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and only if there exists a piecewise smooth vector
X
(
t
)
defined in (7.17) which is a
critical solution for the variational problem
min
J
(
X
)=
t
f
t
0
X
(
t
)
H
(
X
(
t
)
,
,
t
)
dt
(7.19)
with the boundary conditions
⎧
⎨
X
x
(
t
0
)=
x
(
t
0
)
,
X
u
(
t
0
)=0
,
,
X
ζ
(
t
0
)=0
⎩
X
λ
(
t
0
)=0
,
X
μ
(
t
0
)=0
,
.
X
x
(
t
f
)=
x
(
t
f
)
X
(
t
)
If
X
(
t
) is a solution for the variational problem (7.19) then, at all points where
is continuous, the Euler-Lagrange equation must hold:
dt
∂
d
H
X
−
∂
H
X
= 0
,
(7.20)
∂
∂
and at each discontinuity point of
X
(
t
), the corner conditions must be satisfied:
∂
=
∂
H
H
t
+
,
X
X
∂
∂
t
−
(7.21)
X
∂
H
=
X
∂
H
H
H
X
−
X
−
t
+
.
∂
∂
t
−
Admissible solution that satisfy the necessary conditions (7.20,7.21) are called crit-
ical solution. It can be shown that if
X
(
t
) is solution of the variational problem,
then
t
f
(
H
X
z
+
H
X
z
)
dt
= 0
(7.22)
t
0
holds for all admissible piecewise smooth function
z
(
t
) that satisfies the boundary
conditions (
H
X
and
H
X
denote the partial derivatives of
H
with respect to
X
and
X
).
7.4
Solving the Variational Problem
Conventionally, the extremum of the variational problem are obtained by solving
the Euler-Lagrange equation (7.20). However, these equations are only valid at the
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