Biomedical Engineering Reference
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where ı
1
is an admissible variation. We find
!
Z
X
K
u
C
b
r
u
C
u
C
u
3
ı
1
Ǜ
i
f
i
D 0:
i
D
1
Since ı
1
is arbitrary, from this equation we promptly obtain the state prob-
lem (
6.30
). Similarly,
LJ
LJ
LJ
LJ
2
Z
"
L
.
u
;˛;;
1
;
2
/
D
L
@
2
@
1
.ı
2
/ D
lim
"
!
0
.
u
;˛;;
1
;
2
C"ı
2
/
L
ı
2
u
D0
(6.41)
leading to (
6.31
).
Let us write explicitly now the adjoint equation
LJ
LJ
LJ
LJ
u
@
@
u
1
"
.
.ı
u
/ D
lim
"
!
0
L
.
u
C "ı
u
;˛;;
1
;
2
/
L
.
u
;˛;;
1
;
2
//
Z
Z
1
ı
u
C
b
rı
u
C ı
u
C 3
u
2
ı
u
D
.
u
d/ı
u
Z
2
ı
u
D 0:
(6.42)
Let us factor out the arbitrary variation ı
u
. If we integrate by parts the second and
first order terms, we get
Z
ı
u
u
d C
1
C
b
r
1
1
3
u
2
1
Z
Z
C
1
rı
u
n
.r
1
n
C
1
b
n
C
2
/ı
u
D 0:
Because ı
u
is arbitrary, this equation is equivalent to
1
b
r
1
C
1
C 3
u
2
1
D
u
d in
1
D 0
on
2
D
b
r
1
n
1
n
on :
Notice that
2
does not affect the solution of the problem, therefore in the following
we drop the last equation because we are not interested in the particular value
assumed by
2
. Finally, we compute the derivative with respect to the control
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