Nonlinear Control Sy.. - Free

3.9. REGION OF ATTRACTION 95
Figure 3.6: System trajectories in Example 3.21.
In general, the exact determination of this region can be a very difficult task. In this section
we discuss one way to "estimate" this region. The following theorem, which is based entirely
on the LaSalle's invariance principle (Theorem 3.8), outlines the details.
Theorem 3.9 Let xe be an equilibrium point for the system (3.1). Let V : D -4 IR be a
continuous differentiable function and assume that
(i) M C D is a compact set containing xe, invariant with respect to the solutions of (3.1).
(ii) V is such that
V < 0 VX 34 X, E M.
V = 0 ifx=xe.
Under these conditions we have that
M C RA.
In other words, Theorem 3.9 states that if M is an invariant set and V is such that V < 0
inside M, then M itself provides an "estimate" of RA.
Proof: Under the assumptions, we have that E _ {x : x E M, and V = 0} = xe. It then
follows that N = largest invariant set in E is also xe, and the result follows from LaSalle's
Theorem 3.8.