34
D. Parra-Guevara and Y.N. Skiba
solution g. Indeed, let us take
p(r, t) =
1
τ |Ω i | , r ∈ Ω i and t ∈ (T − τ, T )
0, otherwise
where |Ω i | denotes the volume of oil-polluted zone, and τ is the time required for the
nutrient to reach its critical concentration in the zone. Then the use of this formula
in (2.18) leads to
J i (φ) =
N
j=1
T
0
g i (r j , t)Q j (t) dt +
D
g i (r, 0)φ
0
(r ) dr,
(2.25)
also known as the duality principle. Provided that φ 0 (r ) = 0 for the first discharge
of nutrient, the last formula is reduced to
J i (φ) =
N
j=1
T
0
g i (r j , t)Q j (t) dt.
(2.26)
The use of (2.26) in (2.2) for each zone Ω i (i = 1, . . . , N ), transforms the
variational problem (2.1)–(2.3) to a more convenient form for the analysis:
minimize m(Q 1 , . . . , Q N ) =
1
2
N
j=1
T
0
Q
2
j (t) dt
(2.27)
subject to: c i − α i ≤
N
j=1
T
0
g i (r j , t)Q j (t)dt ≤ c i + β i , 1 ≤ i ≤ N (2.28)
0 ≤ Q j (t),
0 ≤ t ≤ T,
1 ≤ j ≤ N .
(2.29)
Note that problem (2.27)–(2.29) uses N adjoint functions g i (r, t), which, when
restricted to the discharge points r j , j = 1, . . . , N , generate N 2 temporal influence functions g i (r j , t). Each function g i (r j , t) compresses dynamical information
necessary to estimate how a signal emitted at point r j impacts the zone Ω i . As a
consequence, the duality principle (2.26) quantifies the total effect on zone Ω i due
to the signals emitted at points r j , j = 1, . . . , N .
However, if a repeated discharge of nutrient is needed for degrading oil-residuals,
then the nonzero initial concentration of the nutrient must be taken into account (see
(2.25)). It should be noted that, due to microbial intake of nutrient in the oil-polluted
D. Parra-Guevara and Y.N. Skiba
solution g. Indeed, let us take
p(r, t) =
1
τ |Ω i | , r ∈ Ω i and t ∈ (T − τ, T )
0, otherwise
where |Ω i | denotes the volume of oil-polluted zone, and τ is the time required for the
nutrient to reach its critical concentration in the zone. Then the use of this formula
in (2.18) leads to
J i (φ) =
N
j=1
T
0
g i (r j , t)Q j (t) dt +
D
g i (r, 0)φ
0
(r ) dr,
(2.25)
also known as the duality principle. Provided that φ 0 (r ) = 0 for the first discharge
of nutrient, the last formula is reduced to
J i (φ) =
N
j=1
T
0
g i (r j , t)Q j (t) dt.
(2.26)
The use of (2.26) in (2.2) for each zone Ω i (i = 1, . . . , N ), transforms the
variational problem (2.1)–(2.3) to a more convenient form for the analysis:
minimize m(Q 1 , . . . , Q N ) =
1
2
N
j=1
T
0
Q
2
j (t) dt
(2.27)
subject to: c i − α i ≤
N
j=1
T
0
g i (r j , t)Q j (t)dt ≤ c i + β i , 1 ≤ i ≤ N (2.28)
0 ≤ Q j (t),
0 ≤ t ≤ T,
1 ≤ j ≤ N .
(2.29)
Note that problem (2.27)–(2.29) uses N adjoint functions g i (r, t), which, when
restricted to the discharge points r j , j = 1, . . . , N , generate N 2 temporal influence functions g i (r j , t). Each function g i (r j , t) compresses dynamical information
necessary to estimate how a signal emitted at point r j impacts the zone Ω i . As a
consequence, the duality principle (2.26) quantifies the total effect on zone Ω i due
to the signals emitted at points r j , j = 1, . . . , N .
However, if a repeated discharge of nutrient is needed for degrading oil-residuals,
then the nonzero initial concentration of the nutrient must be taken into account (see
(2.25)). It should be noted that, due to microbial intake of nutrient in the oil-polluted