Editing Lyapunov function (section)

== Basic Lyapunov theorems for autonomous systems==
{{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>:
:<math>\dot{V}(x) = \frac{d}{dt} V(x(t)) = \frac{\partial V}{\partial x}\cdot \frac{dx}{dt} = \nabla V \cdot \dot{x} = \nabla V\cdot f(x).</math>

===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]]).