By the definition of J, there is a control u + du together with a 
trajectory x + 6x such that 
als 
J (Sp + AXn> T+AT) = { R(@5 soar Oey Woe Su) do 
(1.41) 
where u is the control such that 
T 
Js T) = | WW, 35 wy) do (1.42) 
0 
From (1.41) and (1.42) 
J (Xp + AX» T+AT) - J (ips T) 
T 
= { I(@, a OX> War Ow) = F(, x, wu) do 
0 
T+AT 
+ N@, + OX, Ww OW) do 
T 
Ie 
{ (F 6x + F du) dt + FAT 
0 x — nm — 
Now from (1.23), the equation for p is 
Hence 
J (Kp + Axon» TFL) = J (Xs Ty) 
aE 
fa [tq + pte) ox + F bul do + FAT 
0 SK Ww = 
T 
peGL) soscGh) ace J p'(£, 6x - OkS)ch F ou do + FAT 
ZAM 
