Nonlinear Control Sy.. - Free

80 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
2 2
x2 = -21 - 2201 + 22).
To study the equilibrium point at the origin, we define V (x) = x2 + x2. We have
(2) = 7f(x)
2 2[xlix2][x2 -21(21
-2(x1 + 22)2
2
+22),-21 -22(21 -f-22)]T
Thus, V(x) > 0 and V(x) < 0 for all x E R2. Moreover, since is radially unbounded,
it follows that the origin is globally asymptotically stable.
3.6 Positive Definite Functions Revisited
We have seen that positive definite functions play an important role in the Lyapunov theory.
We now introduce a new class of functions, known as class 1C, and show that positive definite
functions can be characterized in terms of this class of functions. This new characterization
is useful in many occasions.
Definition 3.9 A continuous function a : [0, a) -> R+ is said to be in the class K if
(i) a(0) = 0.
(ii) It is strictly increasing.
a is said to be in the class KQO if in addition a : ]EF+ -+ IIt+ and a(r) -+ oo as r -+ oo.
In the sequel, B,. represents the ball
Br= {xER":IxII R is positive definite if and only if there exists class K functions a1
and a2 such that
al(II2II) 5 V(x) < a2(IIxII) Vx E Br C D.
Moreover, if D = 1R' and is radially unbounded then a1 and a2 can be chosen in the
class K.
Proof: See the Appendix.