Nonlinear Control Sy.. - Free

3.8. THE INVARIANCE PRINCIPLE 89
and by (3.12), we obtain
thus x2 = 0 Vt = x2 = 0
k - x2 and thus x2 = 0 = sin x1 = 0
0 = sinxl 9
-n
restricting xl to the interval xl E (-7r, Tr) the last condition can be satisfied if and only if
xl = 0. It follows that V (x) does not vanish identically along any solution other than x = 0,
and the origin is (locally) asymptotically stable by Theorem 3.6.
Proof of Theorem 3.6: By the Lyapunov stability theorem (Theorem 3.1), we know that
for each e > 0 there exist b > 0
1x011 < b => lx(t)II < E
that is, any solution starting inside the closed ball Bs will remain within the closed ball B,.
Hence any solution x(t, x0i t0) of (3.1) that starts in B6 is bounded and tends to its limit
set N that is contained in B, (by Lemma 3.4). Also V(x) is continuous on the compact set
B, and thus is bounded from below in B,. It is also non increasing by assumption and thus
tends to a non negative limit L as t -* oo. Notice also that V (x) is continuous and thus,
V(x) = L Vx in the limit set N. Also by lemma 3.5, N is an invariant set with respect to
(3.1), which means that any solution that starts in N will remain there for all future time.
But along that solution, V (x) = 0 since V (x) is constant (= L) in N. Thus, by assumption,
N is the origin of the state space and we conclude that any solution starting in R C B6
converges to x = 0 as t -4 oo.
Theorem 3.7 The null solution x = 0 of the autonomous system (3.1) is asymptotically
stable in the large if the assumptions of theorem 3.6 hold in the entire state space (i.e.,
R = R), and V(.) is radially unbounded.
Proof: The proof follows the same argument used in the proof of Theorem 3.3 and is
omitted.
Example 3.19 Consider the following system:
xl = x2
x2 = -x2 - axl - (x1 + x2)2x2.
To study the equilibrium point at the origin we define V (x) = axe + x2. We have
(x) = 8- f (x)
2[ax1, x2] [x2, -x2 - ax1 - (x1 + x2)2x2]T
-2x2[1 + (x1
+X2)2].