AN INVARIANCE PRINCIPLE IN THE THEORY OF STABILITY

set in Rn , we say that V is a Liapunov function on G for
equation (3) if fl = (grad V). f does not change sign on G .
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Define E = ( x ; i(x) = 3 , x ir, G 1 , where G is ihe
closure of G . Let M be the largest invariant set in E . M
will be a closed set.
autonomous systems is then the following:
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The fundamental stability theorem for
THEORFM 2. If V is a Liapunov function on G for (3), then
each solution x(t) of (3) that remains in G for all t > 0
(t < 0) approaches M* = M U (m) as t -+ m (t -m). If M is
bounded, then either x(t) -+M or x(t) +m as t -+m (t -m) .
This one theorem contains all of the usual Liapunov like
theorems on stability and instability of autonomous systems. Here
however, there are no conditions of definiteness for V or V ,
and it is often possible to obtain stabilitjr information about a
system witii these more general types of Liapunov functions.
first corollary below is a stability result which for applications
has been quite useful aid the second illustrates how one obtains
infomatton on instability.
v
Cetaev's instability theorem is
similarly an immediate consequence of Theorem 2 (see section 3).
COROLLARY 2. Let G be a component of Q = ( x ; V(x) < 7 ) .
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Ass-me tliai G is bon-ided, V 6 0 on G , and M = M n G c G .
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Then M is an attractor and G is in its region of attraction.
If, in addition, V is constant on the boundary of Mo , then
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