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.
Definition
(Liapunov Function)
.
. Let
X
be a
C
r
vector field 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
defined on a neighborhood
U
of
x
e
, di
ff
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 field on
R
n
,
r
≥
1
, and let
x
e
be an equilibrium point for
X
, that is,
X
(
x
e
) = 0
. 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 fix
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 define 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 indefinitely 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 infinite dimensions, one needs to augment (i) by the additional condition that
there is an
ε
>
0 such that for all 0
<
ε
≤
ε
,
inf
{
L
(
ϕ

1
(
x
))

x

x
e
=
ε
}
>
0
.
In finite dimensions, this condition is automatic by the compactness of spheres.
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1.7 Liapunov Functions
83
boundary with a well defined inside and outside, is called a
positively
trapping region
if for initial conditions that start in
R
, the solution stays
in
R
for
t
≥
0. If the given vector field is pointing inwards (or is tangent)
to the boundary of
R
, this is a su
ffi
cient condition for
R
to be a trapping
region. If
R
is bounded, necessarily the dynamical system is positively
complete on
R
. Often sublevel sets
R
=
{
x
∈
R
n

L
(
x
)
≤
C
}
of Liapunov
functions provide such trapping regions because the vector field is inwards
corresponds to the condition that
L
is decreasing along solutions at the
boundary.
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 Fall '09
 Marsden
 Stability theory, Liapunov, liapunov function

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