Nonlinear Control Sy.. - Free

3.8. THE INVARIANCE PRINCIPLE 91
bounded), but it is not required to be positive definite. Perhaps more important, LaSalle's
result applies not only to equilibrium points as in all the Lyapunov theorems, but also to
more general dynamic behaviors such as limit cycles. Example 3.20, at the end of this
section, emphasizes this point.
Before looking at some examples, we notice that some useful corollaries can be found
if is assumed to be positive definite.
Corollary 3.1 Let V : D -4 R be a continuously differentiable positive definite function in
a domain D containing the origin x = 0, and assume that V (x) < 0 E D. Let S = {x E
D : V (x) = 0} and suppose that no solution can stay identically in S other than the trivial
one. Then the origin is asymptotically stable.
Proof: Straightforward (see Exercise 3.11).
Corollary 3.2 If D = Rn in Corollary 3.1, and is radially unbounded, then the origin
is globally asymptotically stable.
Proof: See Exercise 3.12.
Example 3.20 [68] Consider the system defined by
it = x2 + xl(Q2 - x1 - x2)
i2 = -xi + x2(02 - xi - x2)
It is immediate that the origin x = (0, 0) is an equilibrium point. Also, the set of points
defined by the circle x1 + x2 = (32 constitute an invariant set. To see this, we compute the
time derivative of points on the circle, along the solution of i = f (x):
T [x2 + x2 - Q2] = (2x1, 2x2)f (x)
2(xi + x2)(b2 - xi - x2)
for all points on the set. It follows that any trajectory initiating on the circle stays on the
circle for all future time, and thus the set of points of the form {x E R2 : xl + x2 = 32}
constitute an invariant set. The trajectories on this invariant set are described by the
solutions of
i = f(x)I. ztx2=R2 r x1 = x2
x.x 1 2 l x2 = -xl.