guidance, flight mechanics and trajectory optimization

Hence, the boundary condition
OAI f’“,jd =o (2.4.51)
results. Setting a_R
/ y
du If
=,& an: substituting
JR
-- /& 22 =o
into (2.4.51) provides
z -- ~~=O
ON p,jd =o (2.4.52)
av Y
which indicates that the gradient of@,x,y) and the gradient of ~~x,Y/I
are co-linear along the curve t(x,~j=O.
Eqs. (2.4.46), (2.4.49), (2.4.51) and (2.4.52) are different but
equivalent representations of the boundary condition that theg function
must satisfy when the terminal point is required to lie on the curve
ptx,p=o * From this boundary condition the transversality condition
which the Calculus of Variations requires can be derived. This is shown
next.
From Eq. (2.4.45B) it follows that the optimal slope must satisfy
af
T
sle
+y
=O
(2.4.53)
at all points (x,t/) including the terminal point. Using this equation,
Eq. (2.4.45B) becomes
/=o (2.4.54)
and must also hold at every point including the terminal point. Combining
Eqs. (2.4.51), (2.4.53) and (2.4.54) provides
which is the transversality condition which the optimal solu%ion must satisfy;
that is, Eq. (2.4.55) s p ecifies which of the points along pl.~,#)=Ois the
point for which the integral J is a minimum.
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