Nonlinear Control Sy.. - Free

3.8. THE INVARIANCE PRINCIPLE 87
Definition 3.12 Let x(t) be a trajectory of the dynamical system ± = f (x). The set N is
called the limit set (or positive limit set) of x(t) if for any p E N there exist a sequence of
times {tn} E [0, oc) such that
or, equivalently
x(t,,)->p as t -*oc
urn IJx(tn) -p1l = 0.
Roughly speaking, the limit set N of x(t) is whatever x(t) tends to in the limit.
Example 3.16 An asymptotically stable equilibrium point is the limit set of any solution
starting sufficiently near the equilibrium point.
Example 3.17 A stable limit cycle is the positive limit set of any solution starting sufficiently
near it.
Lemma 3.4 If the solution x(t, x0i to) of the system (3.1) is bounded for t > to, then its
(positive) limit set N is (i) bounded, (ii) closed, and (iii) nonempty. Moreover, the solution
approaches N as t - oc.
Proof: See the Appendix.
The following lemma can be seen as a corollary of Lemma 3.4.
Lemma 3.5 The positive limit set N of a solution x(t, x0i to) of the autonomous system
(3.1) is invariant with respect to (3.1).
Proof: See the Appendix.
Invariant sets play a fundamental role in an extension of Lyapunov's work produced
by LaSalle. The problem is the following: recall the example of the pendulum with friction.
Following energy considerations we constructed a Lyapunov function that turned out to
be useful to prove that x = 0 is a stable equilibrium point. However, our analysis, based
on this Lyapunov function, failed to recognize that x = 0 is actually asymptotically stable,
something that we know thanks to our understanding of this rather simple system. LaSalle's
invariance principle removes this problem and it actually allows us to prove that x = 0 is
indeed asymptotically stable.
We start with the simplest and most useful result in LaSalle's theory. Theorem 3.6
can be considered as a corollary of LaSalle's theorem as will be shown later. The difference