Biomedical Engineering Reference
In-Depth Information
The inequalities of Eqs. (7.29) and (7.30) are analogues of the inequalities
of Eqs. (7.18) and (7.19) for the case under consideration. From these
inequalities it follows that the lower bound of the variable
x(t)
for any
t
is
defined by
k
1
−
8
+
2
3
/
5
k
x(t)
=
k)
1
k
3
/
5
−
5
k)
8
/
5
,
(7.31)
+
(t
−
−
8
(t
−
k < t
≤
(k
+
1
)
and is attained at
k
3
/
5
k)
3
/
5
,
v(t)
=
+
(t
−
(7.32)
k < t
≤
(k
+
1
),
k
=
0
,
1
,
2
,...,
which corresponds to the control
3
5
1
(t
−
k)
2
/
5
,k<t
1
.
(7.33)
The maximum of the expression of (7.31) is attained at the time instant
u(t)
=−˙
v(t)
=−
≤
(k
+
1
),
0
<
≤
=
−
3
/
5
+
1
−
−
3
/
5
3
/
5
5
/
3
T()
(7.34)
and is defined by
k
1
−
8
+
2
k
3
/
5
+
8
1
k
3
/
5
8
/
3
,
=
−
3
/
5
.
(7.35)
1
3
x(T )
=
−
k
The brackets with the missing upper horizontal bar (
) denote the integer
part of the corresponding expression.
Hence, the control
⎧
⎨
⎩
3
5
1
(t
−
k)
2
/
5
,k<t
−
≤
+
≤
(k
1
),
t
T,
u(t)
=
(7.36)
0
,
t > T,
is optimal if the inequality of Eq. (7.16) holds for this control. This will be
proved.
Proof.
Let
t
2
≤
T
and, hence,
t
1
<T
. The function
v(t)
for
t
≤
T
has the
form of Eq. (7.32). According to the constraint 0
<t
2
−
t
1
≤
, the instants
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