Let

d
-- x = f(x)
dt

an

autonomous ordinary differential equation in

**R**^{n} with
f:

**R**^{n} ->

**R**^{n} a

vector field on

**R**^{n}.

A

function V:

**R**^{n} ->

**R** is called a

**Ljapunov function** iff for all x in

**R**^{n}
/ d \
| -- V(x) , f(x) | <= 0
\ dx /

where the brackets denote the

standard inner product on

**R**^{n}.

The above

condition means that the

values of the Ljapunov function are

constant or

decreasing
(

monotonically decreasing) 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.