Nonlinear Control Sy.. - Free

3.5. ASYMPTOTIC STABILITY IN THE LARGE 79
(i) V(0) = 0.
(ii) V(x)>0 Vx#0.
(iii) V(x) is radially unbounded.
(iii) V (x) < 0 Vx # 0.
then x = 0 is globally asymptotically stable.
Figure 3.5: The curves V(x) = 0.
Proof: The proof is similar to that of Theorem 3.2. We only need to show that given an
arbitrary /3 > 0, the condition
S2p = {x E Rn : V (X) < /3}
defines a set that is contained in the ball B,. = {x E R' : IIxII < r}, for some r > 0. To see
this notice that the radial unboundedness of implies that for any 0 > 0, 3r > 0 such
that V(x) > 0 whenever IIxII > r, for some r > 0. Thus, S20 C B,., which implies that S20
is bounded.
Example 3.8 Consider the following system
2
xl = x2 - x1(xl + X2