Decomposition of homogeneous vector fields of degree one and representation of the flow
Ancona, Fabio
Annales de l'I.H.P. Analyse non linéaire, Tome 13 (1996), p. 135-169 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)
Publié le : 1996-01-01
@article{AIHPC_1996__13_2_135_0,
     author = {Ancona, Fabio},
     title = {Decomposition of homogeneous vector fields of degree one and representation of the flow},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {13},
     year = {1996},
     pages = {135-169},
     mrnumber = {1378464},
     zbl = {0843.34007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_1996__13_2_135_0}
}

Ancona, Fabio. Decomposition of homogeneous vector fields of degree one and representation of the flow. Annales de l'I.H.P. Analyse non linéaire, Tome 13 (1996) pp. 135-169. http://gdmltest.u-ga.fr/item/AIHPC_1996__13_2_135_0/

[1] A.A. Agrachev, R.V. Gamkrelidze and A.V. Sarychev, Local invariants of smooth control systems, Acta Applicandae Mathematicae, Vol. 14, 1989, pp. 191-237. | MR 995286 | Zbl 0681.49018

[2] F. Ancona, Homogeneous normal forms for vector fields with respect to an arbitrary dilation, Ph. D. Dissertation, University of Colorado, Boulder, December 1993.

[3] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1982. | Zbl 0507.34003

[4] R.M. Bianchini and G. Stefani, Graded approximations and controllability along a trajectory, SIAM J. Control Optim., Vol. 28, 1990, pp. 903-924. | MR 1051629 | Zbl 0712.93005

[5] A. Bressan, Local asymptotic approximations of nonlinear control systems, Internat. J. Control., Vol. 41, 1985, pp. 1331-1336. | MR 792946 | Zbl 0565.93031

[6] K.T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math., Vol. 85, 1963, pp. 693-722. | MR 160010 | Zbl 0119.07505

[7] C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet and G. Iooss, A simple global characterization for normal forms of singular vector fields, Physica, Vol. 29D, 1987, pp. 85-127. | Zbl 0633.58020

[8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci., Springer-Verlag, New York, Heidelberg, Berlin, 1983. | MR 709768 | Zbl 0515.34001

[9] H. Hermes, Nilpotent approximations of control systems and distributions, SIAM J. Control Optim., Vol. 24, 1986, pp. 731-736. | MR 846379 | Zbl 0604.93031

[10] H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, in: Differential Equations, Stability and Control (S. Elaydi, Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 127, pp. 249-260, Dekker, New York, 1991. | MR 1096761 | Zbl 0711.93069

[11] H. Hermes, Asymptotically stabilizing feedback controls and the nonlinear regulator problem, SIAM J. Control Optim., Vol. 29, 1991, pp. 185-196. | MR 1088226 | Zbl 0738.93061

[12] H. Hermes, Nilpotent and higher order approximations of vector field systems, SIAM Review, Vol. 33, 1991, pp. 238-264. | MR 1108590 | Zbl 0733.93062

[13] H. Hermes, Asymptotically stabilizing feedback controls, J. Diff. Eqns., Vol. 92, 1991, pp. 76-89. | MR 1113589 | Zbl 0736.93069

[14] H. Hermes, Resonance and continuous asymptotically stabilizing feedback controls, Proc. IFAC NOLCOS 95 (to appear). | MR 1113589

[15] J.E. Humphreys, Introduction to Lie algebras and Representation Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1972. | MR 323842 | Zbl 0254.17004

[16] M. Kawski, Stabilization of nonlinear systems in the plane, System Control Lett., Vol. 12, 1989, pp. 169-175. | MR 985567 | Zbl 0666.93103

[17] P.J. Olver, Applications of Lie groups to Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1993. | MR 1240056 | Zbl 0785.58003

[18] L.P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., Vol. 137, 1976, pp. 247-320. | MR 436223 | Zbl 0346.35030

[19] G. Stefani, Polynomial approximations to control systems and local controllability, Proc. 24th IEEE Conference on Decision and Control, Vol. I, 1985, pp. 33-38.

[20] H.J. Sussmann, A general theorem on local controllability, SIAM J. Control Optim., Vol. 25, 1987, pp. 158-194. | MR 872457 | Zbl 0629.93012

[21] A. Vanderbauwhede, Center manifolds, normal forms and elementary bifurcations, in Dynamics Reported, U. Kirchgraber and H. O. Walther Ed., Vol. 2, pp. 89-169, Teubner, Stuttgart and Wiley, Chichester, 1989. | MR 1000977 | Zbl 0677.58001

[22] J.C. Van Der Meer, Nonsemisimple 1 : 1 resonance at an equilibrium, Cel. Mech., Vol. 27, 1982, pp. 131-149. | MR 666434 | Zbl 0485.70025

[23] J.C. Van Der Meer, The Hamiltonian-Hopf Bifurcation, Lecture Notes in Math., Vol. 1160, Springer-Verlag, New York, Heidelberg, Berlin, 1985. | MR 815106 | Zbl 0585.58019

[24] V.S. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Springer-Verlag, New York, Heidelberg, Berlin, 1984. | MR 746308 | Zbl 0955.22500