two-point, boundary-value problem. The vector q is introduced here as a 
Lagrange variable. Now the problem of minimizing the cost function (14) 
is replaced by the problem of finding the unconstrained minimum of 
iP 
Ce @ = { F(o, x, u) + q'(x - £(0, x, u) do 
0 
+ IMGs T) + C(x LD) AGES 29) 
The boundary condition (1.31) has been inserted into the cost function 
by means of the Lagrange multiplier uw. Suppose v(t) is the control 
which minimizes Gao For a variation du to the control v, let x(t) 
denote the new state variable, and let t = T+AT be the time at the new 
terminal point. The main difference from the previous argument in this 
section is that the terminal time is T+AT rather than AT. The new cost 
is given by 
T+AT 
Cw = N@_ Bsr Os, War Ow) GY]! sw OR = (> Bw Ok Wo Su) do 
+ G(x(T + AT), T+ AT) + u'M(x(T + AT), T + AT) 
Hence the increase in cost C2 - {2 is given as: 
ue 
DOE Bs a eel 
J [F dx + Fou + q'6x q'f ox q £ oul do 
ca = OD) 
37 y 
+ G Ax, + GAT + u"(M Ax, + MAT) 
cee 
T+AT 
| (On xe) tach = igi (O75) x.) tu) ido 
2 
where x = z + 6x 
EN ree T 
q' Ace dt = q'(T) 6x(T) - q' 6x do 
Once a iF aaa a 
16 
