National Laboratory for Scientific Computation - LNCC

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1 p. 1/6 National Laboratory for Scientific Computation - LNCC Theory of Optimal Control Applied to Problems of Ambiental Pollution Santina de Fátima Arantes Jaime Edilberto Muñoz Rivera Supported by FAPERJ/Brazil

2 Model Formulation: R We consider Ω as a rectangular region with boundary Γ = Γ 1 Γ 2 Γ 3 Γ 4 with pointwise source: F(x, t) = v(t)δ(x b), as in the Figure below y Γ 4 Γ 1 β b ω F(x, t) Ω Γ 3 Γ 2 x The state is defined by the solution z of system ( A1, A2, A3 ) z λ z+β z=v(t)δ(x b) t in Ω (, T) eq.1a z(x, ) = in Ω eq.1b z(x, t) = on Γ 1 (, T) eq.1c z ν (x, t) = f(x, t) on (Γ 2 Γ 3 Γ 4 ) (, T). eq.1d p. 2/6

3 p. 3/6 z(x, t) stands for the pollutant s concentration at (x, t) Ω [, T]. λ > is the diffusion coefficient. β is the velocity vector with β i C 1 (Ω). v(t) is the pointwise control. δ(x b) is the Dirac mass at b, where b determines the point of pollution entrance. T > is given. Condition (1) b states that in the initial time the pollution is null. Condition (1) c states that in any time t (, T) in boundary Γ 1 there is no pollution. Condition (1) d represents the diffusive flow of pollution through boundaries Γ 2 Γ 3 Γ 4 by the function f, where f is given in L 2(, T; H 1 2 (Γ 2 Γ 3 Γ 4 ) ). z/ ν = z ν, where ν is the normal unitary vector exterior to Ω.

4 p. 4/6 Cost functional We want to minimize the cost functional J : L 2 (, T) IR J(v) = z(, T; v) zd 2 dx + N T v 2 dt eq.1e Ω where z d L 2 (Ω) defines a standard (objective) function and the cost constant N > is given. The cost functional J is well defined, if z(, T; v) L 2 (Ω). Given cost functional J, the control problem summarizes in solving the following two problems: P1: To find a unique u U ad such that J(u) = inf { J(v); v U ad }. P.P1 Such u is called optimal control. P2: To characterize optimal control u in such a way that it is possible the numerical calculation of the problem. P.P2

5 Application p. 5/6 We study the case of the contamination of a river or lagoon by mercury (Hg). In this case, by the Brazilian Legislation, the acceptable level z d of pollution of mercury (Hg) in drinking water is given by: z d = mg/l, effluent is given by: z d = mg/l. We will consider in 1a : Water without movement: β =. Water in movement: β. Our problem consists of determining ways (through the controls) in such a way to minimize the effect caused by the debris of pollutant.

6 p. 6/6 Objective: To solve the problems P1 and P2. Stages to follow : To show that the state system given by 1b admits a unique solution. To show the existence of a unique optimal control u that minimizes the functional J on a set of admissible functions U ad. Being the most important aspect of the mathematical point of view. To get a characterization for the optimal control u and from this characterization the optimality system that allows the numerical calculation of the problem. Being the most important aspect of the numerical point of view. To get a numerical solution of the problem by using stabilized method SUPG of semidiscrete finite elements. To plot the numerical solution.

7 p. 7/6 Decomposition To analyse the problem we decompose 1c considering z = y + w, where y and w are, respectively, solutions of systems y λ y+β y=v(t)δ(x b) t in Ω (, T) eq.2a y(x, ) = in Ω eq.2b y(x, t) = on Γ 1 (, T) eq.2c y ν (x, t) = on (Γ 2 Γ 3 Γ 4 ) (, T) eq.2d and w t λ w+β w= in Ω (, T) eq.3a w(x, ) = in Ω eq.3b w(x, t) = on Γ 1 (, T) eq.3c w ν (x, t) = f(x, t) on (Γ 2 Γ 3 Γ 4 ) (, T). eq.3d

8 Solution of the problem p. 8/6 The cost functional J given in 1e can be rewritten as J(v) = Ω y(, T; v) [z d w(, T)] 2 dx + T v 2 dt. Here, so that the cost functional J is well defined, we need that y(, T; v) L 2 (Ω) and w(, T) L 2 (Ω). eq.4a Consider and f L 2(, T; H 1 2 (Γ2 Γ 3 Γ 4 ) ) V = { v H 1 (Ω); v = on Γ 1 }, with norm v = ( n k=1 Ω ) 1 2 dx x k v 2.

9 Solution of the problem p. 9/6 Problem 3a can be solved using standard methods for parabolic partial differential equations. In this case there exists a unique w solution of this system, such that w L 2(, T; V H 2 (Ω) ) C (, T; H 1 (Ω) ) C 1 (, T; L 2 (Ω)). Therefore, w(, T) L 2 (Ω) and the functional J is well defined in L 2(, T ). On the other hand, the problem 2a is not a usual one, the weak solution is not simple because of the Dirac Delta. In this case, the functions space where the solution is defined depends on the dimension n of the spatial variable.

10 Solution of the problem p. 1/6 For n = 1. By Sobolev imbedding Theorem, we obtain δ V H 1 (Ω). Therefore, for v L 2 (, T), there exists a unique y of 2b, such that y L 2(, T; V ) and y t L2(, T; V ). Thus, y(, T, v) makes sense in L 2 (Ω) and the functional J is well defined in L 2(, T ).

11 p. 11/6 Solution of the problem For n = 2 or n = 3. We have that δ H 2 (Ω), which makes the problem more delicate from the mathematical point of view. Using the transposition method as in Lions 2, we have that for v L 2 (, T) there exists only one y(v) L 2 (, T; L 2 (Ω)) ultraweak solution of 2c, such that y satisfies Ω T y ϕ dxdt = T v(t)φ(b, t) dt, for all ϕ L 2 (, T; L 2 (Ω)) and φ solution of φ t λ φ β φ = ϕ Ω (, T) φ(x, T) = Ω φ(x, t) = (Γ 1 Γ 3 ) (, T) φ (x, t) = ν (Γ 2 Γ 4 ) (, T).

12 Solution of the problem p. 12/6 We have that φ satisfies φ L 2(, T; H 2 (Ω) V ) L (, T; V ), φ t L2(, T; L 2 (Ω) ), T and y is such that φ(b, t) 2 dt C ϕ 2 ( ) L 2,T;L 2 (Ω) v y(v) is continuous from L 2 (, T) L 2(, T; L 2 (Ω) ) and t y(t) is continuous from [, T] H 1 (Ω). The main problem here is that for n = 2, 3, we have y C([, T];H 1 (Ω)), therefore y(, T; v) makes sense only in H 1 (Ω) and not in L 2 (Ω) as required for the cost functional ( see 4a ).

13 Space U The question is as to define the set of admissible controls U, such that v U y(., T; v) L 2 (Ω)? Lions in 21, studies a similar problem to 1d with β = and homogeneous Dirichlet boundary conditions, i.e. this equation y λ y = v(t)δ(x b). t He shows using Fourier transform that in this case is possible to define the set of admissible controls as U = { v L 2 (, T); y(, T; v) L 2 (Ω) } eq.5a and to characterize it of the following form { U = v L 2 (, T); T T (2T t s) n/2 v(t)v(s)dtds < }. eq.5b In our case with β, this is more complicated, due to the term β y y t λ y + β y = v(t)δ(x b). p. 13/6

14 p. 14/6 Space U To overcome this difficulty, we rewrite the system 2d considering y = y 1 + y 2, thus we get and y 1 t λ y 1 = v(t)δ(x b) in Ω (, T) y 1 (x,) = in Ω eq.5c y 1 (x, t) = on Γ 1 (, T) y 1 ν (x, t) = on (Γ 2 Γ 3 Γ 4 ) (, T) y 2 t λ y 2 + β y = in Ω (, T) y 2 (x,) = in Ω eq.5d y 2 (x, t) = on Γ 1 (, T) y 2 ν (x, t) = on (Γ 2 Γ 3 Γ 4 ) (, T).

15 p. 15/6 Space U In the problem 5c, we use the Lions result. In problem 5d, of the theory of parabolic equations, we have that y 2 L 2(, T; V ) C ( [, T];L 2 (Ω) ) and y 2 t L 2(, T; H 1 (Ω) ). Therefore, y C (, T; L 2 (Ω)) and in particular, y(, T) L 2 (Ω). According to this, we can define the space U of admissible controls and its characterization of the same form that Lions as in 5a and 5b respectively. Which with the norms v 2 U = T v 2 dt + Ω y(, T; v) 2 dx v 2 U = T v 2 dt + T T respectively, are Hilbert spaces. The norms given above are equivalents. (2T t s) n/2 v(t)v(s)dtds

16 p. 16/6 Closed convex subset U ad Here, we can consider closed convex subset U ad of U as being, for example: U ad = U or U ad = { v U; v ψ a.e. in (, T) } where ψ is given in H 1 (, T). We introduce the function ψ for generality. eq.6 With the definition of U and its characterization, J is well defined in U and one can show that J is strictly convex, continuous and coercive on U ad. So that, we have the following Theorem: There exists a unique element u U ad that minimizes the functional J, where u is the optimal control. Here, we had solved the problem P1.

17 p. 17/6 Properties of the Space U: (see 22 ) U does not depend on Ω. U does not depend on the boundary conditions. C (, T) U. L 2 (, T)= { v L 2 (, T); v in (T ǫ, T), ǫ } is dense in U, by the regularizing effect. Space D(, T) is dense in U. Therefore, if U denotes the dual space of U, then D(, T) U L 2 (, T) U D (, T).

18 Characterization of the Optimal Control p. 18/6 Theorem: Through the Gâteaux derivative of J, the optimal control u is characterized, v U ad and u U ad, by Ω {y(, T; u) [z d w(, T)]}y(, T; v u)dx+n T u(v u)dt. This estimate is not appropriate to numerical analysis, because does not describe u in explicit form. To get a more adequated characterization of u, we introduce the adjoint operator of y. q λ q β q = in Ω (, T) t q(x, T; u) = y(x, T; u) [ z d w(x, T) ] in Ω eq.8 q(x, t) = on Γ 1 (, T) q(x, t) λ ν eq.7 + q(x, t)(β ν) = on (Γ 2 Γ 3 Γ 4 ) (, T), where q L 2(, T; V ) C ([, T];L 2 (Ω)) and satisfies q C ([, T) Ω); therefore, we can define q(b, t) for t < T.

19 Better characterization p. 19/6 Using adjoint system 8 we arrive at a similar Lions 23 result. Theorem: One has q(b, t) U and condition 7 is equivalent to T ( ) q(b, t)+ Nu (v u)dt, v U ad, u U ad, where the integral above denotes duality between U and U.

20 p. 2/6 Explicit form of u With this better characterization, we get u in explicit form for some convex closed U ad. For example: If U ad = U, the optimal control is given by u = 1 N q(b, t). If U ad = { v U; v ψ a.e. in (, T)}, with ψ given in H 1 (, T), we obtain and u = ψ + u = ( 1 N ( 1 N ) q(b, t) + ψ, as ψ ) q(b, t), as ψ, where f =max{, f}. For applications, the controls must be positive, therefore we consider the closed convex subset U ad given by 6. With the explicit form of u, we already solved the problem P2.

21 Optimality system p. 21/6 The optimal control u that minimizes J, is characterized by the unique solution {u, y, q} of optimality system y t λ y + β y = u(t)δ(x b) in Ω (, T) eq.9a q λ q β q = t in Ω (, T) y(x, ) = in Ω eq.9b q(x, T; u) = y(x, T; u) [ z d w(x, T) ] in Ω y(x, t) = q(x, t) = on Γ 1 (, T) eq.9c y ν (x, t) = on (Γ 2 Γ 3 Γ 4 ) (, T) q(x, t) λ + q(x, t)(β ν) = on (Γ 2 Γ 3 Γ 4 ) (, T) ν ( ) 1 with u(t) = ψ(t) + q(b, t) + ψ(t) = P ( 1N ) N q(b, t), v U ad, u U ad. Where P is the projection operator of L 2 (, T) on U ad.

22 p. 22/6 Remarks We have some problems: The numerical solution is obtained from optimaly system 9a, which is not a Cauchy problem and is coupled. Still, this system depends on solution w of system 3b in the final time. To solve numerically the system 9a, we need to uncouple it and later show the convergence of the uncoupled system to coupled system 9a, That is not simple for n = 2, 3, because δ H 2 (Ω). Moreover, because of the lack of regularity of the delta, we need to regularize the system 9a.

23 Regularizing effect of parabolic equations p. 23/6 For the regularization of system 9b, we introduce a sequence (φ η ) η IN of C (Ω) functions satisfying ( supp{φ η } B, 1 ) η = { x Ω ; x < 1 } η and φ η, Ω φ η (x)dx = 1, η IN. Let denote by (φ η ) η IN the sequence of functions in C (Ω Ω) defined as φ η (x, b) = φ η (x b); then supp{φη } B(b, 1 η ). Consider ε be given with < ε < T. By the regularizing effect of parabolic equations, for any η IN we rewrite the system 9b as the η-ε-approximated problem

24 Regularization for the non-regularized problem p. 24/6 (y ε ) η t λ (y ε ) η +β (y ε ) η =χ [,T ε] (u ε ) η φ η in Ω (, T) (qε ) η λ (q ε ) η β (q ε ) η = t in Ω (, T) eq.1a (y ε ) η (x,) = in Ω (q ε ) η (x, T;(u ε ) η )=(y ε ) η (x, T;(u ε ) η ) [ z d w(x, T) ] in Ω (y ε ) η (x, t) = (q ε ) η (x, t) = on Γ 1 (, T) eq.1b (y ε ) η ν =λ (qε ) η ν (u ε ) η (t) = (ψ ε ) η + +(q ε ) η (β ν)= on (Γ 2 Γ 3 Γ 4 ) (, T) ( ) 1 (q ε ) η φ η dx+(ψ ε ) η = P N Ω ( 1 N Ω ) (q ε ) η φ η dx (y ε ) η (x,t;(u ε ) η ) ε yη (x,t;u η ) η y(x,t;u) strong in H1 (Ω) V and φ η δ(x b) strong in H 2 (Ω), as η. Note that this system still is uncouple.

25 Convergence of the non-regularized problem p. 25/6 The system 1a is important because it implies in Theorem: The solution of system 1a satisfies (y ε ) η (., T),(q ε ) η (., T) H 1 (Ω), provided that z d H 1 (Ω). We have the following: Theorem For z d H 1 (Ω), the solution {(u ε ) η, (y ε ) η, (q ε ) η } of system 1a converge to solution of system 9c, as ε and η.

26 Uncoupling of the system 1b (we omit index ε and η) Let n IN, then the uncoupled system 1b is given by y n t λ y n + β y n = χ [,T ε] [ψ + q n 1 t λ q n 1 β q n 1 = y n (x,) = ( 1 N q n 1 (x, T) = y n 1 (x, T; u) [ z d w(x, T) ] Ω ) ] φq n 1 dx + ψ φ eq.11a y n (x, t) = q n 1 (x, t) = eq.11b y n (x, t) = λ qn 1 ν ν y (x, T) = given. + q n 1 (β ν) = eq.11c Theorem: Consider z d H 1 (Ω) and N large enough. Then, the solution of this system converges to the solution of system 1b, provided that n. p. 26/6

27 Remark: Multiple pollutant sources If we consider m pollutant sources, 1 is given by z m t λ z + β z = j=1 v j (t)δ(x b j ) in Ω (, T). The control is given now by vector v = (v 1,..., v m ) (L 2 (, T)) m and points b 1,..., b m Ω define the m distinct places of pollution entrance. Cost functional is given by J(v) = Ω ] [z 2 y(x, T; v) d w(x, T) dx + where N j > are given constants. m T N j j=1 v j 2 dt, The optimal control is characterized by m j=1 T (q(b j, t)+ N j u j ) (v j u j )dt, v j U ad, u j U ad. p. 27/6

28 Remark: Multiple pollutant sources p. 28/6 Therefore, the optimality system is y m t λ y + β y = j=1 [ ψ j + ( 1 N j q(b j, t) + ψ j ) ] δ(x b j ) q λ q β q = t y(x,) = ; q(x, T; u) = y(x, T; u) [ z d w(x, T) ] y(x, t) = q(x, t) = y t) (x, t) = λ q(x, ν ν + q(x, t)(β ν) =. Regularization, uncoupling and convergence follow of similar form what it was made before.

29 p. 29/6 Remark: Varying point x = b of pollutant entrance Varying point x = b of pollutant entrance, 2 not change, and for each point b, among optimal controls u (that depend on point b), exists only one optimal control u b U ad, such that J ( u b ) = inf { J ( u ) ; u Uad }. This means that in Ω, apart from optimal control, there exists only one point b o (called strategic point or optimal point) where pollution is even less harmful to the environment.

30 p. 3/6 The numerical solution of 11a So that the numerical problem represents a practical situation and makes physical sense, we take f = in boundaries Γ 2 Γ 3 Γ 4 3. Thus, in 3c, w. First, we analyse cases of only one pollutant source and we vary the place of pollution entrance. Then, we analyse cases of multiple pollutant sources. In all the cases we consider either water without movement or with movement. We use a combination of spatial discretization by stabilized semidiscrete finite element method and time discretization by Euler implicit method of finite differences, as in 24.

31 p. 31/6 The numerical solution of 11b Acceptable level z d of mercury (Hg) in drinking water is z d = sen(πx/l) sen(πy/l) mg/cm 3 (see ). Diffusion coefficient of Hg is λ = cm 2 /s. Velocity field β = 1 4 ( 1 (( y (L/2) ) /(L/2) ) 2, ) cm/s. Cost constant N = Tinal time T = 864 s. We use implemented code in Fortran 9 and graphics are presented in Maple. Without loss of generality, we consider ψ in 11b.

32 Computational domain p. 32/6 We consider a square domain Ω = (, L) (, L), where L = 1 cm and point (,) corresponds to boundaries Γ 1 Γ 2. We use a mesh of 1 equal-length quadratic elements with x = y = 1 cm. 4 steps in time with t = 216 s fixed. Γ4 β b Γ 1 Γ 3 ω F = v(t)φ l (x) Ω Γ 2

33 p. 33/6 Sequence of functions The sequence of functions φ l, of compact support in ω Ω 2, is chosen as 1 l se x ω φ l (x) = 2 se x / ω where ω a square of side l centered in point b Ω, represents the place of pollutant entrance. We plot the cases of internal pollutant source F(x, t) = v(t)φ l (x). Letting l to get graphics for pointwise source F(x, t) = v(t)δ(x b).

34 p. 34/6 One source and β = (, )cm/s 2e 11 2e 6 1.8e 11 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) x (cm) y (cm) 8 1 Figure 1: Control u(t) Concentration y, ω =(46, 54) (46, 54)

35 p. 35/6 One source and β = (, )cm/s 2e 11 2e 6 1.8e 11 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) x (cm) y (cm) 8 1 Figure 2: Control u(t) Concentration y, ω =(49, 51) (49, 51)

36 p. 36/6 One source and β = (, )cm/s eq.fig 3 2e 11 2e 6 1.8e 11 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) 2 4 x (cm) y (cm) 8 1 Figure 3: Control u(t) Concentration y, ω=(49.9, 5.1) (49.9, 5.1)

37 Comments: p. 37/6 In this case (water without movement), we note that initially we can pour certain amount of pollution, such that in elapsing time, this amount diminishes so that the pollution concentration in the final time y(x, T; u(t)) does not exceed the allowed value z d. We observe that the control is decreasing and, through the graphics of the concentration, we see that as the square region ω diminishes, that is, as l, we move from located source profile F(x, t) = u(t)φ l (x) to pointwise source profile F(x, t) = u(t)δ(x b), characterizing the numerical convergence of the sequence φ l (x) for δ(x b).

38 p. 38/6 One source and β = (, )cm/s 2e 11 2e 6 1.8e 11 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) x (cm) y (cm) 8 1 Figure 4: Control u(t) Concentration y, ω=(79.9, 8.1) (69.9, 7.1)

39 p. 39/6 One source and β = (, )cm/s 2e 11 2e 6 1.8e 11 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) x (cm) y (cm) 8 1 Figure 5: Control u(t) Concentration y, ω=(19.9, 2.1) (14.9, 15.1)

40 Comments: Note that in graphics (3), (4) and (5), we choose the region ω of pollution entrance, with same area.2.2 cm 2, in different positions in domain Ω. p. 4/6 We observe that the poured amount of pollution depends on the localization of the source. We noted that in the central region of the domain, the pollutant has more freedom of spreading, then a greater amount of pollution can be poured in this place and consequently the pollution concentration at the final time reaches a greater index, close to z d. Now, as the pollutant source moves from the central region, the poured amount of pollution diminishes and therefore the pollution concentration also diminishes. Here also as time passes, the amount of pollution must diminish, so that the pollution concentration in the final time y(x, T; u(t)) does not exceed the value of z d. Finally, we observe that the control is also decreasing.

41 p. 41/6 One source, β (, )cm/s (water in movement) 2e 11 2e 6 1.8e 11 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) x (cm) y (cm) 8 1 Figure 6: Control u(t) Concentration y, ω =(38, 42) (48, 52)

42 p. 42/6 One source, β (, )cm/s (water in movement) eq.fig 7 2e 11 2e 6 1.5e 6 uo (mg/cm^3) 1.5e 11 1e 11 Y (mg/cm^3) 1e 6 5e 7 5e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) 2 4 x (cm) y (cm) 8 1 Figure 7: Control u(t) Concentration y, ω=(39.9, 4.1) (49.9, 5.1)

43 Comments: p. 43/6 Here, we see that the amount of pollution starts being poured in a defined level, is increased and later diminished in elapsing time. In relations to pollution concentration, we note similar behavior to the one of water without movement, when we diminish square region ω.

44 p. 44/6 Three sources and β = (, )cm/s 2e 11 2e 6 1.8e 11 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) x (cm) y (cm) 8 1 Figure 8:Control u Concentration y : u 1 region ω=(19.9, 2.1) (14.9, 15.1) : u 3 region ω=(79.9, 8.1) (69.9, 7.1) +++++: u 2 region ω=(49.9, 5.1) (49.9, 5.1)

45 Three sources, β (, )cm/s (water in movement) p. 45/6 2e e 11 2e 6 uo (mg/cm^3) 1.6e e e 11 1e 11 8e 12 6e 12 Y (mg/cm^3) 1.5e 6 1e 6 5e 7 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (seconds) x (cm) y (cm) 8 1 Figure 9:Control u Concentration y : u 1 region ω=(15.9, 16.1) (25.9, 26.1) : u 3 region ω=(74.9, 75.1) (85.9, 86.1) +++++: u 2 region ω=(42.9, 43.1) (55.9, 56.1)

46 Comments: p. 46/6 We observe that either the control or the pollution concentration behave similarly to the case of only one source. Note that in figure (8), on purpose we choose the same places of pollution entrance of figures (3), (4) and (5). We made repeated simulations varying the parameters of the model, and in all the cases we evidence the convergence of the method.

47 p. 47/6 Arbitrary Control: v = t/864 mg/cm 3. 2e e e e e 11 v (mg/cm^3) 1e 11 8e 12 6e 12 4e 12 2e 12 2e+6 4e+6 6e+6 8e+6 t (segundos) fig 3 fig 7

48 p. 48/6 One source, β = (, ) ω =(49.9, 5.1) (49.9, 5.1) Concentration y 2e 6 3.5e 6 3e 6 1.5e 6 2.5e 6 Y (mg/cm^3) 1e 6 5e 7 Y (mg/cm^3) 2e 6 1.5e 6 1e 6 5e x (cm) y (cm) x (cm) y (cm) 8 1 Figure 1: To Optimal Control J(u o ) = To Arbitrary Control J(v) =

49 p. 49/6 One source, β (, ) ω =(39.9, 4.1) (49.9, 5.1) Concentration y 2e 6 2e 6 1.5e 6 1.5e 6 Y (mg/cm^3) 1e 6 Y (mg/cm^3) 1e 6 5e 7 5e x (cm) y (cm) x (cm) y (cm) Figure 11: To Optimal Control J(u o ) = To Arbitrary Control J(v) =

50 Comments: p. 5/6 Note that in both graphics, we choose the same places of pollution entrance. We observe that the value of J is lesser when the control is optimal, what in fact it would have to occur. Comparing these graphics of the pollution concentration, we observe that the concentration reaches higher indexes, when we use an arbitrary control. What, also, it is in accordance with our theory. We made repeated simulations varying the parameters of the model, and in all the cases we evidence the convergence of the method.

51 p. 51/6 References RR 1 Lions, J. L., Function Spaces and Optimal Control of Distributed Systems. Rio de Janeiro, 198. L.2 2 Lions, J. L., Optimal Control of Systems Gouverned by Partial Differential Equations, Springer-Verlag, New York, L.21 3 Lions, J. L., Some Methode in the Mathematical Analysis of Systems and their Control, Science Press, Beijing, L.22 4 Lions, J. L., Some Aspects of the Optimal Control of Distributed Parameter Systems. Université de Paris and I.R.I.A., Regional Conference Series in Applied Mathematics, 1972, pp. 92. L.23 5 T. J. R. Hughes, The finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, INC. Mineola, New York, 2. H.24

52 References p. 52/6 6 CONAMA, Conselho Nacional do Meio Ambiente, D. O. da União-3/7/1986, Res. N. 2 de 18 de junho, 1-2, C.25 7 Pichard, A., Mercury and its Derivatives, Institut National de L Environnement Industriel et des Risques - INERIS, Version No. 1, 2, P.26 8 Banks. H. Thomas, Control and estimation in Distributed Parameter System. Masson, Paris, K. W. Morton, Numerical Solution of Convection - Diffusion Problems. Oxford, UK, P. Neittaanmaki, D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications

53 p. 53/6 THANK YOU

54 Variational formulation p. 54/6 The variational formulation of 11c, consists of finding such that y n, q n V = { v H 1 (Ω) ; v = on Γ 1 }, ( y n t, v) + λ ( y n, v ) + (β y n, v) = (F n 1, v) eq.12 ( q n 1 t, v ) + λ ( q n 1, v ) (β q n 1, v) = eq.12 v V. With the initial, final conditions and final datum" y n (x,)=, q n 1 (x, T)=y n 1 (x, T; u) z d, y (, T; u)= given in H 1 (Ω) where F n 1 = [ ( 1 ψ+ N Ω ) ] q n 1 φdx+ψ φ.

55 Finite dimensional subspace p. 55/6 The approximated problems, semidiscrete and completely discretized by finite elements are defined in usual way. We consider V h (Ω) C (Ω) the finite dimensional subspace given by V h = { v h V ; v e h P 1 (Ω e ) }, where v e h is the restriction of v h to element e, and P 1 (Ω e ) is the set of linear polynomials defined in Ω e. We will apply the formulation of Streamline-Upwind/Petrov-Galerkin - SUPG in this subspace.

56 Time discretization p. 56/6 For time discretization, we use the Euler implicit method of finite difference, to approximate terms y t and q t. We divide interval [, T] into subintervals [t j 1, t j ], where t j = j T, j = 1,...,k with t = and t k = T. The approximations of terms y t and q t are given by y(x, t j ) t = y j y j 1 t and q(x, t j ) t = q j+1 q j t.

57 Completely discretized problems 12 consist of: Given j = 1,..., k, find y n h,j, qn h,j V h V, such that ( y n ) ) ) h,j t, v N e h +λ ( yh,j, n v h + (β yh,j, n v h + = ( F n 1 h,j + yn h,j 1 t, v h ) + ( q n 1 ) ( ) h,j t, v h + λ q n 1 h,j, v h N e ( q n 1 h,j t λ qn 1 h,j β qn 1 h,j e=1 Ω e N e e=1 e=1 Ω e Ω e ( F n 1 h,j ( ) β q n 1 h,j, v h ( y n ) h,j t λ yn h,j+β yh,j n τβ v h dω e + yn h,j 1 t ) ( q n 1 τβ v h dω e h,j+1 = t ) τβ v h dω e N e, v h ) e=1 Ω e q n 1 h,j+1 t τβ v h dω e v h V h. With the initial, final conditions and final datum" y n h, (x) =, q n 1 h, k (x) = yn 1 h, k (x; u) z d, yh, k( ) = given in H 1 (Ω) where parameter τ is given by τ = h 2 β. p. 57/6

58 p. 58/6 Resolution algorithm: Given y h, k. For n = 1,...,N p. q n 1 h, k = yn 1 ( q n 1 h, j, v h h, k z d. For j = k 1,...,, find qh, n j V h, such that ) ( ) ( ) + λ t q n 1 h, j, v h t β q n 1 h, j, v h N e ( ) q n 1 h, j λ t qn 1 h, j t β qn 1 h, j τβ v h dω e e=1 Ω e = ( ) q n 1 h, j+1, v h N e e=1 yh, =. For j = 1,...,k, find yn h, j V h, such that ( ) ( ) ( ) yh, n j, v h + λ t yh, n j, v h + t β yh, n j, v h N e ( + yh, n j λ t yh, n j + t β yh, n j e=1 Ω e ( ) = t F n 1 h, j + yh, n j 1, v N e ( h + e=1 Ω [ e ( v h V h, where F n 1 h, j = ψ j + Ω e q n 1 h, j+1 τβ v h dω e, v h V h. ) τβ v h dω e ) t F n 1 h, j + yh, n j 1 τβ v h dω e, 1 N Ω qn 1 h, j φ hdx + ψ j ) ]φ h.

59 p. 6/6 One source, β = (, ) ω =(49.9, 5.1) (49.9, 5.1) Concentration y Figure 13: To Optimal Control J(u o ) = To Arbitrary Control J(v) =