-- x = f(x)
an autonomous ordinary differential equation
a vector field
is called a Ljapunov function iff
for all x in Rn
/ d \
| -- V(x) , f(x) | <= 0
\ dx /
where the brackets denote the standard inner product
The above condition
means that the values
of the Ljapunov function are constant
) on any trajectory
of the ODE
You can also restrict the definition
of Ljapunov functions to open sets
Ljapunov functions are useful for reason about the dynamics of the ODE without knowing any exact solutions.
An example for this is LaSalle's invariance principle.
For a given ODE there is no algorithmic way of determining a Ljapunov function, you get one usually but the ancient mathematical principle of guessing.
You can of course make this definition
on any Hilbert space
(o.k. in fact anywhere where you can define vector fields and derivatives
). But I don't know if you would get any useful results
in such spaces