82
Introduction
Liapunov Functions
Besides the Liapunov spectral theorem, there is another basic method of
proving stability that is a generalization of the energy method we have seen
in the introductory mechanical examples.
Defnition
(Liapunov Function)
.
. Let
X
be a
C
r
vector feld on
R
n
,
r
≥
1
, and let
x
e
be an equilibrium point For
X
, that is,
X
(
x
e
)=0
.
A
Liapunov function
For
X
at
x
e
is a continuous Function
L
:
U
→
R
defned on a neighborhood
U
oF
x
e
, di±erentiable on
U
\{
x
e
}
, and satisFying
the Following conditions:
(i)
L
(
x
e
) = 0
and
L
(
x
)
>
0
iF
x
±
=
x
e
;
(ii)
The directional derivative oF
L
along
X
, denoted
X
[
L
]
≤
0
on
U
\{
x
e
}
;
this means that
d
dt
L
≤
0
along solution curves oF
X
.
The Liapunov Function
L
is said to be
strict
, iF
(ii)
is replaced by the
condition
(ii)
±
:
X
[
L
]
<
0
in
U
\{
x
e
}
.
Condition (i) is sometimes called the
potential well hypothesis
.
7
By
the Chain Rule for the time derivative of
V
along integral curves, condition
(ii) is equivalent to the statement:
L
is nonincreasing along integral curves
of
X
.
Theorem
(Liapunov Stability Theorem.)
.
Let
X
be a
C
r
vector feld on
R
n
,
r
≥
1
, and let
x
e
be an equilibrium point For
X
, that is,
X
(
x
e
. IF
there exists a Liapunov Function For
X
at
x
e
, then
x
e
is stable.
ProoF.
Since the statement is local, we can assume that
x
e
= 0. By the
local existence theory, there is a neighborhood
U
of 0 such that all solutions
starting in
U
exist for time
t
∈
[

δ, δ
], with
δ
depending only on
X
and
U
, but not on the solution. Now ±x
ε >
0 and an open ball
D
ε
(0) that
is included in
U
. Let
ρ
(
ε
)
>
0 be the minimum value of
L
on the sphere
of radius
ε
, and de±ne the open set
U
±
=
{
x
∈
D
ε
(0)

L
(
x
)
< ρ
(
ε
)
}
. By
(i),
U
±
±
=
∅
,0
∈
U
±
, and by (ii), no solution starting in
U
±
can meet the
sphere of radius
ε
(since
L
is decreasing on integral curves of
X
). Thus all
solutions starting in
U
±
never leave
D
ε
(0)
⊂
U
and therefore by uniformity
of time of existence, these solutions can be extended inde±nitely in time.
This shows that 0 is stable.
±
There is a more global concept that is related to this circle of ideas that
we discuss somewhat informally. Namely, a region
R
⊂
R
n
with a (smooth)
7
In infnite dimensions, one needs to augment (i) by the additional condition that
there is an
0 such that For all 0
< ε
±
≤
ε
,
inF
{
L
(
ϕ

1
(
x
))
 ±
x

x
e
±
=
ε
±
}
>
0
.
In fnite dimensions, this condition is automatic by the compactness oF spheres.