Aubin and A. Cellina, Differential Inclusions Set-Valued Maps and Viability Theory. Berlin: Springer-Verlag, 1984.
2. Z. Artstein, "Stabilization with relaxed controls," Nonlinear Anal., TMA, vol. 7, pp. 1163-1173, 1983.
3. R. W. Brockett, "Asymptotic stability and feedback stabilization," in Differential Geometric Control Theory, R. W. Brockett, R. S. Millman, and H. J. Sussmann, Eds. Boston: Birkhauser, 1983, pp. 181-191.
4. F. H. Clarke, Methods of Dynamic and Nonsmooth Optimization, vol. 57 of CBMS-NSF Regional Conf. Ser. Appl. Math. Philadelphia: SIAM, 1989.
5. F. H. Clarke, Yu. S. Ledyaev, and P. Wolenski, "Proximal analysis and minimization principles," J. Math. Anal. Appl., vol. 196, pp. 722-735, 1995.
6. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. Wolenski, "Qualitative properties of trajectories of control systems: A survey," J. Dynamical Contr. Syst., vol. 1, pp. 1-48, 1995.
7. F. H. Clarke, Yu. S. Ledyaev, and A. I. Subbotin, "Universal feedback strategies for differential games of pursuit," SIAM J. Contr., to be published.
8. _, "Universal feedback via proximal aiming in problems of control and differential games," Centre de Recherches Math., Univ. deMontréal, Rapport CRM-2386, 1994.
9. ppush J.-M. Coron, "Global asymptotic stabilization for controllable systems without drift," Math. Contr., Signals, Syst., vol. 5, pp. 295-312, 1992.
10. _, "Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws," SIAM J. Contr. Optim., vol. 33, pp. 804-833, 1995.
11. J.-M. Coron and L. Rosier, "A relation between continuous time-varying and discontinuous feedback stabilization," J. Math. Syst., Estimation, Contr., vol. 4, pp. 67-84, 1994.
12. M. Crandall and P.-L. Lions "Viscosity solutions of Hamilton-Jacobi equations," Trans. Amer. Math. Soc., vol. 277, pp. 1-42, 1983.
13. A. F. Filippov, "Differential equations with discontinuous right-hand side," Matem. Sbornik, vol. 5, pp. 99-127, 1960;
14. English trans. in Amer. Math. Translations, vol. 42, pp. 199-231, 1964.
15. W. H. Fleming, and H. M. Soner, Controlled Markov Processes and Viscosity Solutions. New York: Springer-Verlag, 1993.
16. H. Hermes, "Discontinuous vector fields and feedback control," in Differential Equations and Dynamic Systems, J. K. Hale and J. P. La Salle, Eds. New York: Academic, 1967.
17. _, "On the synthesis of stabilizing feedback control via Lie algebraic methods," SIAM J. Contr. Optim., vol. 18, pp. 352-361, 1980.
18. A. Isidori, Nonlinear Control Systems, 3rd ed. London: Springer-Verlag, 1989.
19. M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wlley, 1995.
20. N. N. Krasovskii, and A. I. Subbotin, Positional Differential Games. Moscow: Nauka, 1974 (in Russian);
21. Jeux Differentiels (in French). Mir: Moscow, 1979;
22. revised English translation Game-Theoretical Control Problems. New York: Springer-Verlag, 1988.
23. Yu. S. Ledyaev and E. D. Sontag, "A remark on robust stabilization of general asymptotically controllable systems," in Proc. Conf. Inform. Scis. Syst., Johns Hopkins, Baltimore, MD, Mar. 1997, pp. 245-251.
24. E. P. Ryan, "On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback," SIAM J. Contr. Optim., vol. 32, pp. 1597-1604, 1994.
25. E. D. Sontag, "Nonlinear regulation: The piecewise linear approach," IEEE Trans. Automat. Contr. vol. AC-26, pp. 346-358, 1981.
26. _, "A Lyapunov-like characterization of asymptotic controllability," SIAM J. Contr. Optim., vol. 21, pp. 462-471, 1983.
27. _, "A 'universal' construction of Artstan's theorem on nonlinear stabilization," Syst. Contr. Lett., vol. 13, pp. 117-123, 1989.
28. _, Mathematical Control Theory, Deterministic Finite Dimensional Systems. New York: Springer-Verlag, 1990.
29. _, "Control of systems without drift via generic loops," IEEE Trans. Automat. Contr. vol. 40, pp. 1210-1219, 1985.
30. E. D. Sontag and H. J. Sussmann, "Remarks on continuous feedback," in Proc. IEEE Conf. Decision Contr., Albuquerque, NM, Dec. 1980. New York: IEEE, pp. 916-921.
31. _, "Nonsmooth control-Lyapunov functions," in Proc. IEEE Conf. Decision Contr., New Orleans, LA, Dec. 1995. New York: IEEE, 1995, pp. 2799-2805.
32. A. I. Subbotin, Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspective. Boston: Birkhaüser, 1995.
33. H. J. Sussmann, "Subanalytic sets and feedback control," J. Differential Equations, vol. 31, pp. 31-52, 1979.