Summary: 74 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32. NO. 1, JANUARY 1987
dx, f)=zIle(x, 0, 0 1 1 .
a) A symmetrical control structure for p(x, t) is implied by (10). This
more restrictive constraint, in comparing with  and , arises since
only some functional properties on +(.) are assumed and utilized. A
particular example of (10) is
b) The condition on the uncertainty as shown in (3) is sometimes
referred toas a matching condition . Discussions on mismatched
uncertainty are in , [131, [141.
c) The practicality of (A.3) is preempted by the matching condition.
This is so, since one can always choose an asymptotically stable nominal
part f1(x, t ) and then assume
f(x, t1-h (X, t ) E ~ ( B ( x ,t)) (12)
where Ui @(x, t))denotes the range space of B(x, t).
d) As an example, if y(llull) = b llull q, b > 0, q > 1, then $(p) =
Proof of Theorem: As a consequence of the Carathedory assump-
tions on the functions on the right-hand side of (3), one canreadily show.
using elementary results from the theories of continuous and measurable