The vector f and the cost function F are twice continuously differentiable 
with respect to x and continuously differentiable with respect to u. 
The Lagrange multipliers will be used here to reduce this problem 
to a two-point, boundary-value problem. As in (1.32), the differ- 
ential equation is introduced into the cost function by means of a 
Lagrange multiplier p. 
T 
I@, 35 BW) py Ce = 2@5 &5 W)) ae 
0 
os (u) 
which yields the variational equation 
ap 
i 2 te 1! Ow ar)? Oe = jo)! 
0 gia tae aa ay 
Pe) 
So. £ ox Die Su] do 
T 
= ' =a iT 
= p'(T)éx(T) + J [@, - B' - p'£,) ox 
as ' 
+ = pie) & do 2 O (3.3) 
The differential in the cost is greater than or equal to zero since 
it is assumed that the variation du is around an optimal control, a 
control which minimizes the cost. 
Because of the constraint (3.1), the vector du is not free. 
For instance, suppose that for t between ty and tos the trajectory z(t) 
due to the optimal control v(t) is along the boundary of the allow- 
able region; see Figure 3. One cannot freely choose the variation Su 
in the control vector fort, <t<t 
1 and still expect to remain in the 
2 
allowable region R. 
For the optimal trajectory z and control v, there are at most a 
finite number of intervals t, < t < t, + 1 such that equality holds for 
k k 
any of the equations in (3.1).* On such an interval, the conditions 
(3.1) can be split into two sets 
*The proof of the statement is topological and beyond the scope of 
these notes. 
27 
