Nonlinear Control Sy.. - Free

86 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
properties of the function V we assumed that the variables xl and x2 are independent. Those
variables, however, are related by the pendulum equations and so they are not independent
of one another. An extension of Lyapunov's theorem due to LaSalle studies this problem in
great detail. The central idea is a generalization of the concept of equilibrium point called
invariant set.
Definition 3.11 A set M is said to be an invariant set with respect to the dynamical system
i = f(x) if'
x(0)EM = x(t)EM VtER+.
In other words, M is the set of points such that if a solution of i = f (x) belongs to M at
some instant, initialized at t = 0, then it belongs to M for all future time.
Remarks: In the dynamical system literature, one often views a differential equation as
being defined for all t rather than just all the nonnegative t, and a set satisfying the definition
above is called positively invariant.
The following are some examples of invariant sets of the dynamical system i = f (x).
Example 3.11 Any equilibrium point is an invariant set, since if at t = 0 we have x(0) =
x, then x(t) = xe Vt > 0.
Example 3.12 For autonomous systems, any trajectory is an invariant set.
Example 3.13 A limit cycle is an invariant set (this is a special case of Example 3.12).
Example 3.14 If V(x) is continuously differentiable (not necessarily positive definite) and
satisfies V(x) < 0 along the solutions of i = f (x), then the set Sgt defined by
SZi={xE1R' :V(x)