Nonsmooth optimal regulation and discontinuous stabilization

Abstract and Applied Analysis

Volumn 2003, Issue 20, 2003, Pages 1159-1195

  Bacciotti, A ^a   Ceragioli, F ^a  

^a POLITECNICO DI TORINO   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1842449635     PISSN: 10853375     EISSN: None     Source Type: Journal    
DOI: 10.1155/S1085337503304014     Document Type: Review

References (38)

1. F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization, ESAIM Control Optim. Calc. Var. 4 (1999), 445-471.
2. B. D. O. Andersen and J. B. Moore, Linear Optimal Control, Prentice-Hall, New Jersey, 1971.
3. Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal. 7 (1983), no. 11, 1163-1173.
4. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Massachusetts, 1990.
5. A. Bacciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions, ESAIM Control Optim. Calc. Var. 4 (1999), 361-376.
6. _, Optimal regulation and discontinuous stabilization, Proceedings NOLCOS '01, Elsevier Sci., St. Petersburg, 2001, pp. 36-41.
7. M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Massachusetts, 1997.
8. D. S. Bernstein, Nonquadratic cost and nonlinear feedback control, Internat. J. Robust Nonlinear Control 3 (1993), no. 3, 211-229.
9. R. W. Brockett, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory (Houghton, Mich, 1982) (R. W. Brockett, R. S. Millman, and H. J. Sussmann, eds.), Progr. Math., vol. 27, Birkhäuser Boston, Massachusetts, 1983, pp. 181-191.
10. P. Cannarsa and G. Da Prato, Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations, SIAM J. Control Optim. 27 (1989), no. 4, 861-875.
11. P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var. Partial Differential Equations 3 (1995), no. 3, 273-298.
12. P. Cannarsa and H. M. Soner, On the singularities of the viscosity solutions to Hamilton-Jacobi-Bellman equations, Indiana Univ. Math. J. 36 (1987), no. 3, 501-524.
13. F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1983.
14. F. H. Clarke, Yu. S. Ledyaev, L. Rifford, and R. J. Stern, Feedback stabilization and Lyapunov functions, SIAM J. Control Optim. 39 (2000), no. 1, 25-48.
15. F. H. Clarke, Yu. S. Ledyaev, E. D. Sontag, and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control 42 (1997), no. 10, 1394-1407.
16. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998.
17. F. H. Clarke, L. Rifford, and R. J. Stern, Feedback in state constrained optimal control, ESAIM Control Optim. Calc. Var. 7 (2002), 97-133.
18. R. Conti, Linear Differential Equations and Control, Institutiones Mathematicae, vol. 1, Academic Press, London, 1976.
19. F. Da Lio, On the Bellman equation for infinite horizon problems with unbounded cost functional, Appl. Math. Optim. 41 (2000), no. 2, 171-197.
20. A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications (Soviet Series), vol. 18, Kluwer Academic, Dordrecht, 1988.
21. H. Frankowska, Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation, Appl. Math. Optim. 19 (1989), no. 3, 291-311.
22. S. T. Glad, On the gain margin of nonlinear and optimal regulators, IEEE Trans. Automat. Control 29 (1984), no. 7, 615-620.
23. H. Hermes, Stabilization via optimization, New Trends in Systems Theory (Genoa, 1990) (G. Conte, A.-M. Perdon, and B. Wyman, eds.), Progr. Systems Control Theory, vol. 7, Birkhäuser Boston, Massachusetts, 1991, pp. 379-385.
24. _, Resonance, stabilizing feedback controls, and regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations, Math. Control Signals Systems 9 (1996), no. 1, 59-72.
25. D. H. Jacobson, Extensions of Linear-Quadratic Control, Optimization and Matrix Theory, Mathematics in Science and Engineering, vol. 133, Academic Press, London, 1977.
26. M. Janković, R. Sepulchre, and P. V. Kokotovic, CLF based designs with robustness to dynamic input uncertainties, Systems Control Lett. 37 (1999), no. 1, 45-54.
27. M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995.
28. E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967.
29. P. J. Moylan and B. D. O. Anderson, Nonlinear regulator theory and an inverse optimal control problem, IEEE Trans. Automat. Control 18 (1973), no. 5, 460-465.
30. B. E. Paden and S. S. Sastry, A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators, IEEE Trans. Circuits and Systems 34 (1987), no. 1, 73-82.
31. L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions, SLAM J. Control Optim. 39 (2000), no. 4, 1043-1064.
32. _, On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients, ESAIM Control Optim. Calc. Var. 6 (2001), 593-611.
33. R. Sepulchre, M. Jankovic, and P. V. Kokotovic, Constructive Nonlinear Control, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1997.
34. D. Shevitz and B. E. Paden, Lyapunov stability theory of nonsmooth systems, IEEE Trans. Automat. Control 39 (1994), no. 9, 1910-1914.
35. E. D. Sontag, Mathematical Control Theory, 2nd ed., Texts in Applied Mathematics, vol. 6, Springer-Verlag, New York, 1998.
36. _, Stability and stabilization: discontinuities and the effect of disturbances, Nonlinear Analysis, Differential Equations and Control (Montreal, Quebec, 1998) (F. H. Clarke, R. J. Stern, and G. Sabidussi, eds.), NATO Sci. Ser. C Math. Phys. Sci., vol. 528, Kluwer Academic Publishers, Dordrecht, 1999, pp. 551-598.
37. J. N. Tsitsiklis and M. Athans, Guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators, IEEE Trans. Automat. Control 29 (1984), no. 8, 690-696.
38. M. Valadier, Entraînement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques [Unilateral driving, descent curves, nonpathological Lipschitzian functions], C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 8, 241-244 (French).