The optimal control and the resulting trajectory in the phase plane 
can now be obtained by reversing the above procedure. If (a, b) is 
above the switching curve, proceed with the control u=- 1. The 
optimal trajectory will be along the circle 
(x + 1)? + v = (at 1)? + be 
in the direction of that part of the switching curve which lies to the 
right of x = 0. For (a, b) on the switching curve, use u = - 1 if 
x < Q0oru=1if x > 0. If (a, b) lies below the switching curve, 
start with u = 1 and change to u = - 1 at the switching curve. 
Change the sign of u at each intersection with the switching curve. 
When u = 1, the optimal trajectory lies on a circle with center at 
(1, 0); when u=- 1, it is on a circle around (-l1, 0). 
Suppose only one switch in u is needed to reach the origin from 
(a, b). Because of the symmetry of the problem geometry in the phase 
plane, it is necessary to consider only those cases for which a = 1 
after the switch. The origin is then approached along the trajectory 
1 - cos (T - T) 
* 
Ul 
y = = sim (= Tt) (3.24) 
2 
which is on the semicircle {(x, y)|(x - Ly? yn —tollva Ole Met 
Ge be the time at which the switch occurs. The optimal trajectory 
for T Sous is given by 
== 1 Asim (G0) 
A cos (tT + a) (@EZ5)) 
< 
i] 
37 
