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Consider the following nonlinear perturbed dynamical system:

$\dot{x} = f(t,x)+g(t,x)$,

such that $x \in \mathbb{R}^n$, $g(t,x)$ is a perturbation term such that

$g(t,0) = 0$

and

$x=0$ is an uniformly asymptotically stable equilibrium point of the nominal system

$\dot{x} = f(t,x)$.

Consider also that $g(t,x)$ has a polynomial form in $x$ of order N.

With the information given, can you take any conclusions about the stability of the zero equilibrium point of the nonlinear perturbed system?

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  • $\begingroup$ Maybe you need something along the lines of $$\partial_x \, g(t,x){|_{x=0}} = 0 $$ for all $t$? $\endgroup$ – Futurologist Apr 18 '17 at 6:32
  • $\begingroup$ Since $g(t,x)$ is a polynomial with $g(t,0)=0$ then it can be bounded by $\|g(t,x)\|\leq K\|x\|$ locally for some small ball. Then using the vanishing perturbation theorems in Khalil's book, nonlinear-systems-chapter 9, you can conclude asymptotic stability for sufficiently small neighboring ball. $\endgroup$ – Bilal Jafar Karaki Aug 29 '17 at 16:42

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