Third-order nilpotency, nice reachability and asymptotic stability

Journal of Differential Equations

Volumn 233, Issue 1, 2007, Pages 136-150

  Sharon, Yoav ^a   Margaliot, Michael ^a  

^a TEL AVIV UNIVERSITY   (Israel)

Author keywords

Arbitrary switching; Bang bang control; Chen series; First and second order maximum principle; Global asymptotic stability; Lie bracket; Nonlinear switched systems; Optimal control; P. Hall basis; Singular control; Sussmann's product expansion

Indexed keywords

EID: 34447127569     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2006.10.011     Document Type: Article

References (35)

1. Liberzon D. Lie algebras and stability of switched nonlinear systems. In: Blondel V., and Megretski A. (Eds). Unsolved Problems in Mathematical Systems and Control Theory (2004), Princeton Univ. Press 203-207
2. Sussmann H.J., and Nohel J.A. Commentary on Norman Levinson's paper 'Minimax, Lyapunov and Bang-Bang'. In: Nohel J.A., and Sattinger D.H. (Eds). Selected Papers of Norman Levinson, vol. 2 (1998), Birkhäuser, Boston 463-475
3. Agrachev A.A., and Sachkov Y.L. Control Theory from the Geometric Viewpoint. Encyclopaedia Math. Sci. vol. 87 (2004), Springer-Verlag
4. Artstein Z. Discrete and continuous bang-bang and facial spaces or: Look for the extreme points. SIAM Rev. 22 (1980) 175-185
5. Liberzon D. Switching in Systems and Control (2003), Birkhäuser, Boston
6. Liberzon D., and Morse A.S. Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. (1999) 59-70
7. Boyd S., Ghaoui L.E., Feron E., and Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. SIAM Stud. Appl. Math. vol. 15 (1994), SIAM, Philadelphia
8. Pyatnitskiy E.S., and Rapoport L.B. Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems. IEEE Trans. Circuits Syst. I 43 (1996) 219-229
9. Margaliot M. Stability analysis of switched systems using variational principles: An introduction. Automatica 42 (2006) 2059-2077. http://www.eng.tau.ac.il/~michaelm
10. Rapoport L.B. Asymptotic stability and periodic motions of selector-linear differential inclusions. In: Garofalo F., and Glielmo L. (Eds). Robust Control via Variable Structure and Lyapunov Techniques. LNCIS vol. 217 (1996), Springer 269-285
11. Margaliot M., and Gitizadeh R. The problem of absolute stability: A dynamic programming approach. Automatica 40 (2004) 1247-1252
12. Margaliot M., and Langholz G. Necessary and sufficient conditions for absolute stability: The case of second-order systems. IEEE Trans. Circuits Syst. I 50 (2003) 227-234
13. Holcman D., and Margaliot M. Stability analysis of switched homogeneous systems in the plane. SIAM J. Control Optim. 41 (2003) 1609-1625
14. Gurvits L. Stability of discrete linear inclusion. Linear Algebra Appl. 231 (1995) 47-85
15. Liberzon D., Hespanha J.P., and Morse A.S. Stability of switched systems: A Lie-algebraic condition. Systems Control Lett. 37 (1999) 117-122
16. Agrachev A.A., and Liberzon D. Lie-algebraic stability criteria for switched systems. SIAM J. Control Optim. 40 (2001) 253-269
17. Mancilla-Aguilar J.L. A condition for the stability of switched nonlinear systems. IEEE Trans. Automat. Control 45 (2000) 2077-2079
18. Jurdjevic V. Geometric Control Theory (1996), Cambridge Univ. Press
19. Margaliot M., and Liberzon D. Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions. Systems Control Lett. 55 1 (2006) 8-16
20. Sussmann H.J. A product expansion for the Chen series. In: Byrnes C.I., and Lindquist A. (Eds). Theory and Applications of Nonlinear Control Systems (1986), Elsevier 323-335
21. Krener A.J. The high order maximal principle and its applications to singular extremals. SIAM J. Control Optim. 15 (1977) 256-293
22. Bressan A. A high order test for optimality of bang-bang controls. SIAM J. Control Optim. 23 (1985) 38-48
23. Agrachev A.A., and Gamkrelidze R.V. Symplectic geometry for optimal control. In: Sussmann H.J. (Ed). Nonlinear Controllability and Optimal Control (1990), Marcel Dekker, New York 263-277
24. Kawski M. Chronological algebras: Combinatorics and control. J. Math. Sci. 103 (2001) 725-744
25. Margaliot M. On the reachable set of nilpotent control systems. http://www.eng.tau.ac.il/~michaelm
26. Hermes H. Control systems which generate decomposable Lie algebras. J. Differential Equations 44 (1982) 166-187
27. Sussmann H.J. A bang-bang theorem with bounds on the number of switchings. SIAM J. Control Optim. 17 (1979) 629-651
28. Filippov A.F. Differential Equations with Discontinuous Righthand Sides (1988), Kluwer, Dordrecht
29. Sussmann H.J. The Markov-Dubins problem with angular acceleration control. Proc. 36th IEEE Conf. on Decision and Control. San Diego, CA (1997) 2639-2643
30. Zelikin M.I., and Borisov V.F. Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering (1994), Birkhäuser
31. 3. SIAM J. Control Optim. 42 (2003) 513-531
32. Sharon Y., and Margaliot M. Third-order nilpotency, finite switchings and asymptotic stability. Proc. 44th IEEE Conf. on Decision and Control (2005), Seville, Spain 2639-2643
33. Sussmann H.J. The bang-bang problem for certain control systems in GL (n, R). SIAM J. Control 10 (1972) 470-476
34. Lafferriere G., and Sussmann H.J. Differential geometric approach to motion planning. In: Li Z., and Canny J.F. (Eds). Nonholonomic Motion Planning (1993), Kluwer Academic 235-270
35. Hermes H. Nilpotent and high-order approximations of vector field systems. SIAM Rev. 33 (1991) 238-264