Stability analysis of planar systems with nilpotent (non-zero) linear part

A simple necessary and sufficient condition for asymptotic stability of the origin is derived for a large class of planar nonlinear systems that cannot be studied by means of their linearization, which is nilpotent and not zero. The condition is based on the use of Darboux polynomials, to derive a Lyapunov function for the first approximation of the system with respect to a vector function representing a dilation. Results by Andreev (1958), on the qualitative behaviour of equilibrium points, and the Belitskii normal form are used in the proofs. The proposed condition is applied to the asymptotic stabilization of the origin, for a class of planar systems whose linearization cannot be stabilized.

[1]  G. Belitskii.  C∞-Normal Forms of Local Vector Fields , 2002 .

[2]  J. Giné.  On some open problems in planar differential systems and Hilbert’s 16th problem , 2007 .

[3]  Henry Hermes,et al.  Nilpotent and High-Order Approximations of Vector Field Systems , 1991, SIAM Rev..

[4]  G. Gaeta.  Poincaré Normal and Renormalized Forms , 2001, math-ph/0106027.

[5]  S. Sastry,et al.  Stabilization of nonlinear systems with uncontrollable linearization , 1988 .

[6]  Laura Menini,et al.  On the use of semi-invariants for the stability analysis of planar systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[7]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[8]  Jaume Llibre,et al.  Qualitative classification of singular points , 2005 .

[9]  L. Rosier.  Homogeneous Lyapunov function for homogeneous continuous vector field , 1992 .

[10]  A. Bacciotti,et al.  Liapunov functions and stability in control theory , 2001 .

[11]  I. Bendixson.  Sur les courbes définies par des équations différentielles , 1901 .

[12]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[13]  Jaume Llibre,et al.  Qualitative Theory of Planar Differential Systems , 2006 .

[14]  F. Takens.  Forced oscillations and bifurcations , 2001 .

[15]  G. Vegter,et al.  Global Analysis of Dynamical Systems: Festschrift dedicated to Floris Takens for his 60th birthday , 2001 .

[16]  J. P. LaSalle,et al.  Recent Advances in Liapunov Stability Theory , 1964 .