2 A Strategy for Bioremediation of Marine Shorelines …
43
Proof To show this we must prove that F = F [10]. Let Q 0 be an element of F.
Then there is a sequence {Q k } ∞
k=1 in F such that
Q k − Q 0 → 0 as k → ∞
Assume that Q 0 (t) < 0 in some interval I ⊂ (0, T ) of positive measure |I | > 0.
Then
Q k − Q 0
2
=
T
0
(Q k − Q 0 )
2 dt ≥
I
(Q k − Q 0 )
2 dt ≥
I
Q
2
0 dt = l > 0
The last inequality contradicts the convergence of sequence {Q k } ∞
k=1 in H , and hence,
Q 0 is a non-negative function in (0, T ).
On the other hand, applying the Schwarz inequality we get
c 1 −
T
0
Q 0 g 1 (r 1 , t)dt
=
T
0
(Q k − Q 0 ) g 1 (r 1 , t)dt
≤ Q k − Q 0 g 1 (r 1 , t) → 0 as k → ∞
and therefore
T
0 Q 0 g 1 (r 1 , t) dt = c 1 , that is Q 0 ∈ F. The lemma is proved.
Note that the zero function q(t) ≡ 0, 0 ≤ t ≤ T, does not belong to the feasible
set F. Indeed, the constraint (2.51) is not satisfied for such function because c 1 > 0.
This remark allows us to establish the most important result of this section.
Theorem 2.2 The variational problem (2.50)–(2.52) has non-trivial unique solution
in the space H .
Proof By Lemma 2.1, the space H is a uniformly convex Banach space. Besides,
by Lemmas 2.1, 2.2 and 2.3, the feasibility space F is a non-empty closed convex
set in H . Therefore, due to Theorem 2.1, there is a unique function Q ∗ ∈ F that
minimizes the distance between the set F and the point q ≡ 0. That is according to
(2.50), function Q ∗ minimizes the objective functional m(Q). Finally, we observe
that Q ∗ = 0 because q /
∈ F, and hence, the unique solution of problem (2.50)–(2.52)
is non-trivial. The theorem is proved.
It is shown in the next section that function Q ∗ , mentioned in Theorem 2.2, is
precisely the function (2.55).
2.6.1.2 Optimal Discharge Parameters and the Adjoint Functions
The analytical expression for the optimal discharge rate Q ∗ , namely, the solution
of variational problem (2.50)–(2.52), can be obtained by means of the method of
43
Proof To show this we must prove that F = F [10]. Let Q 0 be an element of F.
Then there is a sequence {Q k } ∞
k=1 in F such that
Q k − Q 0 → 0 as k → ∞
Assume that Q 0 (t) < 0 in some interval I ⊂ (0, T ) of positive measure |I | > 0.
Then
Q k − Q 0
2
=
T
0
(Q k − Q 0 )
2 dt ≥
I
(Q k − Q 0 )
2 dt ≥
I
Q
2
0 dt = l > 0
The last inequality contradicts the convergence of sequence {Q k } ∞
k=1 in H , and hence,
Q 0 is a non-negative function in (0, T ).
On the other hand, applying the Schwarz inequality we get
c 1 −
T
0
Q 0 g 1 (r 1 , t)dt
=
T
0
(Q k − Q 0 ) g 1 (r 1 , t)dt
≤ Q k − Q 0 g 1 (r 1 , t) → 0 as k → ∞
and therefore
T
0 Q 0 g 1 (r 1 , t) dt = c 1 , that is Q 0 ∈ F. The lemma is proved.
Note that the zero function q(t) ≡ 0, 0 ≤ t ≤ T, does not belong to the feasible
set F. Indeed, the constraint (2.51) is not satisfied for such function because c 1 > 0.
This remark allows us to establish the most important result of this section.
Theorem 2.2 The variational problem (2.50)–(2.52) has non-trivial unique solution
in the space H .
Proof By Lemma 2.1, the space H is a uniformly convex Banach space. Besides,
by Lemmas 2.1, 2.2 and 2.3, the feasibility space F is a non-empty closed convex
set in H . Therefore, due to Theorem 2.1, there is a unique function Q ∗ ∈ F that
minimizes the distance between the set F and the point q ≡ 0. That is according to
(2.50), function Q ∗ minimizes the objective functional m(Q). Finally, we observe
that Q ∗ = 0 because q /
∈ F, and hence, the unique solution of problem (2.50)–(2.52)
is non-trivial. The theorem is proved.
It is shown in the next section that function Q ∗ , mentioned in Theorem 2.2, is
precisely the function (2.55).
2.6.1.2 Optimal Discharge Parameters and the Adjoint Functions
The analytical expression for the optimal discharge rate Q ∗ , namely, the solution
of variational problem (2.50)–(2.52), can be obtained by means of the method of