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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 9, Number 3, July 2013

doi:10.3934/jimo.2013.9.505 pp. 505–524

CONTROL PARAMETRIZATION AND FINITE ELEMENT METHOD FOR CONTROLLING MULTI-SPECIES REACTIVE TRANSPORT IN A CIRCULAR POOL

Heung Wing Joseph Lee, Chi Kin Chan and Karho Yau Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong, China

Kar Hung Wong and Colin Myburgh School of Computational and Applied Mathematics University of the Witwatersrand Johannesburg, South Africa

(Communicated by Kok Lay Teo) Abstract. In this paper, we consider an optimal control problem for a cleaning program involving effluent discharge of several species in a circular pool. A computational scheme combining control parametrization and finite element method is used to develop a cleaning program to meet the environmental health requirements. A numerical example is solved to illustrate the efficiency of our method.

1. Introduction. The “natural environmental destruction potential” due to contaminant in many enclosed areas such as fish tanks [15], aquifers [1, 9, 16], and underground tunnels [2, 3, 4, 13] has been widely studied. Previous field studies in these references have shown the importance of understanding the “natural environmental destruction potential” before a contaminant cleanup action or a long term monitoring program is deployed. In several instances, the above self-decaying process may be adequate to preclude the need for additional source removal or plume cleaning activities [1]. In this paper, we consider a cleaning program involving effluent discharge in a circular pool. During the entire cleaning program from t = 0 to t = tf , effluent consisting of different species is introduced into the pool from a small arc of a circular block given by {r = ε1 , θ ∈ [0, ε2 ]}, where ε1 is the radius and ε2 is the angle subtending this arc. The effluent is released into the environment from another arc opening on the circumference of the pool given by {r = R, θ ∈ [π − ε3 , π + ε3 ]}, where R is the radius and 2ε3 is the angle subtending that arc. The schematic diagram is briefly sketched in Figure 1. To operate the whole process optimally, we wish to discharge as much effluent as possible into the pool. However, due to strict environmental health requirements, we may not release more than a certain amount of such chemical species into the 2010 Mathematics Subject Classification. Primary: 49M15, 65M60; Secondary: 35Q92. Key words and phrases. Species concentration, circular pool, optimal control, Galerkin Scheme, control parametrization, non-negativity requirement of state variables.

505

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH

environment at any time. Therefore, we set up an optimal problem whereby we wish to maximize the total amount of effluent discharge, subject to the condition that the amount of each species being released into the environment at any time t ∈ [0, tf ] does not exceed the upper limit set up by the health department. ........................................... ............. ......... ......... ........ ........ ....... ...... . ...... . . . . ...... .... . . . . ..... ...... ..... . . . . .. ........... ..... . . . . ... ... .. ........ . ... . ..... .. ... . . . ..... ... ... ..... . ... ..... .. ... . ..... . . . . . . . . . . . . . . . . . . . ... . . .. . . . . . . . . . . . ..... ...... ....... . ... . . . . . . . . 1 2 . . . . .... ........ ... . . ... . . ... ... ......... 3 . ... ..... ..... .. . ... . ..... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . ...... . .... ... . . . . . . . . . ... . . . . . . . . . . . .. .. .... ................................................................ . . .. . ..... . ..... ... .. . . .. . . .. ... . .. ...... . . . . ... . . . . . .... . ... 1 ... .... 1 .... ... . ..... .... ... . 3 ...... ...... ..... .. ......... .............. ............. ... .. . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... . . ..... . .... ..... ..... ..... ..... ..... ..... ...... ...... ...... . . . . . . ....... ....... ........ ........ .......... ..........................................................

R

•

(R, π −  ) •

•

(R, π +  ) •



( ,  )

•

( , 0)

Figure 1. The schematic diagram of the circular wastewater treatment pool. In [25], a similar problem involving the effluent dispersal program in a rectangular diffuser unit is considered. Both the method used in [25] and in this paper involve the reduction of a distributed optimal control problem with path constraints to a lumped optimal control problem with path constraints by the finite element method, together with the solution of the lumped optimal control problem by the control parametrization method. (For the solution of control problems with path constraints, see [5, 22] for details. For the control parametrization method, see [7, 8, 10, 11, 12, 14, 17, 18, 19, 20, 21, 23, 24, 25] for details.) However, the distributed system for the circular pool problem considered in this paper is more complicated than that for the rectangular diffuser problem considered in [25]. Although both systems consist of second order partial differential equations, together with Neumann and Dirichlet boundary conditions, the system for the circular pool problem involves additional boundary conditions arising from the fact that (r, 0) and (r, 2π) correspond to the same point in a circular pool for all r ∈ [ε1 , R]. Due to the existence of these additional boundary conditions, the method of reduction of a distributed system to a lumped system for the circular pool problem should be modified as follows: For the rectangular diffuser problem, we convert the distributed system to a lumped system by multiplying each partial differential equation by root functions Ri (x, y) and integrating over the whole domain of (x, y) (See Section 3 of [25] for details.); for the circular pool problem, we need to multiply each partial differential equation, first ¯ i (r, θ), and then by sum of two root functions R ¯ j (r, θ)+ R ¯ k (r, θ) by root functions R and integrate over the whole domain of (r, θ). (See Section 3 of this paper for details.) Moreover, from the numerical example solved in Section 7 of this paper and that solved in Section 8 of [25], we show that the circular pool is more effective than the rectangular diffuser unit in terms of the amount of effluent that can be discharged into their respective domains.

CONTROLLING N SPECIES IN A CIRCULAR POOL

507

In Section 2, we formulate the problem as mentioned in the previous paragraph. In Section 3, a finite element method using the Galerkin Scheme is used to convert the partial differential equations into a set of ordinary differential equations involving additional state variables. In Section 4, the well-known control parametrization method is used for solving the problem obtained in Section 3 . In Section 5, owing to the fact that the amount of species concentration in real-life situation cannot be negative, we devise a method for modifying the state equations obtained in Section 3 so that the species concentration at any position and at any time will not become negative during any numerical calculation. For this purpose, we need to approximate the non-smooth state equations which reflect the real-life situation by some smooth state equations with a small parameter δ. The convergence result of the approximated system of the state equations to that of the real-life situation as δ → 0 is discussed in Section 6. A numerical example is solved in Section 7. Conclusions and some suggestions for further study are given in Section 8. 2. Problem formulation. In this model, we assume that the effluent consists of n sequentially decaying chemical species. Each species, except species n, undergoes a first order chemical reaction. After the chemical reaction has occurred, a certain amount of each species i(i = 1, ..., n − 1) decays to produce species i + 1. Species n is more stable or chemically inert and does not produce any further species. The circular pool is described in polar coordinates by the region {r ∈ [0, R], θ ∈ [0, 2π]} . For ease of numerical calculations and practical issues, we change the above circular domain to the following domain: {r ∈ [ε1 , R], θ ∈ [0, 2π]} , where ε1 is a small number. One of the advantages for the above consideration is that the singular point in the domain represented by {r = 0} can be avoided in all calculations. The general governing differential equations for this problem, which can be easily derived from those given in [3, 13, 25] for the case that the domain is a rectangle, can be expressed as follows:   2 ∂Ci ∂ 2 Ci 2 ∂Ci 2 ∂ Ci 2 ∂Ci + γr −D r +r + Mi r ∂t ∂r ∂r2 ∂r ∂θ2 (1) 2 = r (zi Mi−1 ki−1 Ci−1 − Mi ki Ci ) , r ∈ [ε1 , R] , θ ∈ [0, 2π] , Mi < ∞,

i = 1, . . . , n,

with M0 = 0, k0 = 0, C0 = 0, −3

(2)

where Ci is the concentration of species i [M L ] (mole per liter), zi is the effective yield factor which describes the mass of a species i produced from another species [M M−1 ] (mole per mole), ki is the first order contaminant decay rate constant of species i [T−1 ] (per day), γ is the seepage velocity [L T−1 ] (meters per day), D is the dispersion coefficient [L2 T−1 ] (square meters per day), Mi is the retardation factor for species i [M M−1 ] (mole per mole) and n is the total number of species.

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH

As shown in Figure 1, to introduce the species into the circular wastewater treatment pool, a small opening in the form of an arc near the center of the pool is available. Suppose that the location of this opening is {(r, θ) | r = ε1 , θ ∈ [0, ε2 ]} .

(3)

Furthermore, to allow the species flow out of the circular treatment pool, there is a larger opening in the form of an arc on the circumference of the pool. Suppose that the location of this opening is {(r, θ) | r = R, θ ∈ [π − ε3 , π + ε3 ]} .

(4)

Now, we can introduce the following Neumann boundary conditions for the concentration at every point inside the circumference of the pool, except for the small opening given by (3). ∂Ci (ε1 , θ, t) = 0, i = 1, . . . , n, θ ∈ [ε2 , 2π] , ∂r ∂Ci (R, θ, t) = 0, i = 1, . . . , n, θ ∈ [0, 2π] \ [π − ε3 , π + ε3 ] , ∂r ∂Ci (R, θ, t) ¯ i (R, θ, t) , i = 1, . . . , n, θ ∈ [π − ε3 , π + ε3 ] , = −kC ∂r

(5) (6) (7)

where k¯ is some real-valued constant. Since (r, 0) and (r, 2π) correspond to the same point in a circular pool, we need the following additional boundary conditions Ci (r, 0, t) = Ci (r, 2π, t) ,

i = 1, . . . , n,

r ∈ [ε1 , R] ,

(8)

∂Ci (r, 0, t) ∂Ci (r, 2π, t) = , i = 1, . . . , n, r ∈ [ε1 , R] . (9) ∂θ ∂θ As for the small opening defined by (3) for the inlet concentration, we introduce the following Dirichlet boundary condition Ci (ε1 , θ, t) = ui (t) ,

i = 1, . . . , n,

θ ∈ [0, ε2 ] ,

(10)

where ui (t) is the control variable which corresponds to the inlet concentration of species i being introduced into the pool at time t at the small opening. ui (t) , i = 1, . . . , n, are chosen to be piecewise linear functions. Usually, we need to set a maximum allowable amount of contaminants being released into the environment at the opening given by (4) at any time t as follows: Z π+ε3 RCi (R, θ, t) dθ ≤ c¯i , i = 1, . . . , n, t ≥ 0. (11) π−ε3

Initially, the pool does not contain any contaminants. That is, r ∈ [ε1 , R] ,

Ci (r, θ, 0) = 0,

θ ∈ [0, 2π] .

(12)

Next, we need to introduce an objective function. We want to maximize the total amount of contaminants being discharged into the circular treatment pool, with the requirement that the rate of discharging the species into the pool is both bounded and non-decreasing for all time. As shown in Figure 1, the arc length of the inlet opening is ε1 ε2 . Therefore, we want to solve the following problem n Z tf X max J (u) = (ε1 ε2 ) ui (t) dt (13) i=1

0

CONTROLLING N SPECIES IN A CIRCULAR POOL

509

subject to umax ≥ u˙ i (t) ≥ 0

i = 1, . . . , n,

t ≥ 0,

(14)

where tf and umax are given. The optimal control problem (P) is stated as follows: Problem (P). Subject to the system (1), together with the boundary conditions (5) - (9), the boundary constraint (11) and the discharge constraint (14), we want to find piecewise linear functions u1 (t) , u2 (t) , . . . , un (t) which maximize J (u) given by (13) over the set of all piecewise linear control functions.

3. Approximated problem. 3.1. Transforming the variables from circular domain to rectangular domain. In order to convert the problem involving a distributed system into one involving a lumped system, we need to first transform the variables from the circular domain to a rectangular domain represented by {(r, θ) | r ∈ [ε1 , R] , θ ∈ [0, 2π]}. Moreover, we divide the rectangular domain into a finite number of sub-regions as shown in Figure 2. For illustrative purposes, we divide the domain into only 24 sub-regions, where each sub-region is also a rectangle. The method remains the same if we divide the domain into more sub-regions. Thus, in Figure 2, the segment {r = ε1 , θ ∈ [0, ε2 ]} at the left-hand boundary of region S11 corresponds to the opening for the inlet of the species into the pool and the segment {r = R, θ ∈ [π − ε3 , π + ε3 ]} at the right-hand boundary of region S43 and S44 corresponds to the opening for the outlet of the species into the environment.

(1 , 2π) • (1 , 2π − 2 ) •

( R4 , 2π) ( R2 , 2π) ( 3R 4 , 2π) • • • S16 S26 S36 S46 S15

S25

S35

S45

(1 , π + 3 ) •

• (R, π + 3 ) S14

S24

S34

S44

(1 , π) •

• (R, π) S13

S23

S33

S43

S12

S22

S32

S42

S31

S41 • ( 3R 4 , 0)

(1 , π − 3 ) • (1 , 2 ) • (1 , 0) •

• (R, 2π) • (R, 2π − 2 )

• (R, π − 3 )

S11

• ( R4 , 0)

S21

• ( R2 , 0)

• (R, 2 ) • (R, 0)

Figure 2. The illustrative diagram of transforming the circular domain into a rectangular domain.

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH

3.2. Generating the approximated problem. We use the Galerkin Scheme finite element method to generate the Approximated Problem. The method is as follows. ¯ i, j (r, θ) Corresponding to each node point (ri , θj ), we construct a root function R as follows:  (r−ri−1 )(θ−θj−1 ) ¯    (ri −ri−1 )(θj −θj−1 ) if (r, θ) ∈ Si j ,   (r−r )(θ−θ ) i+1 j−1   (ri −ri+1 )(θj −θj−1 ) if (r, θ) ∈ S¯(i+1) j ,  (r−ri−1 )(θ−θj+1 ) ¯ i, j (r, θ) = (15) R if (r, θ) ∈ S¯i (j+1) , (ri −ri−1 )(θj −θj+1 )   (r−ri+1 )(θ−θj+1 )   if (r, θ) ∈ S¯(i+1) (j+1) ,  (ri −ri+1 )(θj −θj+1 )   0 otherwise, where S¯r denotes the closure of Sr . Remark 1. From (15), it is clear that ( ¯ i, j (r, θ) = 1 at node point (ri , θj ) , R 0 at other node points.

(16)

¯ i, j (r, θ) is always Since Sij is only defined when 1 ≤ i ≤ 4 and 1 ≤ j ≤ 6, R well-defined as ri , i = 0, ..., 4, and θj , j = 0, ..., 6, are given in Figure 2. (Note ¯ 0,j , R ¯ 4,j , R ¯ i,0 , R ¯ i,6 that we do not need r−1 , r5 , θ−1 , θ7 in the definition of R respectively.) ¯ i, j (r, θ) by R ¯ k (r, θ), where k = Next, to simplify our notation, we replace R ¯ i + 5j + 1, j = 0, 1, . . . , 6 and i = 0, 1, . . . , 4. Here, Rk (r, θ) represents the k th global node point of the whole domain. We can approximate the concentration of the species i, i = 1, . . . , n, by the concentration function Ci35 (r, θ, t) =

35 X

¯ j (r, θ) . Ti,j (t) R

(17)

j=1

From (17), it is clear that the Dirichlet boundary condition (10) becomes Ti,1 (t) = ui (t) ,

i = 1, . . . , n,

(18)

Ti,6 (t) = ui (t) ,

i = 1, . . . , n.

(19)

Moreover, the constraint (11) can by approximated by Rε3 [0.5Ti,15 (t) + Ti,20 (t) + 0.5Ti,25 (t)] ≤ c¯i ,

i = 1, . . . , n,

t ≥ 0.

(20)

Note that the constraints (18) - (20) are called path constraints. The initial condition (12) becomes Ti,j (0) = 0,

i = 1, . . . , n,

j = 1, . . . , 35.

(21)

In view of (17), the boundary condition (8) becomes Ti,31 (t) = Ti,1 (t) ,

i = 1, . . . , n,

(22)

Ti,32 (t) = Ti,2 (t) ,

i = 1, . . . , n,

(23)

Ti,33 (t) = Ti,3 (t) ,

i = 1, . . . , n,

(24)

Ti,34 (t) = Ti,4 (t) ,

i = 1, . . . , n,

(25)

Ti,35 (t) = Ti,5 (t) ,

i = 1, . . . , n.

(26)

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511

In order to convert the system of partial differential equations (1) into a system of ordinary differential equations, we let β (r, θ) be an arbitrary function in C 1 ([ε1 , R] × [0, 2π]). Multiplying (1) by β (r, θ) and integrating over the region {r ∈ [ε1 , R] , θ ∈ [0, 2π]}, we get Z 2π Z R Z 2π Z R ∂Ci (r, θ, t) ∂Ci (r, θ, t) 2 r2 β (r, θ) r β (r, θ) drdθ + γ drdθ Mi ∂t ∂r ε1 ε1 0 0 Z 2π Z R Z 2π Z R ∂Ci (r, θ, t) ∂ 2 Ci (r, θ, t) rβ (r, θ) r2 β (r, θ) drdθ − D −D drdθ 2 ∂r ∂r ε1 ε1 0 0 Z 2π Z R ∂ 2 Ci (r, θ, t) β (r, θ) drdθ −D ∂θ2 ε1 0 Z 2π Z R =zi Mi−1 ki−1 r2 β (r, θ) Ci−1 (r, θ, t) drdθ 0 2π

Z

Z

ε1 R

− Mi k i 0

r2 β (r, θ) Ci (r, θ, t) drdθ.

ε1

(27) Using integration by parts, we obtain Z 2π Z R Z 2π Z R ∂Ci (r, θ, t) ∂Ci (r, θ, t) 2 drdθ + γ drdθ r2 β (r, θ) Mi r β (r, θ) ∂t ∂r 0 ε1 0 ε1 Z 2π Z 2π ∂Ci (R, θ, t) ∂Ci (ε1 , θ, t) −D R2 β (R, θ) dθ + D dθ ε21 β (ε1 , θ) ∂r ∂r 0 0 Z 2π Z R Z 2π Z R ∂Ci (r, θ, t) ∂β (r, θ) ∂Ci (r, θ, t) +D drdθ + D drdθ 2rβ (r, θ) r2 ∂r ∂r ∂r 0 0 ε1 ε1 Z R Z 2π Z R ∂Ci (r, 2π, t) ∂Ci (r, θ, t) drdθ − D β (r, 2π) dr −D rβ (r, θ) ∂r ∂θ ε1 0 ε1 Z R Z 2π Z R ∂Ci (r, 0, t) ∂β (r, θ) ∂Ci (r, θ, t) +D β (r, 0) dr + D drdθ ∂θ ∂θ ∂θ ε1 0 ε1 Z 2π Z R = zi Mi−1 ki−1 r2 β (r, θ) Ci−1 (r, θ, t) drdθ Z

0 ε1 2π Z R

r2 β (r, θ) Ci (r, θ, t) drdθ.

− M i ki 0

ε1

(28) Using the boundary conditions (5) - (7) and combining the fifth term and the seventh term of (28), we get Z 2π Z R Z 2π Z R ∂Ci (r, θ, t) ∂Ci (r, θ, t) 2 drdθ + γ r2 β (r, θ) drdθ Mi r β (r, θ) ∂t ∂r 0 ε1 0 ε1 Z π+ε3 Z ε2 ∂Ci (ε1 , θ, t) ¯ + kD R2 β (R, θ) Ci (R, θ, t) dθ + D ε21 β (ε1 , θ) dθ ∂r π−ε3 0 Z 2π Z R Z 2π Z R ∂Ci (r, θ, t) ∂β (r, θ) ∂Ci (r, θ, t) rβ (r, θ) r2 +D drdθ + D drdθ ∂r ∂r ∂r 0 ε1 0 ε1 (29)

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH R

Z −D

β (r, 2π) ε1 2π

Z

Z

∂Ci (r, 2π, t) dr + D ∂θ

Z

R

β (r, 0) ε1

∂Ci (r, 0, t) dr ∂θ

R

∂β (r, θ) ∂Ci (r, θ, t) drdθ ∂θ ∂θ ε1 0 Z 2π Z R r2 β (r, θ) Ci−1 (r, θ, t) drdθ =zi Mi−1 ki−1 +D

ε1

0

Z

2π

Z

R

r2 β (r, θ) Ci (r, θ, t) drdθ.

− M i ki ε1

0

¯ m for Ci and β in (29) for all m = 7, 8, . . . , 30, we Then, substituting Ci35 and R get

Mi

35 X

T˙i,j (t) Aj,m =

35 X

Ti,j (t) Gj,m

j=1

j=1

+ zi Mi−1 ki−1

35 X

Ti−1,j (t) Aj,m ,

m = 7, . . . , 30, (30)

j=1

where   ¯ j,m + Cˆj,m + A¯j,m + B ¯j,m − Mi ki Aj,m , Gj,m = −γ Aˆj,m − D kB Z 2π Z R ¯ j (r, θ) R ¯ m (r, θ) drdθ, Aj,m = r2 R 0

¯ m (r, θ) ¯ j (r, θ) ∂ R ∂R drdθ, ∂r ∂r 0 ε1 Z 2π Z R ¯ j (r, θ) ∂R ¯ m (r, θ) drdθ, R = r2 ∂r 0 ε1 Z π+ε3 ¯ j (R, θ) R ¯ m (R, θ) dθ, = R2 R

A¯j,m = Aˆj,m Bj,m

Z

r2

(32) (33) (34) (35)

π−ε3 Z 2π Z R

¯ m (r, θ) ¯ j (r, θ) ∂ R ∂R drdθ, ∂θ ∂θ 0 ε1 Z 2π Z R ¯ j (r, θ) ∂R ¯ m (r, θ) drdθ, = r R ∂r 0 ε1

¯j,m = B Cˆj,m

ε1 2π Z R

(31)

(36) (37)

and T0,j (t) = 0,

j = 1, . . . , 35.

(38)

¯ m for β in (29) for all m = Remark 2. The reasons that we do not substitute R ¯ m (ε1 , θ) is 1, . . . , 6 and m = 31, . . . , 35 are as follows. In the fourth term of (29), R ¯ m (r, 2π) not identically zero when m = 1 and m = 6. In the seventh term of (29), R ¯ m (r, 0) is is not identically zero when m = 31, . . . , 35. In the eighth term of (29), R not identically zero when m = 1, . . . , 5 .

CONTROLLING N SPECIES IN A CIRCULAR POOL

513

¯m + R ¯ m+30 for Ci and β in (29) On the other hand, by substituting Ci35 and R and applying the boundary condition (9) for m = 2, . . . , 5 successively, we get

Mi

35 X

T˙i,j (t) (Aj,m + Aj,m+30 ) =

j=1

35 X

Ti,j (t) (Gj,m + Gj,m+30 )

j=1

+ zi Mi−1 ki−1

35 X

Ti−1,j (t) (Aj,m + Aj,m+30 ) ,

m = 2, . . . , 5, (39)

j=1

where Gj,m and Aj,m are same as defined in (31) - (37). Furthermore, due to equations (22) - (26) and (18) - (19), for each i = 1, . . . , n, we can replace the state variables Ti,1 (t), Ti,2 (t), Ti,3 (t), Ti,4 (t), Ti,5 (t) and Ti,6 (t) by Ti,31 (t), Ti,32 (t), Ti,33 (t), Ti,34 (t), Ti,35 (t) and Ti,31 (t) respectively. Hence, the number of state variables Ti,j (t) can be reduced from 35n to 29n, counting from Ti,7 (t) to Ti,35 (t) for each i = 1, . . . , n. Let J1 = {7, . . . , 30} and J2 = {2, . . . , 5}. Then, from (30) and (39), we get 35 X

T˙i,j (t) ψ˜m,i,j =

35 X

Ti,j (t) ψ¯m,i,j +

j=7

j=7

35 X

Ti−1,j (t) ψˆm,i,j , i = 1, . . . , n, m ∈ J1 ∪ J2 , (40)

j=7

where for j = 7, . . . , 30, ( ψ˜m,i,j = ( ψ¯m,i,j = ( ψˆm,i,j =

Mi Aj,m , Mi (Aj,m + Aj,m+30 ) ,

m ∈ J1 , m ∈ J2 ,

(41)

Gj,m , m ∈ J1 , Gj,m + Gj,m+30 , m ∈ J2 , zi Mi−1 ki−1 Aj,m , zi Mi−1 ki−1 (Aj,m + Aj,m+30 ) ,

(42) m ∈ J1 , m ∈ J2 ,

(43)

for j = 31

ψ˜m,i,j

ψ¯m,i,j

ψˆm,i,j

 Mi (A1,m + A6,m + Aj,m ) , m ∈ J1 , = Mi (A1,m + A6,m + Aj,m + A1,m+30 (44)  , m ∈ J2 , +A6,m+30 + Aj,m+30 ) ( G1,m + G6,m + Gj,m , m ∈ J1 , = (45) G1,m + G6,m + Gj,m + G1,m+30 + G6,m+30 + Gj,m+30 , m ∈ J2 ,  zi Mi−1 ki−1 (A1,m + A6,m + Aj,m ) , m ∈ J1 , = zi Mi−1 ki−1 (A1,m + A6,m + Aj,m + A1,m+30 (46)  , m ∈ J2 , +A6,m+30 + Aj,m+30 )

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH

for j = 32, . . . , 35 ( Mi (Aj−30,m + Aj,m ) , m ∈ J1 , (47) ψ˜m,i,j = Mi (Aj−30,m + Aj,m + Aj−30,m+30 + Aj,m+30 ) , m ∈ J2 , ( Gj−30,m + Gj,m , m ∈ J1 , ¯ ψm,i,j = (48) Gj−30,m + Gj,m + Gj−30,m+30 + Gj,m+30 , m ∈ J2 , ( zi Mi−1 ki−1 (Aj−30,m + Aj,m ) , m ∈ J1 , ψˆm,i,j = (49) zi Mi−1 ki−1 (Aj−30,m + Aj,m + Aj−30,m+30 + Aj,m+30 ) , m ∈ J2 . Let ˜ ψ˜m,i = ψ˜m,i,7 ψ¯m,i = ψ¯m,i,7

ψ˜m,i,8 ψ¯m,i,8

···

ˆ ψˆm,i = ψˆm,i,7

ψˆm,i,8

···

 ˜m = diag ψ˜ ˜m,1 φ

¯m = diag ψ¯m,1 φ

˜ ψ˜m,2

ψ¯m,2

···

···

···


ψ˜m,i,35 ,
ψ¯m,i,35 ,
ψˆm,i,35 ,

˜ ψ˜m,1    029 ˜ ψ˜m,n =   ..  . 029 ¯ ψm,1 
 029 ψ¯m,n =  .  ..

ˆm = subdiag ψˆ ˆm,2 φ

ˆ ψˆm,3

···

(50)

m ∈ J1 ∪ J2 ,

(51)

m ∈ J1 ∪ J2 ,

(52)

029 ˜ ψ˜m,2 .. .

···

···

029

029 ψ¯m,2 .. .

··· ··· .. . 029

···

029 



m ∈ J1 ∪ J2 ,

029  ˆˆ ψ   m,2 0 ˆ 29 ψˆm,n =   .  ..  029

··· .. .

029



 029   ..  , .  ˜ ψ˜m,n  029 029   ..  , .  ¯ ψm,n

(53)

(54)



029

···

029

029 ˆˆ ψm,3 .. .

···

 029   029  , ..  .   029

··· .. . ˆˆ ψm,n

···

(55)

where 029 is a zero row vector of dimension 29. Therefore, from (40) and (21) , we get   ˜T˙ = φ ¯+φ ˆ T, φ (56) T (0) = 0,

(57)

where  ˜= φ ˜T7 φ  ¯= φ ¯T7 φ  ˆ= φ ˆT7 φ

˜T8 φ ¯T8 φ ˆT8 φ

··· ··· ···

˜T30 φ ¯T30 φ ˆT30 φ

˜T2 φ ¯T2 φ ˆT2 φ

˜T3 φ ¯T3 φ ˆT3 φ

˜T4 φ ¯T4 φ ˆT4 φ

˜T5 φ

T

¯T5 φ

T

ˆT5 φ

T

,

(58)

,

(59)

,

(60)

CONTROLLING N SPECIES IN A CIRCULAR POOL

515

and T = T1,7

T1,8

···

T1,35

···

Tn,7

Tn,8

···

Tn,35


T

.

(61)

˜ φ ¯ and φ ˆ are all 28n by 29n matrices. Moreover, in view of (22), Note that φ, the Dirichlet boundary condition (18) can be written as Ti,31 (t) = ui (t) ,

i = 1, . . . , n.

(62)

Thus, we have transformed the Problem (P) into the Problem (Q). Problem (Q). Subject to the system (56) with initial condition (57), the Dirichlet boundary condition (62), the discharge constraint (14) and the continuous constraint (20), we want to find the piecewise linear functions u1 (t) , . . . , un (t) which maximize the objective function J (u) given by (13). The convergence result of Problem (Q) to Problem (P) can be state as follows: Both the optimal control u∗i (t) , i = 1, ..., n and the associated optimal concentration Ci∗ (r, θ, t) , i = 1, ..., n of Problem (Q) converge to that of Problem (P) when the number of partitions of the domain {(r, θ) | r ∈ [ε1 , R] , θ ∈ [0, 2π]} tends to infinity. 4. Control parametrization method for the approximated Problem (Q). Due to the fact that the state equation (56) does not involve any control variables, the approximated Problem (Q) is not a standard optimal control problem and so cannot be solved directly by any optimal control softwares. Hence we now convert the Problem (Q) into a standard optimal control problem so that it can be solved by the Control Parametrization method. Firstly, we divide the time interval [0, tf ] into np equal subintervals so that the piecewise linear control ui (t) has a constant slope at each subinterval of [0, tf ]. In this way, the control ui (t) can be completely specified by the following equations dui (t) = vi (t) , i = 1, . . . , n, dt ui (0) = 0, i = 1, . . . , n,

(63) (64)

where vi (t) =

np X

vij χj (t) ,

i = 1, . . . , n,

(65)

j=1

( 1, t ∈ [(j − 1) ∆, j∆) , χj (t) = 0, otherwise,

j = 1, . . . , np and ∆ =

tf . np

(66)

Obviously, the constraint (14) becomes umax ≥ vi (t) ≥ 0.

(67)

In view of (62), we obtain from (63) and (64) another n new state equations as follows: T˙i,31 (t) = vi (t) , i = 1, . . . , n, (68) Ti,31 (0) = 0,

i = 1, . . . , n,

(69)

where vi (t) are now regarded as control functions. Note that the equations (57) and (69) are consistent. Incorporating the n new state equations given in (68) into (56), we get ˆ ¯T˙ = φT + φv, ˆ φ (70)

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH

where ¯= φ

! ˜ φ , ˜ ˜ φ

(71)

 ˜ ˜ = eT25,29n eT25+29,29n φ   ¯+φ ˆ φ φ= , 0n,29n   ˆ ˆ = 028,n , φ In

···

eT25+29(n−1),29n

T

,

(72) (73) (74)

ej,k is an elementary row vector in Rk with the jth coordinate equal to 1, 0j,k is ¯ and φ ¯ are the j × k zero matrix and In is the n × n identity matrix. Note that φ ˆ ˆ is a 29n × n matrix. 29n × 29n square matrices and φ From (70) and (57), we get ˆ ¯−1 φT + φ ¯−1 φv, ˆ T˙ = φ

(75)

T (0) = 0.

(76)

From (13) and (62), the new objective function becomes n Z tf X max J¯ (v) = (ε1 ε2 ) Ti,31 (t) dt. i=1

(77)

0

Definition 4.1. The step function v (t) which satisfies the equation (67) is called an admissible control of a problem and U is the set of all admissible controls.
¯ can be stated as The new optimal control problem, denoted by Problem Q follows: ¯ Subject to the equations (75) - (76) and the continuous constraint Problem (Q). (20), we want to find an admissible control v (t) which maximizes J¯ (v) over U.
¯ can be solved by the package MISER. (See [6] for deThe above Problem Q
¯ , it is clear from (62) and (17) that the tails). After having solved the Problem Q approximated optimal control for the original Problem (P) is given by ∗ u∗i (t) = Ti,1 (t) ,

(78)

and the optimal concentration Ci∗ (r, θ, t) can be obtained by Ci∗ (r, θ, t) =

35 X

∗ ¯ i, j (r, θ) , Ti,j (t) R

r ∈ [ε1 , R] ,

θ ∈ [0, 2π] ,

(79)

j=7


∗ ¯ . where Ti,j (t) are the optimal state variables acquired by solving Q 5. Modifying the state equations to handle the non-negativity requirement
for the speices concentration. During the process of solving the Problem ¯ by the control parametrization method, we may encounter a situation where Q some state variables Ti,j (t) generated by evaluation of the state equations have negative values. This implies that the concentration also takes on negative values, which is clearly physically
impossible. In fact, the state equations governed by (75) ¯ are valid provided that all Ti,j (t) remain non-negative all - (76) for Problem Q

CONTROLLING N SPECIES IN A CIRCULAR POOL

517


¯ in the time. We should amend the state equations (75) - (76) for the Problem Q the following manner: For the sake of simplicity, we first let ˆ ¯−1 φT + φ ¯−1 φv. ˆ f (T, v) = φ

(80)
¯ are as Then the state equations reflecting the real-life situation for Problem Q follows.   T˙i,j = ξ Ti,j , f (T, v)i,j , i = 1, . . . , n, j = 1, . . . , 29, (81) Ti,j (0) = 0,

(82)

where the function ξ : R2 → R is defined as ( z, ξ (y, z) = max (z, 0) ,

y > 0, y ≤ 0.

(83)


¯ reflect the Remark 3. Although the state equations (81) - (83) for Problem Q real-situation more accurately, they
can exhibit rapid transients when any of the ¯ approaches zero from above since the function state variables Ti,j for Problem Q ξ is non-smooth. Thus, we encounter computational difficulties due to the stiffness of the differential equations. In order to overcome the above difficulties, we replace the non-smooth function ξ in (83) by a sufficiently smooth function ξδ , where ξδ : R2 → R is defined by   y > 0, z, (84) ξδ (y, z) = Iδ (y) z + (1 − Iδ (y)) maxδ (z) , −δ ≤ y ≤ 0,   maxδ (z) , y < −δ, where maxδ (z) is the function used for smoothing max (z, 0) defined by   0, 2 z ≤ −δ, maxδ (z) = (z+δ) 4δ , −δ ≤ z ≤ δ,   z, z > δ,

(85)

and Iδ (y) = −2

 y 3

 y 2

+ 1, −δ ≤ y ≤ 0, (86) δ δ is a real number between 0 and 1.
¯ δ , to be defined below, Therefore, the state equations for the new problem Q are   T˙i,j = ξδ Ti,j , f (T, v)i,j , i = 1, . . . , n, j = 1, . . . , 29, (87) Ti,j (0) = 0.

−3

(88)

Remark 4. Since
Iδ (y) ∈ C ∞ [−δ, 0] and maxδ (z) ∈ C 1 (R), it is clear that ξδ (y, z) ∈ C 1 R2 . Although ξδ (y, z) does not converge pointwise to ξ (y, z) as δ → 0, we are able to prove in the next section that as δ → 0, the values of the state variable obtained from the evolution of (87) - (88) and (84) - (86) converge to those obtained by the evolution of (81) - (83).

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH


¯ δ may Remark 5. Although the new state equations (87) - (88) for Problem Q also sometimes generate state variables with values Ti,j (t) ∈ [−δ, ∞) instead of Ti,j (t) ∈ [0, ∞), it does not exhibit rapid transients when any of the state variables Ti,j (t) approach zero due to the reason given for Remark 4. Therefore, by replacing the state equations (75)
- (76) by the new state equations ¯ δ as follows. (87) - (88), we have the modified Problem Q ¯ δ ). Subject to the state equations (87) - (88), the continuous constraint Problem (Q (20),we want to find a control v (t) which maximizes J¯ (v) given by (77) over U.
¯ δ . We first 6. Convergence results of the state variables of Problem Q
¯ as follows: rewrite the state equations (81) - (82) for the Problem Q

where ξ¯ (T, v) ∈ R

29×n

T˙ = ξ¯ (T, v) ,

(89)

T (0) = 0,

(90)

is defined as  
ξ¯ (t, v) i,j = ξ Ti,j , f (T, v)i,j .

(91)
¯δ Similarly, we also rewrite the state equations (87) - (88) for the Problem Q as follows: T˙ = ξ¯δ (T, v) , (92) T (0) = 0, where ξ¯δ (T, v) ∈ R29×n is defined as  
ξ¯δ (t, v) i,j = ξδ Ti,j , f (T, v)i,j .

(93)

(94)

For each v ∈ U, let T (v) and Tδ (v) be the solution of (89) - (90) and the solution of (92) - (93) respectively. We wish to prove that Tδ (v) converges to T (v) in the L∞ norm uniformly with respect to v ∈ U as δ → 0. Let |·| denote both the Euclidean norm of a vector and the Frobenius norm of a matrix. Definition 6.1. Let [T1 , T2 ] be any finite sub-interval of [0, ∞). For any piecewise continuous function τ : [T1 , T2 ] → Rn , let kτ k[T1 ,T2 ] = ess sup |τ (t)| . t∈[T1 ,T2 ]

The following theorem is a direct consequence of Theorem 7.1 of [25]. Theorem 6.2. For all v ∈ U, we have lim kTδ (v) − T (v)k[0,tf ] = 0.

δ→0

(95)

7. Numerical results. In this section, for the purpose of comparing the effectiveness of a circular pool and a rectangular diffuser unit [25] in terms of the amount of effluent that can be discharged into their respective domains, we solve a numerical problem using exactly the same parameters as those used in [25] as follows: (However, the size of the inlet opening and the outlet opening for the species concentration considered in this paper and those considered in [25] are not exactly the same.)

CONTROLLING N SPECIES IN A CIRCULAR POOL

519

Suppose the radius of a circular pool is 40m. Suppose the location of the small opening for the inlet of the species into the pool and the outlet of the species into the environment are (in polar coordinates), respectively, {(r, θ) | r = 0.1m, θ ∈ [0, 0.3]} , {(r, θ) | r = 40m, θ ∈ [π − 0.15, π + 0.15]} . Then, we first divide the circular domain into 24 sub-regions exactly
the same as ¯ δ twice, with described in Figure 2. We then solve the transformed Problem Q δ = 0.00001 and np (number of partitions of the time interval [0, tf ] for the control parametrization method ) = 10 first, and then with δ = 0.00001 and np = 20 by the package MISER, using the following set of parameters. Transport and reaction parameters used in the example problem: Dispersion coefficient Retardation coefficient Efficient yield factor of species 1 Efficient yield factor of species 2 Efficient yield factor of species 3 Seepage velocity Decay rate of species 1 Decay rate of species 2 Decay rate of species 3 Constant in the Neumann boundary condition
¯δ Final time for Problem Q

D Mi (i = 1, 2, 3) z1 z2 z3 γ k1 k2 k3 k¯ tf

0.01m2 /day 1.0 1.0 0.75 0.5 0.13 m/day 0.0025/day 0.0017/day 0.0007/day 1.0 90 days

Constraint parameters used in the example problem: The upper limit for the rate of the species being discharged into the pool at any time (umax ) = 0.017 mole/liter/day. The upper limit for each amount of contaminants being released into the environment at any time (¯ ci , i = 1, 2, 3) = 0.5 mole/liter × dm.
¯ δ can be summarized as follows. The numerical result for Problem Q
¯ δ with np = 10 and np = 20 are The results obtained by solving Problem Q almost the same. From Table 1 - Table 9, it is clear that for all t = 36 days, 72 days and 90 days, the optimal concentrations for all the species 1, 2, 3 are monotonically decreasing almost everywhere with respect to r. The existence of places where the concentration is not decreasing with respect to r is probably due to the coarse nature of the Galerkin Scheme. The total amount of contaminants being discharged into the circular pool from t = 0 to t = 90 days is 4.759 mole/liter × dm. From Table 1, Table 2 and Table 3, it is clear that the maximum amount of contaminants being released into the environment for species 1 occurs at t = 90 days, which is approximately equal to 400 × 0.15 × 0.00833 = 0.4998 mole/liter × dm. From Table 4, Table 5 and Table 6, it is clear that the maximum amount of contaminants being released into the environment for species 2 also occurs at t = 90 days, which is also approximately equal to 400 × 0.15 × 0.00833 = 0.4998 mole/liter × dm. From Table 7, Table 8 and Table 9, it is clear that the maximum amount of contaminants being released into the environment for species 3 also occurs at t = 90

520

H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH

days, which is approximately equal to 400 × 0.15 × 0.00712 = 0.4272 mole/liter × dm. Thus, the constraint for the amount of contaminants being released into the environment is almost binding at t = 90 days for both species 1 and 2, but is not binding for species 3 for all t. Remark 6. Using an inlet opening and outlet opening of size 0.03 meter and 12 meter respectively, the total amount of effluent that can be discharged into the circular pool for 90 days is 4.759 mole/liter × dm. Using an inlet opening and outlet opening of size 2 meter and 40 meter respectively, the total amount of dispersal that can be discharged into the rectangular diffuser unit is 2.42 mole/liter × dm. (See [25] for details.) Hence, the circular pool can discharge more effluent than the rectangular diffuser unit into their respective domains, even though the size of both the inlet opening and outlet opening of the circular pool is smaller than that of the rectangular diffuser unit. Hence, we conclude that the circular pool is more effective than the rectangular diffuser unit, in terms of the amount of effluent that can be discharged into their respective domain. θ/r 0.1m 10m 0 0.01200 0.00001 0.3 0.01200 0 π − 0.15 0 0.00049 π 0.01344 0 π + 0.15 0 0.00021 2π − 0.3 0.00062 0 Table 1. Concentration of the

20m 30m 40m 0.00005 0 0 0.00015 0 0.00002 0 0.00002 0 0.00012 0 0.00001 0 0.00001 0 0.00001 0 0 species 1 at time 36 days (M)

θ/r 0.1m 10m 20m 30m 40m 0 0.40810 0 0.02514 0 0.00168 0.3 0.40810 0 0.04079 0 0.00658 π − 0.15 0 0.07574 0 0.00512 0 π 0.28060 0 0.02495 0 0.00301 π + 0.15 0 0.02977 0 0.00134 0 2π − 0.3 0.04162 0 0.00186 0 0.00032 Table 2. Concentration of the species 1 at time 72 days (mole/liter)

8. Conclusions and suggestions for further study. A computational scheme using combined control prametrization and finite element method has been developed for solving the problem of controlling multi-species transport in a circular pool. The extension of this method for solving problems involving a three-dimensional environment is quite a challenging task, because it involves solving a system with much more state variables and control variables, which cannot be easily handled by the existing optimization or optimal control software. Thus, this provides us with a very interesting research area in the near future.

CONTROLLING N SPECIES IN A CIRCULAR POOL

θ/r 0.1m 0 0.70346 0.3 0.70346 π − 0.15 0 1.83267 π π + 0.15 0 2π − 0.3 0.33779 Table 3. Concentration

10m 20m 30m 40m 0.03264 0.02054 0.00131 0.00076 0 0.06513 0 0.01166 0.16739 0 0.01021 0 0 0.07013 0 0.00833 0.16343 0 0.01019 0 0 0.02162 0 0.00193 of the species 1 at time 90 days (mole/liter)

θ/r 0.1m 0 0.11961 0.3 0.11961 0 π − 0.15 π 0.07363 π + 0.15 0.01166 2π − 0.3 0 Table 4. Concentration

10m 20m 30m 40m 0 0.01014 0 0.00076 0 0.01181 0 0.00157 0.02560 0 0.00118 0 0 0.00180 0 0.00026 0.00010 0.00006 0 0 0.00373 0 0.00011 0 of the species 2 at time 36 days (mole/liter)

θ/r 0.1m 0 0.67883 0.3 0.67883 π − 0.15 0 π 0.47801 π + 0.15 0 2π − 0.3 0.06844 Table 5. Concentration

10m 20m 30m 40m 0.00028 0.02951 0.00001 0.00243 0 0.04235 0 0.01113 0.13719 0 0.00869 0 0 0.02771 0 0.00514 0.03949 0 0.00233 0 0 0.00379 0 0.00060 of the species 2 at time 72 days (mole/liter)

θ/r 0.1m 0 1.28431 0.3 1.28431 π − 0.15 0 π 1.87695 π + 0.15 0 2π − 0.3 0.43121 Table 6. Concentration

10m 20m 30m 40m 0.06441 0.02706 0.00208 0.00120 0 0.05918 0 0.01627 0.22260 0 0.01360 0 0 0.07768 0 0.00833 0.19754 0 0.00973 0 0 0.02039 0 0.00208 of the species 2 at time 90 days (mole/liter)

521

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H. W. J. LEE, C. K. CHAN, K. YAU, K. H. WONG AND C. MYBURGH

θ/r 0.1m 0 0.60000 0.3 0.60000 π − 0.15 0 0.34673 π π + 0.15 0.03529 0 2π − 0.3 Table 7. Concentration

10m 20m 30m 40m 0 0.03131 0 0.00432 0 0.03577 0 0.00958 0.05770 0 0.00720 0 0 0.01213 0 0.00172 0.00341 0 0.00012 0 0.02020 0 0.00059 0 of the species 3 at time 36 days (mole/liter)

θ/r 0.1m 0 1.20000 0.3 1.20000 0 π − 0.15 π 0.76010 π + 0.15 0 2π − 0.3 0.00083 Table 8. Concentration

10m 20m 30m 40m 0 0.04743 0 0.00630 0 0.08774 0 0.01945 0.25244 0 0.01482 0 0 0.02690 0 0.00541 0.03062 0 0.00159 0 0.01191 0.00142 0.00025 0.00035 of the species 3 at time 72 days (mole/liter)

θ/r 0.1m 10m 20m 30m 40m 0 1.50000 0 0.04816 0 0.00638 0.3 1.50000 0 0.10162 0 0.02048 π − 0.15 0 0.30863 0 0.01887 0 π 1.00570 0 0.04137 0 0.00712 π + 0.15 0 0.06214 0 0.00345 0 2π − 0.3 0.05583 0 0.00490 0 0.00090 Table 9. Concentration of the species 3 at time 90 days (mole/liter)

CONTROLLING N SPECIES IN A CIRCULAR POOL

523

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Received October 2011; 1st revision August 2012; 2nd revision October 2012. E-mail E-mail E-mail E-mail E-mail

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Heung Wing Joseph Lee: [email protected] Chi Kin Chan: [email protected] Karho Yau: [email protected] Kar Hung Wong: [email protected] Colin Myburgh: [email protected]