Since X > 0, it follows from the last of these equations that the sign 
of q is the same as the sign of u. Hence, if q changes from positive to 
negative, the optimal control must switch from +1 to -l. It switches 
from -l1 to +l if q changes from negative to positive. 
In a neighborhood of the origin, the optimal trajectory satisfies 
@ ED Sy =1 
Hence, its final segment is either on the circle of radius 1 about 
(-1, 0), or it is on the circle of radius 1 about (1, 0); see Figure 4. 
Suppose for the sake of argument that there is an € > O such that 
HG) se Foruh ie) San er themlasitysesment of ‘the joptimalsstragector 
is on the semicircle {(x + i)? + a = ils © < Wes 
Between (0, 0) and (-2, 0), the parameter T would change along this 
semicircle by the amount 1; hence, the sign of q must change somewhere 
on this semicircle. At the point S, where q changes sign, the sign of u 
must also change, and u switches cee, -l to 1. The optimal path continues 
backward on the circle of radius ry around (1, 0) until either (a, b) is 
reached or q changes sign. But q does not change sign until the point 
S5 is reached since the time between Sy and S5 I Wo | ANE So» the control 
would switch to -1l and the optimal trajectory would continue back on the 
circle of radius 5 around (-1, 0). This process is continued until the 
point (a, b) is reached. In the process, one switches control each time 
one of the following semicircles is intercepted: 
2 2 
be = Gme I) say = lo VS 05 MED SISZ5 e060 (@e22)) 
or 
ik @n oie Sr Sie SO BeO,,.4. GL25) 
The curve formed by these semicircles is called the switching curve; see 
Figure 5. 
35 
