Let
d
-- x = f(x)
dt
an
autonomous ordinary differential equation in
Rn with
f:
Rn ->
Rn a
vector field on
Rn.
A
function V:
Rn ->
R 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 on
Rn.
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.