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