guidance, flight mechanics and trajectory optimization

characteristics associated with Eqs. (2.4.73) and (2.4.74) are
(2.4.75)
(2.4.76)
(2.4.77)
(2.4.78)
The meaning of the first two equations is obvious. They are simply a restatement
of the definitions of 4’ and Rtx!, ). The last equation, when coupled
with Eq. (2.4.74), reduced to the Eu f er-Lagrange equation
Equation (2.4.77) is also equivalent to the Euler-Lagrange equation. This
equivalence can be shown by differentiating (2.4.73) with respect to x
and using (2.4.74). Thus, the characteristic directions associated with
the Bellman equation are determined by solving the Euler-Lagrange equation.
Since the value of R at the point (&,,fO ).and the associated curve y (x)
(i.e., the curve emanting from the point (r,,p )) is of primary interest,
it is only necessary to solve for one characteristic; namely, that one
starting at ( 7L,,y0 ). Thus, the solution to the problem of minimizing the
integral
(2.4.79)
can be achieved by integrating Eq. (2.4.79) to determine the optimum curve
y(x), and then substituting this value back into (2.4.80) to evaluate J.
This is the normal procedure and is followed in the Calculus of Variations.
It should be mentioned that the solution to the Euler-Lagrange equation
cannot be accomplished directly due to the two-point boundary nature of the
problem (i.e., curve y(x) must connect the two points (zr,,+, ) and (&, ~ )
while the determination of this curve by numerical integration of Fq. ( Iii .4.79)
requires a knowledge of the slope $f foL* 1 . Hence, it may be more
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