where A and @ are constants defined by 
Asinag=adz+l1 
A cos @ = b 
By (3.24) and (3.25), the switching time must satisfy 
1 - cos (T - ue) 1 +A sin (T. + a) 
- sin (T - i) A cos (T. + a) 
Elimination of ue from these equations yields a relationship between the 
terminal time T and the initial point (a, b), namely, 
2 F 2 
(A cP iL sp Cos Tt)” a (> + otm WwW) = 4 (3.26) 
By definition, time fronts are the curves which connect initial 
points having the same terminal time T. Equation (3.26) can be used to 
determine the time fronts for T Soin LES Ole the eimesrrontetsms imple, 
the origin; if there are no switches in the control, the initial point 
is an endpoint of the curve connecting all initial points from which the 
origin is reached with one switch in time T. More than one switch would 
require T > 7. From (3.26), the time fronts for 0 < T < 7 are segments 
of the circle of radius 2 around the point (-l - cos T, - sin T); see 
Figure 6. It is the segment of the circle which lies above the switch- 
ing path. At the switching path, the time front is tangent to the 
vertical line x = constant for x > 0; at the opposite end, it is tangent 
to the switching curve. For T = 7, the time front is a circle of radius 
2 around the origin. 
38 
