Environmental Engineering Reference
In-Depth Information
2.6.1.1 Existence and Uniqueness
In this section the existence and uniqueness of solution to variational problem ( 2.50 )-
( 2.52 ) is proved. To this end, we remind some properties of the Hilbert space H
=
L 2 (
0
,
T
)
together with a strong result of approximation theory (a minimum distance
theorem).
Theorem 2.1 ([ 5 ]) A non-empty closed convex set in a uniformly convex Banach
space possesses a unique point closest to a given point.
Lemma 2.1 ([ 5 ]) The space H
=
L 2 (
0
,
T
)
is a uniformly convex Banach space.
We point out that the meaning of condition ( 2.50 ) is the minimization of the norm
(distance) in the space H . It is for this reason Theorem 2.1 is useful in proving the
existence and uniqueness. We now consider the specific set and point in space H for
which Theorem 2.1 is applied.
Definition 2.1 The feasible space F for variation problem ( 2.50 )-( 2.53 ) is given as
follows
Q
c 1
T
F
=
H
;
Q
(
t
)
0
,
0
t
T
,
and
Q
(
t
)
g 1 (
r 1 ,
t
)
dt
=
(2.54)
0
Lemma 2.2
The feasible space F is a non-empty set in space H .
Proof Because the adjoint solution g 1 (
r 1 ,
t
)
is a non-negative square-integrable func-
tion, we have that
c 1 g 1 (
r 1 ,
t
)
Q (
t
) =
(2.55)
T
0
g 1 (
r 1 ,
t
)
dt
is a function in H that fulfils constraints ( 2.51 ) and ( 2.52 ). Therefore, Q (
t
)
belongs
to the feasible space F . The lemma is proved.
The meaning and usefulness of function Q (
t
)
defined by ( 2.55 ) is established in
the next section.
Lemma 2.3
The feasible space F is a convex set in H .
Proof In fact, let Q 1 , Q 2
F and
ʻ (
0
,
1
)
. Then, evidently,
ʻ
Q 1 + (
1
ʻ)
Q 2
0
,
0
t
T . Besides,
T
0
Q 1 + (
1
ʻ)
Q 2 )
g 1 (
r 1 ,
t
)
dt
= ʻ
c 1 + (
1
ʻ)
c 1 =
c 1
and hence, F is a convex set in H . The lemma is proved.
Lemma 2.4
The feasible space F is a closed set in H .
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