guidance, flight mechanics and trajectory optimization

thus, verifying that the solution is a straight line with slope given by
Eq. (2.4.34).
2.4.4 Additional Properties of the Optimal Solution
The solution to the problem of minimizing the integral
(2.4.35)
is usually developed by means of the Calculus of Variations with the
development consisting of the establishment of certain necessary conditions
which the optimal solution must satisfy. In this section, it will be shown
that four of these necessary conditions resulting from an application of
the Calculus of Variations can also be derived through Dynamic Programming.
In the previous sections it was shown that the function R(x,f)defined
by
% fi
R(Z,y) = hh’
Y’
f +-tz,qc,y*)d$
‘9%
(2.4.36)
satisfies the partial differential equation
flx,y,y’) + g f akz
’ = 0 (2.4.3-i’)
Setting the first derivative with respect to .# ' to zero in this equation
provides the additional condition
Also, if y' is to minimize the bracketed quantity in (2.4.37), then the
second derivative
or equal to zero.
(2.4.38)
of this quantity with respect to f" must be greater than
Hence the condition,
(2.4.39)
must be satisfied along the optimai solution. This condition is referred
to as the Legendre condition in the Calculus of Variations.
A slightly stronger condition than that in (2.4.39) can be developed
by letting u& denote the optimal solution and Y’ denote any other
solution. Then from (2.4.37)
aR aR ,
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