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{{main article|Lyapunov stability}}
Let <math>x^* = 0</math> be an equilibrium point of the autonomous system
:<math>\dot{x} = f(x).</math>
and use the notation <math>\dot{V}(x)</math> to denote the time derivative of the Lyapunov-candidate-function <math>V</math>:
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===Locally asymptotically stable equilibrium===
If the equilibrium point is isolated, the Lyapunov-candidate-function <math>V</math> is locally positive definite, and the time derivative of the Lyapunov-candidate-function is locally negative definite:
:<math>\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}(0)\setminus\{0\},</math>
for some neighborhood <math>\mathcal{B}(0)</math> of origin, then the equilibrium is proven to be locally asymptotically stable.
===Stable equilibrium===
If <math>V</math> is a Lyapunov function, then the equilibrium is [[Lyapunov stable]].
===Globally asymptotically stable equilibrium===
If the Lyapunov-candidate-function <math>V</math> is globally positive definite, [[Radially unbounded function|radially unbounded]], the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:
:<math>\dot{V}(x) < 0 \quad \forall x \in \R ^n\setminus\{0\},</math>
then the equilibrium is proven to be [[Stability theory|globally asymptotically stable]].
The Lyapunov-candidate function <math>V(x)</math> is radially unbounded if
:<math>\| x \| \to \infty \Rightarrow V(x) \to \infty. </math>
(This is also referred to as norm-coercivity.)
The converse is also true,<ref name=Massera1949>{{Citation
| author = Massera, José Luis
| year = 1949
| title = On Liapounoff's conditions of stability
| journal = Annals of Mathematics |series=Second Series
| volume = 50
| issue = 3
| pages = 705–721
| doi = 10.2307/1969558
| mr = 0035354
| jstor = 1969558
}}</ref> and was proved by [[José Luis Massera]] (see also [[Massera's lemma]]).
==Example==
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