for all a satisfying a > 0 (Proposition 7.3). In particular, gq is continuous on the closed
interval 0 < ox < 1 and its first derivative exists at every interior point of that interval.
Therefore, if consideration is limited to monotone decreasing control parameter
sequences, then by the mean value theorem the difference between the m components of
any two consecutive vectors in the sequence can be written as
qk+ I )-q(k(m)= dq(x (a(k + 1)- oa(k))
I a=a'(k)
where the value a'(k) satisfies o(k + 1) < ao(k) < oa(k). Consequently,
lqk+(m)-q(m)| = Id( xI o(k + )- (k)l
Sqk,( ) ( a=a'(k)
and
dq (m)
lqk+1(m)-qk(m)= k1 doi x|a(k+ 1)-a(k)j Eq.8.1
SI qk+()-- () do( J= = I I a= C'(k)
k=l k= I a a J
From Propositions 7.3 and 7.8, it is possible to define a function ga(m) which is
continuous in a on the closed interval 0 5 a 5( 1 as follows
I dq,(i) <
Sdo
ga(mi=) dEq. 8.2
lim dq- a=0
a-0+ da
Then, from a fundamental theorem in the calculus of functions, it follows that g,(m) (and
consequently that I g,(m)l) is bounded on the closed interval 0 < a 1. Thus, if
B = sup I g,(m)l, Eq. 8.3
me S',aE [0,11
then it follows from Eq. 8.2 that at every interior point of the interval 0 a c 1
dq,(m)
d d
and application of this result to Eq. 8.1 yields