where 
x(0) = Xo 
pe) = G(T) (1.29) 
There are just enough conditions to determine x, pi, and u. 
The function H contains the variables x, rae u, and of course t. 
Using (1.26) to eliminate u, (1.28) can be expressed in terms of the set 
of dual variables x and p' = Ds where the prime denotes transpose of 
the vector; the resulting system is the familiar canonical form of 
classical mechanics. 
|e 
| 
| 
p=-— isa) 
The boundary conditions are stated in terms of Xo» Xp» and T; for 
instance, both Xp and T might be fixed, or either one might vary while 
the other is fixed. No boundary conditions are specified directly in 
terms of p; the boundary conditions on p are obtained indirectly by 
substitution into (1.29). Equation (1.29) does, however, contain a 
sufficient set of conditions to pose a two-point, boundary-value problem 
for (1.30). 
Another form that the boundary condition at t = T might assume is 
for x, 
T and T to satisfy an end condition of the form 
M(x, T) = 0 (1.31) 
where M is a twice continuously differentiable vector function of both 
its arguments. In this case the method of Lagrange multipliers will be 
used to transform the optimal control problem into a corresponding 
WS) 
