42
D. Parra-Guevara and Y.N. Skiba
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
F =
Q ∈ H ; Q(t) ≥ 0, 0 ≤ t ≤ T, and
T
0
Q(t)g 1 (r 1 , t)dt = c 1
(2.54)
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 function, we have that
Q
∗
(t) =
c 1 g 1 (r 1 , t)
T
0 g 2
1 (r 1 , t) dt
(2.55)
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 λ ∈ (0, 1). Then, evidently, λQ 1 + (1 − λ)Q 2 ≥
0, 0 ≤ t ≤ T . Besides,
T
0
(λQ 1 + (1 − λ)Q 2 )g 1 (r 1 , t) dt = λc 1 + (1 − λ)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 .
D. Parra-Guevara and Y.N. Skiba
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
F =
Q ∈ H ; Q(t) ≥ 0, 0 ≤ t ≤ T, and
T
0
Q(t)g 1 (r 1 , t)dt = c 1
(2.54)
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 function, we have that
Q
∗
(t) =
c 1 g 1 (r 1 , t)
T
0 g 2
1 (r 1 , t) dt
(2.55)
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 λ ∈ (0, 1). Then, evidently, λQ 1 + (1 − λ)Q 2 ≥
0, 0 ≤ t ≤ T . Besides,
T
0
(λQ 1 + (1 − λ)Q 2 )g 1 (r 1 , t) dt = λc 1 + (1 − λ)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 .