A Lyapunov function for the zero solution of an autonomous system 
is a scalar function 
defined on an open neighborhood 
of the origin.
It is a continuously differentiable
function with 
and 
for all 
in 
,
whose derivative along solutions is nonpositive. Assume
is a continuously
differentiable function on 
and 
. The derivative of 
along solutions, written as 
, is the dot product of
its gradient with 
,
 |
(1)
|
The Lyapunov stability theorem guarantees stability of the zero solution when 
on 
. If 
for all 
in 
, then the zero solution is asymptotically stable (Iyer 2026).
For example, given the system
 |  |  |
(2)
|
 |  |  |
(3)
|
and the Lyapunov function 
, we obtain
 |
(4)
|
which is nonpositive, so 
is nonincreasing
along solutions and the zero solution is stable.
