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The 4
×
l
inequalities (7.13) can be converted to the following 2
×
l
constraints:
⎧
⎨
in
2
i
−
2
(
x
)=
u
i
−
a
u
u
i
+
b
u
≤
0
,
i
= 1
···
l
,
(7.14)
⎩
in
2
i
−
1
(
x
)=
v
i
−
a
v
v
i
+
b
v
≤
0
,
i
= 1
···
l
where
⎧
⎨
a
u
=
u
max
+
u
min
b
u
=
u
max
u
min
,
⎩
a
v
=
v
max
+
v
min
,
b
v
=
v
max
v
min
.
The set of inequalities (7.14) can be converted to an equivalent set of equalities by
introducing the 2
l
vector
ζ
1
···
ζ
n
in
]
T
:
ζ
of slack variables
ζ
=[
in
i
(
x
,
u
)
≤
0
⇔
(7.15)
2
i
= 0
eq
i
(
x
,
ζ
)=
in
i
(
x
)+
ζ
,
i
= 0
···
2
l
−
1
.
Now, let us integrate the inequalities (7.15) with vectors of multipliers
μ
onto the
following Hamiltonian:
X
)=
U
T
U
+
T
(
x
T
eq
(
x
H
(
X
,
λ
−
U
)++
μ
,
ζ
)
(7.16)
where
eq
=[
eq
0
···
eq
2
l
−
1
] and
X
is an 18 + 4
l
extended state vector defined as
X
=
X
x
X
u
X
T
ζ
T
X
T
λ
X
T
(7.17)
μ
with
⎧
⎨
X
x
=
x
,
X
u
=
U
,
X
ζ
=[
···
ζ
2
l
]
,
ζ
(7.18)
1
⎩
X
λ
=[
λ
1
···
λ
6
]
,
X
μ
=[
···
μ
2
l
]
.
μ
1
The solutions of the constrained optimal control problem defined by (7.11), (7.12),
(7.13) can be related to those of an unconstrained variational calculus problem as
proved in [13].
Proposition 7.2.
Vectors
x
(
t
)
and
U
(
t
)
satisfy the necessary conditions of the Min-
imum Principle for the optimal control problem defined by (7.11), (7.12), (7.13) if
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