for v, an optimal control. This is the Pontryagin maximal principle 
3 ; T ; : 
which states that for given values of p and x at time t, the optimal 
control v(t) is the control function for which the Hamiltonian H(t, u) 
is a maximum. 
If the control functions are sufficiently smooth, the optimal 
control is that control for which 
je hs Sp eps (1.27) 
u u 
It is assumed that f is differentiable with respect to u; prior to this 
equation, f need only be Lipschitz continuous with respect to u. This 
equation is a system of m equations which could be solved for the 
m control functions (uj5+-+-,u_) in terms of the state variables 
(X)5-+-+-5%)) and the new variables (@iocoesd yc Consequently, the 
optimal control problem has been reduced to a two-point, boundary-value 
problem for an ordinary differential equation: 
oe S CE, Xs u) 
Die hreDe oe 
T 
=- + 
0 i Dp i 
or 
5 oH 
x = = 
op- 
oH 
.ea56 e 16 2S 
p = ( ) 
_ OH 
ons du 
14 
