Trajectory-based local approximations of ordinary differential equations

SIAM Journal on Control and Optimization

Volumn 41, Issue 6, 2003, Pages 1922-1945

  Moreau, Luc ^a   Aeyels, Dirk ^a  

^a GHENT UNIVERSITY   (Belgium)

Author keywords

Asymptotic stability; Averaging; Ordinary differential equations; Perturbations

Indexed keywords

APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; LYAPUNOV METHODS; PERTURBATION TECHNIQUES; SET THEORY;

ASYMPTOTIC METHODS; AVERAGING TECHNIQUES; LIMITING EQUATION;

ASYMPTOTIC STABILITY;

EID: 0346329843     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012900370776     Document Type: Article

References (30)

1. D. AEYELS AND J. PEUTEMAN, On exponential stability of nonlinear time-varying differential equations, Automatica J. IFAC, 35 (1999), pp. 1091-1100.
2. L. BARREIRA AND Y. B. PESIN, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lecture Ser. 23, AMS, Providence, RI, 2002.
3. S.-N. CHOW AND J. MALLET-PARET, Integral averaging and bifurcation, J. Differential Equations, 26 (1977), pp. 112-159.
4. M. M. HAPAEV, Averaging in Stability Theory: A Study of Resonance Multi-Frequency Systems, Math. Appl. 79, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
5. P. HARTMAN, Ordinary Differential Equations, 2nd ed., Birkhäuser Boston, Cambridge, MA, 1982.
6. H. HERMES, Nilpotent and high-order approximations of vector field systems, SIAM Rev., 33 (1991), pp. 238-264.
7. H. HERMES, Asymptotic stabilization via homogeneous approximations, in Geometry of Feedback and Optimal Control, Monogr. Textbooks Pure Appl. Math. 207, B. Jakubczyk and W. Respondek, eds., Marcel Dekker, New York, 1998, pp. 205-218.
8. M. KAWSKI, Homogeneous stabilizing feedback laws, Control-Theory and Advanced Technology (C-TAT), 6 (1990), pp. 497-516.
9. H. KHALIL, Nonlinear Systems, 2nd ed., Prentice-Hall, Upper Saddle River, NJ, 1996.
10. P. KOKOTOVIĆ, H. K. KHALIL, AND J. O'REILLY, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.
11. J. KURZWEIL AND J. JARNÍK, Limit processes in ordinary differential equations, Z. Angew. Math. Phys., 38 (1987), pp. 241-256.
12. W. LIU, An approximation algorithm for nonholonomic systems, SIAM J. Control Optim., 35 (1997), pp. 1328-1365.
13. R. M'CLOSKEY AND P. MORIN, Time-varying homogeneous feedback: Design tools for the exponential stabilization of systems with drift, Internat. J. Control, 71 (1998), pp. 837-869.
14. R. M'CLOSKEY AND R. MURRAY, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback, IEEE Trans. Automat. Control, 42 (1997), pp. 614-628.
15. L. MOREAU AND D. AEYELS, Local approximations and stability: A trajectory-based approach, in Proceedings of the 38th IEEE Conference on Decision and Control (CDC), 1999, pp. 734-739.
16. L. MOREAU AND D. AEYELS, A systematic design tool for asymptotic stabilization of driftless control affine systems, in Proceedings of the 38th IEEE Conference on Decision and Control (CDC), 1999, pp. 861-862.
17. L. MOREAU AND D. AEYELS, Asymptotic methods in the stability analysis of parametrized homogeneous flows, Automatica J. IFAC, 36 (2000), pp. 1213-1218.
18. L. MOREAU AND D. AEYELS, Practical stability and stabilization, IEEE Trans. Automat. Control, 45 (2000), pp. 1554-1558.
19. L. MOREAU, W. MICHIELS, D. AEYELS, AND D. ROOSE, Robustness of nonlinear delay equations with respect to input perturbations: A trajectory based approach, Math. Control Signals Systems, 15 (2002), pp. 316-335.
20. P. MORIN, J.-B. POMET, AND C. SAMSON, Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of Lie brackets in closed loop, SIAM J. Control Optim., 38 (1999), pp. 22-49.
21. P. MORIN AND C. SAMSON, Time-varying exponential stabilization of a rigid spacecraft with two control torques, IEEE Trans. Automat. Control, 42 (1997), pp. 528-534.
22. J. PEUTEMAN AND D. AEYELS, Exponential stability of nonlinear time-varying differential equations and parital averaging, Math. Control Signals Systems, 15 (2002), pp. 42-70.
23. J. PEUTEMAN AND D. AEYELS, Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast time-varying, SIAM J. Control Optim., 37 (1999), pp. 997-1010.
24. J. SANDERS AND F. VERHULST, Averaging Methods in Nonlinear Dynamical Systems, Appl. Math. Sci. 59, Springer-Verlag, New York, 1985.
25. A. V. SARYCHEV, Lie- and chronologico-algebraic tools for studying stability of time-varying systems, Systems Control Lett., 43 (2001), pp. 59-76.
26. E. SONTAG, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed., Texts Appl. Math. 6, Springer-Verlag, New York, 1998.
27. H. SUSSMANN AND W. LIU, Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories, in Proceedings of the 30th IEEE Conference on Decision and Control, 1991, pp. 437-442.
28. A. TEEL, R. MURRAY, AND G. WALSH, Nonholonomic control systems: From steering to stabilization with sinusoids, Internat. J. Control, 62 (1995), pp. 849-870.
29. A. R. TEEL, L. MOREAU, AND D. NEŠIĆ, A unified framework for input-to-state stability in systems with two time scales, IEEE Trans. Automat. Control, submitted.
30. F. VERHULST, Nonlinear Differential Equations and Dynamical Systems, 2nd ed., Springer-Verlag, Berlin, 1996.