A Lyapunov function is a scalar function 
defined on a region 
that is continuous, positive definite, 
for all 
), and has continuous first-order partial
derivatives at every point of 
. The derivative of 
with respect to the system 
, written as 
is defined as the dot product
 |
(1)
|
The existence of a Lyapunov function for which 
on some region 
containing the origin, guarantees the stability of the zero
solution of 
,
while the existence of a Lyapunov function for which 
is negative definite on some region 
containing the origin guarantees the asymptotical stability
of the zero solution of 
.
For example, given the system
 |  |  |
(2)
|
 |  |  |
(3)
|
and the Lyapunov function 
, we obtain
 |
(4)
|
which is nonincreasing on every region containing the origin, and thus the zero solution is stable.
