llD° (tt0 )Ii < me a(tt) (4-23)
for some positive constants m and a. Hence, to show condition (2) of the Poincare-Liapunov theorem and also to conclude stability we must have
sup IIFA(t) -F A L (4-24)
t)O
where L is small enough to insure that
(mL - a) < 0 (4-25)
Equation (4-24) can be verified by using the definitions for F*(t) and FA to obtain the relationship A A
IIAt*Ii * F0 I. * 0IG
IIFA(t) - Foil < lIF (t) - Foil + IIG(t) - Golli.l[Ki, K2]1. (4-26)
Using (i) and (ii) then gives
sup IIFA(t) -oFl< e 2 + II[K1, K2]II. (4-27)
t)O
If 1 and e2 are small enough, then condition (2) of the PoincareLiapunov theorem is satisfied and the system NCT is asymptotically stable.
The final condition which needs to be verified for a solution to the servomechanism problem is the observability of the pair (A, K2). This condition presents no problem, however, since it immediately
follows from Theorem 3.4 that (A, K2) is observable whenever the system