a= 3p 
5 oH 
‘pe as Dee (1.28) 
_ 3H 
Oe du 
The initial condition x(0) = x, together with the terminal conditions 
—0 
M(x, T) = 0 
DCE) ES aaa (uy Mit Ce) Giese) 
Gs mC) = G,  woM. 
provides a sufficient number of conditions to determine x, p, u, and T. 
The last two equations in the system (1.38) are obtained from (1.37). 
The problems of optimal control theory generally reduce to a two- 
point, boundary-value problem for the system of ordinary differential 
equations (1.30). Bailey, Shampine, and Walemanc discuss methods for 
solving such two-point, boundary-value probiems. These problems are 
presently solved either by the shooting method or by solving a sequence 
of simpler boundary value problems whose solutions converge to a solu- 
tion of the given problem. In any case, very few of these problems can 
be solved without the use of electronic computers either digital or 
hybrid. 
The shooting method is the easier, when it works. It consists of 
supplementing the conditions at one end with a sufficient number of 
assumed conditions to yield an initial value problem. The initial value 
yale, P. B., et al., "Nonlinear Two-Point, Boundary-Value Problems," 
Academic Press, Inc., New York (1968). 
19 
