The functions F is twice continuously differentiable with respect to x 
and Lipschitz continuous with respect to u; G represents the cost at the 
terminal point x(T) = x it is twice continuously differentiable with 
Tt? 
respect to Xn 
Suppose that v is an optimal control vector, and consider a slight 
deviation Ou of this control vector. If 
u(t) =v + du 
u(t) is also a control vector, as can be seen from an application of the 
theory of ordinary differential equations. If z is the state vector 
associated with the control v, the new control u yields a new state 
vector x given by 
x(t) = 2 + 6x 
where 6x is an unknown. Moreover, since v minimizes the cost function, 
the new cost function is greater; 
al 
1 (Of: Cs (OM a Cx @ cer ID) 
B =u 7 
IE 
> { D@, 2 w) d@ + G(Z pT) (@>)) 
0 
Since the old state vector satisfies 
Z 
tl 
Fh 
= 
in 
. 
N 
. 
<q 
Vv 
and the new one satisfies 
| 
iT] 
Fh 
oo 
ct 
x 
(ory 
4 
