Biomedical Engineering Reference
In-Depth Information
The number of degrees of freedom in the system of Problem 4.9 is 1
less than that in the system of Problem 4.8.
The solutions of Problems 4.8 and 4.9 are related by a proposition similar
to Proposition 4.4 for Problems 4.5 and 4.6.
J
min
Proposition 4.5.
Let
u
2
(t)
and
˜
be the optimal control and the minimal
1
J
1
(u
2
)
in Problem 4.9. Then:
value of the performance index
(i) The minimum value of the criterion
J
1
in Problem 4.8 is equal to zero
if
D
3
≥
J
min
J
min
J
min
and is equal to
−
D
3
if
D
3
<
,thatis,
1
1
1
0
J
min
≤
if
D
3
,
J
min
1
=
(4.170)
1
J
min
J
min
−
D
3
if
> D
3
,
1
1
(ii) The optimal controls
u
1
(t)
and
u
2
(t)
for Problem 4.8 are defined by
⎧
⎨
i
=
1
¯
n
f
i
(
y
,
J
min
u
2
(t)
˜
−
μ
i
y
)
˙
if
≤
D
3
,
1
y
)
n
D
3
J
min
f
i
(
y
,
˙
u
2
(t)
−
1
μ
i
u
1
(t)
=
(4.171)
⎩
1
i
=
J
min
−
D
3
1
J
min
+
σ(t)
if
> D
3
.
1
J
min
1
u
2
(t)
=
u
2
(t),
(4.172)
The proof of Proposition 4.5 almost completely coincides with that of
Proposition 4.4. The only difference is that, in accordance with Eqs. (4.165),
one should replace Eqs. (4.146) and (4.147) by
n
n
f
i
(
y
,
˙
y
)
=
J
min
f(t)
+
σ(t)
+
f
i
(
y
,
˙
u
2
(t)
=
x
2
(t)
+
1
μ
i
1
μ
i
y
)
1
i
=
i
=
(4.173)
and
n
u
2
(t)
=
J
min
f(t)
+
σ(t)
+
f
i
(
y
,
˙
1
μ
i
y
),
(4.174)
1
i
=
respectively. The relation of Eq. (4.174) is obtained by differentiating
Eq. (4.137) with reference to Eq. (4.165).
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