Global convergence of nonlinear cascade flows with Morse-Bott zero dynamics

Systems and Control Letters

Volumn 58, Issue 6, 2009, Pages 389-394

  Helmke, U ^a  

^a UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

Cascade systems; Double bracket flows; Global asymptotic stability; Invariance principle; Morse decompositions; Nonlinear systems

Indexed keywords

CASCADE SYSTEMS; DOUBLE BRACKET FLOWS; GLOBAL ASYMPTOTIC STABILITY; INVARIANCE PRINCIPLE; MORSE DECOMPOSITIONS;

ASYMPTOTIC ANALYSIS; ASYMPTOTIC STABILITY; CONTROL THEORY; DYNAMICS; NONLINEAR SYSTEMS;

SYSTEM STABILITY;

EID: 64249157348     PISSN: 01676911     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sysconle.2009.01.004     Document Type: Article

References (24)

1. Dunbar S.R., Rybakowski K.P., and Schmitt K. Persistence in models of predator-prey populations with diffusion. J. Differential Equations 65 (1986) 117-138
2. Hirsch M.W. Convergent activation dynamics in continuous time networks. Neural Networks 2 (1989) 331-349
3. Thieme H.R. Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30 (1992) 755-763
4. Sussmann H.J. Limitations of the stabilizability of globally minimum phase systems. IEEE Trans. Automat. Control AC-35 1 (1990) 117-119
5. Aeyels D. Stabilization of a class of nonlinear systems by smooth feedback. Systems Control Lett. 5 (1985) 181-191
6. C.I. Byrnes, A. Isidori, A frequency domain philosophy for nonlinear systems with applications to stabilization and to adaptive control, in: Proc. 23rd IEEE Conf. Decision and Control, pp. 1569-1573, 1984
7. Seibert P., and Suarez R. Global stabilization of nonlinear cascade systems. Systems Control Lett. 14 (1990) 347-352
8. Robinson J.C. The asymptotic completeness of inertial manifolds. Nonlinearity 9 (1996) 1325-1340
9. Curry H.B. The method of steepest descent for nonlinear minimization problems. Quart. Appl. Math. 2 (1944) 258-261
10. Markus L. Asymptotically autonomous differential equations. In: Lefschetz S. (Ed). Contributions to the Theory of Nonlinear Oscillations 111. Ann. of Math. Studies vol. 36 (1956), Princeton University Press, Princeton, NJ 17-29
11. Hirsch M.W., Smith H.L., and Zhao X.-Q. Chain transitivity attractivity, and strong repellors for semidynamical systems. J. Dynam. Differential Equations 13 (2001) 107-131
12. Mischaikow K., Smith H., and Thieme H.R. Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc. 347 5 (1995) 1669-1685
13. Colonius F., and Kliemann W. The Dynamics of Control. Systems and Control: Foundations and Applications (2000), Birkhäuser Publ., Boston
14. Butler G., and Waltman P. Persistence in dynamical systems. J. Differential Equations 63 (1986) 255-263
15. Aulbach B. Continuous and Discrete Dynamics near Manifolds of Equilibria. Lecture Notes in Mathematics vol. 1058 (1985), Springer-Verlag, Heidelberg
16. Shayman M.A. Phase portrait of the matrix Riccati equation. SIAM J. Control Optim. 2 4 (1986) 1-65
17. Brockett R.W. The double bracket equation as the solution of a variational problem. Hamiltonian and Gradient Flows, Algorithms and Control (1994), AMS, Providence 69-76
18. Bloch A.M., and Crouch P.E. Optimal control optimization and analytical mechanics. In: Baillieul J., and Willems J.C. (Eds). Mathematical Control Theory (1999), Springer Publ., New York 268-321
19. R.W. Brockett, Dynamical systems that sort lists diagonalize matrices and solve linear programming problems, in: Proc. IEEE of the 27th Conference on Decision and Control, pp. 799-803, Austin, Texas, 1988, see also Linear Algebra Appl. 146 (1991) 79-91
20. Helmke U., and Moore J. Optimization and Dynamical Systems (1994), Springer, London
21. Brockett R.W. Smooth dynamical systems which realize arithmetical and logical operations. In: Nijmeijer H., and Schumacher J.M. (Eds). Three Decades of Mathematical Systems Theory. Lecture Notes in Control and Information Sciences (1989), Springer-Verlag, Berlin 19-30
22. Bloch A.M., Brockett R.W., and Ratiu T. Double bracket equations and geodesic flows on symmetric spaces. Comm. Math. Phys. 187 (1997) 357-373
23. Hirsch M.W., Pugh C.C., and Shub M. Invariant Manifolds. Lecture Notes in Mathematics vol. 583 (1977), Springer-Publ., Berlin
24. Irwin M.C. Smooth Dynamical Systems (1980), Academic Press, London