Unformatted text preview: ov’s theorem g arantees
that the origin is stable. But can we say more? (To be continued .) ■
Theorem 4.9 (LaSalle’s Invariance Principle). Suppose is an equilibrium for
and
suppose is a weak Lyapunov function on some compact forward invariant neighborhood of
. Let be the set where is not decreasing. Then if is the largest forward invariant subset of , it is asymptotically stable and attracts every point of .
Proof. For any , suppose is a limit point of the trajectory
for all
, since if
we could find a sequence argue . Then
and , for sufficiently large , as in the proof of Lyap nov’s theorem and leading to a contradiction.
Consequently
assumption, must be a forward invariant subset of . Therefore, by
.□ Example. The damped pendulum (continued). Let and 14
Chapter 4A is connected, }. Note that on . Then is closed and bounded and therefore compact. We can see that is forward invariant
since
is always directed out on the boundary of (
has the same sign as
in ) and
. is a weak Lyapunov function on .
is the
subset of where is nondecreasing. If
, the equations of motion give
,
Thus the only subset of that is forward invariant is
is asymptotically stable. ■ . By LaSalle’s invarian e prin iple, Example. The Lorenz system
,
,
.
Consider . Then if we have and for . Furthermore ,
where the square is completed from the first two terms of the second line to get the third line.
When
and
, is a strong Lyapunov function and the Lyapunov function theorem
implies that the origin is an asymptotically stable state. When , on the set and
. However, when the Lorenz equations are evaluated on we
find
unless
. The origin is the largest forward invariant subset of
, and by LaSalle’s invarian e prin iple it is asymptotically stable. ■ Theorem 4.8. If
then the function is an asymptotically stable equilibrium that attracts a neighborhood , 15
Chapter 4A is a strong Lyapunov function on . Do not give proof.
Remark. Lyapunov functions provide a method for showing that equilibria are stable or
asymptotically stable. If the system
has an equilibrium such that all of the
eigenvalues of
have negative real parts, then theorem 4.6 implies that is
asymptotically stable. The challenge is to construct Lyapunov functions when this is not true. If
all the eigenvalues of
do not have negative real parts, there is no general method to
obtain practical expressions for Lyapunov functions....
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 Fall '14
 MarcEvans
 Stability theory, Lyapunov

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