in ea ay Tey 
oH _ 
erat x ap {Ul 
=. oo 
6 5 = « 
eu ey SEE 
4 age Tae 
0 = Beg - A(u- 1) - A(u +1) (3.20) 
Suppose u is an optimal control which reduces the oscillator from 
the state (a, b) to the state (0, 0) in the minimal time T, and suppose 
|u| < 1 for the interval To St Siti Suppose q # O on T) Kt < T,. 
By (3.20), q - 2Au = 0; hence, X # O on (T,> T,)- A consequence of 
dX #0 is that » = 0; hence, if q # 0, it follows that Ju(t) | = lon (T,> 
In other words, one needs to look only for the optimal control among 
those controls for which Ju(t) | = 1. 
Now |u| Seeleetimplete satis —sehaels gunenceP mathe ms out onw ores (3-720) mers 
given as: 
x Ls A sim @ 2 a) 
A cos (t + a) 
< 
il 
> = 3) Sim Ce a Oy) 
) 
Bcos (tT+a 
QQ 
tl 
0 
eas (3.21) 
fe) 
i} 
34 
2) 
