44
D. Parra-Guevara and Y.N. Skiba
Lagrange multipliers [44]. Let
L(Q) =
1
2
T
0
Q
2
(t) dt − λ
T
0
g 1 (r 1 , t)Q(t) dt − c 1
(2.56)
be the Lagrange functional corresponding to problem (2.50)–(2.52), where λ is the
respective Lagrange multiplier. The first variation of L in the sense of Gateaux [44]
is calculated as
δL(Q; δ Q) =
∂
∂ε
L(Q + εδ Q) ε=0 =
T
0
Q(t) − λg 1 (r 1 , t)
δ Q dt
(2.57)
where δ Q is the variation of Q. A necessary condition for Q ∗ to be a minimum is
δL(Q ∗ ; δ Q) = 0, for any δ Q [44]. Therefore, from Eq. (2.57) we get
Q
∗
(t) = λg 1 (r 1 , t),
(2.58)
where the Lagrange multiplier λ is determined by means of the constraint (2.51) in
the way
λ =
c 1
T
0 g 2
1 (r 1 , t) dt
.
(2.59)
The final result is obtained by substituting Eq. (2.59) in (2.58).
Note that, due to Schwarz inequality [17],
0 <
T
0
g 1 (r 1 , t) dt ≤ T
1
2
T
0
g
2
1 (r 1 , t) dt
1
2
and therefore
T
0 g 2
1 (r 1 , t)dt > 0, that is function Q ∗ is well-defined by the
Eqs. (2.58) and (2.59). Besides, since g 1 (r 1 , t) ≥ 0, we conclude that Q ∗ (t) ≥ 0,
0 ≤ t ≤ T .
We now show that Q ∗ , defined by (2.58) and (2.59), also satisfies the sufficient
condition to be a minimum. Indeed, let Q 0 = Q ∗ + δ Q be a feasible discharge rate.
From constraint (2.51) we have
T
0
g 1 (r 1 , t)δ Q dt = 0,
(2.60)
where δ Q = 0 is an arbitrary variation of Q ∗ . Then,
m(Q 0 ) − m(Q
∗
) =
T
0
Q
∗
(t)δ Qdt +
1
2
T
0
δ
2 Q dt.
(2.61)