Biomedical Engineering Reference
In-Depth Information
The constraint
J
2
≤
1, where
J
2
is defined by Eq. (7.9) with
m
=
1, is
equivalent to the inequality
t
2
t
1
|
dτ
5
/
2
1
u(τ )
|
(t
2
−
t
1
)
≤
1
,
0
<t
2
−
t
1
≤
.
(7.13)
t
2
−
t
1
2
5
Raise this inequality to a power of
to represent it in the form
t
2
t
1
|
t
1
)
3
/
5
,
u(τ )
|
dτ
≤
(t
2
−
0
<t
2
−
t
1
≤
.
(7.14)
The last inequality implies
t
2
u(τ ) dτ
≤
(t
2
−
t
1
)
3
/
5
,
−
0
<t
2
−
t
1
≤
,
(7.15)
t
1
or, with reference to Eq. (7.12),
t
1
)
3
/
5
,
v(t
2
)
−
v(t
1
)
≤
(t
2
−
0
<t
2
−
t
1
≤
.
(7.16)
Optimal Control for
≥ 1
First the optimal control will be determined
for
=∞
.Set
t
1
=
0and
t
2
=
t
in Eq. (7.16) to obtain
t
3
/
5
.
v(t)
≤
(7.17)
The relations of Eqs. (7.17) and (7.12) lead to the inequality
t
3
/
5
.
x(t)
˙
≥
1
−
(7.18)
The integration of this inequality from 0 to
t
for the initial condition
x(
0
)
=
0 yields
5
8
t
8
/
5
.
x(t)
≥
t
−
(7.19)
Hence, the lower bound of
x
for any
t
is defined by
5
8
t
8
/
5
x(t)
=
t
−
(7.20)
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