Nonlinear Control Sy.. - Free

A.4. CHAPTER 7 323
With this property in mind, consider a E K. [so that 8, defined in (A.35) is 1C.], and
define
/3(r) = a(B-1(r)), / (r) = &(O-'(r))-
With these definitions, /3(.) and E K. By property 1, there exist a positive definite,
smooth, and nondecreasing function q(-):
Thus
Finally, defining
we have that
q(r)/(r) < /3(r) Vr E [0, oo).
q[9(r)]a(r) < &. (A.38)
a(s) = 1 q(c (s))a(s) (A.39)
q[a(s)]a(s) > 2&(s)
j q[B(s)]a(s) < &(r)
and the theorem is proved substituting (A.40) into (A.37).
Proof of Theorem 7.8: As in the case of Theorem 7.8 we need the following property
(A.40)
Property 2: if /3 = 0(13(r)) as r -f 0+, then there exist a positive definite, smooth, and
nondecreasing function q:
Thus
a(s) < q(s),3(s)
Vs E [0, oc).
Defining /3(r) = 2a[9-1(s)] we obtain /3(r) = d[9-1(s)]. By property 2, there exist q:
Finally, defining
/3 < q(s)Q(s) Vs E [0, oo).
we have that (A.40) is satisfied, which implies (A.37).
-2q[9(s)]a(s) (A.41)
a(r) = q[9(r)]a(s) (A.42)