AN INVARIANCE PRINCIPLE IN THE THEORY OF STABILITY

5
If E
is bounded, then
each solution
of
(1) that remains in G
for t > t 2 0 either
0
approaches E
Thus this theorem explains
or 03
as t+-.
precisely the nature of the
information given by a Liapunov function.
A Liapunov function
relative to a set G defines a set E which under the conditions
of the theorem contains (locates) all the positive limit sets of
P
!
1
1
I
I
I
I
I
solutioils which for positive time remain in
applying the result is to find "good" Liapunov functions.
instance, the zero function V= 0
G. The problem in
is a Liapunov function for tiie
whole space Rn aiid condition (ii) is satisfied but gives no inn
information since E = R . It is trivial but useful for appli-
cations to note that if V, and V, are Liapunov functions on G,
I L
then V = V + V2 is also a Liapunov function and E = E ll E2 .
1
1
If E
is smaller
Liapunov function
"good" as either of the two.
by Yoshizawa.
thm either El or E2 , -then V
For
is a "better"
than either E or E2 and is always at least as
1
Condition (5) of Theorem 1 is essentially the one used
We now look at a simple example where condition (ii)
is satisfied and condition (i) is not.
the conclusion of the theorem is the best possible.
The example also shows that
Consider
*2 + p(t)? + x = 0 where p(t) 2 6 > 0 . Define 2V = x 2 + y 2 ,
2 2
where y = 2 . Then V = -p(t)y 6 - 6y and V is a Liapunov
2 2
function on R2 Now W = 6y and = 2 6 = ~ -2S(xy + p(t)) )
-26xy. Since all solutions are evidently bounded for all t > 0,