publication . Article . Other literature type . Preprint . 2015

Differentiability of Palmer's linearization theorem and converse result for density functions

Castañeda, Alvaro; Robledo, Gonzalo;
Open Access
  • Published: 01 Nov 2015 Journal: Journal of Differential Equations, volume 259, pages 4,634-4,650 (issn: 0022-0396, Copyright policy)
  • Publisher: Elsevier BV
Abstract
We study differentiability properties in a particular case of the Palmer's linearization Theorem, which states the existence of an homeomorphism $H$ between the solutions of a linear ODE system having exponential dichotomy and a quasilinear system. Indeed, if the linear system is uniformly asymptotically stable, sufficient conditions ensuring that $H$ is a $C^{2}$ preserving orientation diffeomorphism are given. As an application, we generalize a converse result of density functions for a nonlinear system in the nonautonomous case.
Subjects
free text keywords: Analysis, Stability theory, Linear system, Differentiable function, Linear differential equation, Homeomorphism, Mathematics, Mathematical analysis, Exponential dichotomy, Converse, Hartman–Grobman theorem, Mathematics - Classical Analysis and ODEs, Mathematics - Dynamical Systems, 34A34, 34D20, 34D23
Related Organizations

{A(t) + Df (t, φ)} ∂φ ∂φ Ψ(0, s)Df (s, φ(s, 0, ξ))Ψ(s, 0) Z(s) Ψ(t, 0){H[0, ξ + Δ] − H[0, ξ]}, Ψ(s, 0) = F (s, ξ) ρ(t, H(t, x)) ∂ nρ(τ + t, H[τ + t, φ(τ + t, t, x)]) ∂η Tr Dg(r, ϕ(r, 0, ξ)) dr > −∞ 2h(u)e−auφ(u, 0, ξ) (1 + φ2(u, 0, ξ))2 = e−auD2f (u, φ(u, 0, ξ)) [1] L. Ya Adrianova. Introduction to Linear Systems od Differential Equations. Translations of Mathematical Monographs. American Mathematical Society. Providence RI, 1995.

[2] M. Akhmet, M. A. Tleubergenova, A. Zafer. Asymptotic equivalence of differential equations and asymptotically almost periodic solutions. Nonlinear Analysis, 67:1870-1877, 2007.

[3] E. Coddington, N. Levinson. Theory of Ordinary Differential Equations. Mc Graw-Hill, New York, 1955.

[4] W. Coppel. Dichotomies in Stability Theory. Lecture notes in mathematics 629, Springer, Berlin, 1978.

[5] J. Fernandez Vasconcelos, A. Rantzer, C. Silvestre, P.J. Oliveira. Combination of Lyapunov and density functions for stability of rotational motion. IEEE Trans.Aut.Cont., 56:2599-2607, 2011.

[6] P. Hartman. Ordinary Differential Equations. SIAM, Philadelphia, 2002.

[7] H. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River NJ, 1996.

[8] G. Meinsma. On Rantzer's density function. Proc. 25th Benelux meeting, Heeze, The Netherlands, Mar. 2006. [OpenAIRE]

[9] P. Monz´on. On necessary conditions for almost global stability. IEEE Trans.Aut.Cont. 48:631-634, 2003.

[10] P. Monz´on. Almost Global Stability of Time-Varying Systems, Congresso Brasileiro de Automa´tica. Bahia, Brasil, pages 198-201, 2006.

[11] K.J. Palmer. A generalization of Hartman's linearization theorem. J.Math.Anal.Appl., 41:753-758, 1973.

[12] A. Rantzer. A dual to Lyapunov's Stability Theorem, Syst.Cont.Lett., 42:161-168, 2001. [OpenAIRE]

[13] A. Rantzer. An converse theorem for density functions. Proceedings of the 41st IEEE Conference on Decision and Control. pages 1890-1891, 2002. [OpenAIRE]

[14] R. Schlanbusch, A. Loria, P.J. Nicklasson. On the stability and stabilization of quaternion equilibria of rigid bodies. Automatica. 48:3135-3141, 2012. [OpenAIRE]

[15] A. Wintner. Linear variation of constants. Amer.J.Math., 68:185-213, 1946. Departamento de Matema´ticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile E-mail address: castaneda@u.uchile.cl,grobledo@u.uchile.cl

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publication . Article . Other literature type . Preprint . 2015

Differentiability of Palmer's linearization theorem and converse result for density functions

Castañeda, Alvaro; Robledo, Gonzalo;