Optimal Control for a Second Grade Fluid System

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1 Optimal Control for a Second Grade Fluid System Conference, Chile-Euskadi Luis Friz Facultad de Ciencias, Departamento de Ciencias Básicas, Universidad del Bío-Bío, Chile Partially supported by Grant FONDECYT December 2014

2 Indice Introduction 1 Introduction 2 3 4

3 We consider the system of second grade fluid: (u α u) ν u + curl(u α u) u + q = f + v, Ω ]0, T[ t divu = 0, Ω ]0, T[ u = 0, Ω ]0, T[, u(0) = u 0, Ω Where u is the velocity, f and v are external forces.

4 The problem is to find an externel force v such that u minimize the functional T T J(u, v) = u u 1 2 dx + v 2 dxdt 0 ω 0 Ω here u 1 defined in ω Ω is a given velocity.

5 Indeed the system of second grade fluid: (u α u) ν u + curl(u α u) u + q = f, Ω ]0, T[, t divu = 0, Ω ]0, T[ u = 0, Ω ]0, T[, u(0) = u 0, Ω Where u is the velocity, f is the external forces and q is a kind of pressure.

6 The study of such fluids was initiated by Dunn and Fosdick, and [1] Fosdick and Rajapogal [2] in order to model, for example, ceramics in a fluid state. The first successful mathematical analysis was found by Cioranescu and Ouazar[1], and subsequently Cioranescu and Girault [3]. This work follows the ideas given by Boldrini, Fernández-Cara and Rojas-Medar [2].

7 Ω R 3 is a simply-connected bounded domain, with boundary Ω of class C 3,1. Define: H = {Ψ L 2 (Ω) : div Ψ = 0, Ψ n = 0 on Ω} V = {v H 1 (Ω) : div v = 0, v = 0, on Ω} H(curl; Ω) = {v L 2 (Ω) : curl v L 2 (Ω)}

8 Ω R 3 is a simply-connected bounded domain, with boundary Ω of class C 3,1. Define: H = {Ψ L 2 (Ω) : div Ψ = 0, Ψ n = 0 on Ω} V = {v H 1 (Ω) : div v = 0, v = 0, on Ω} H(curl; Ω) = {v L 2 (Ω) : curl v L 2 (Ω)}

9 For α R +, we introduce the space: V 2 = { v V : curl (v α v) L 2 (Ω) } Equipped with the norm: u V2 = u + α u + curl (u α u)

10 For α R +, we introduce the space: V 2 = { v V : curl (v α v) L 2 (Ω) } Equipped with the norm: u V2 = u + α u + curl (u α u)

11 The variational formulation is: Given f + v L 2 (0, T; H(curl; Ω)) L (0, T; L 2 (Ω)) and u 0 V 2 to find u L (0, T; V 2 ) (u, w) + α( u, w) + ν( u, w) + u(0) = u 0 for every w V. + b(u; u, w) αb(u; u, w) + αb(w, u, u) = (f + v, w), where b(u; v, w) = 3 v j u i w j dx. Ω x i i,j=1

12 The variational formulation is: Given f + v L 2 (0, T; H(curl; Ω)) L (0, T; L 2 (Ω)) and u 0 V 2 to find u L (0, T; V 2 ) (u, w) + α( u, w) + ν( u, w) + u(0) = u 0 for every w V. + b(u; u, w) αb(u; u, w) + αb(w, u, u) = (f + v, w), where b(u; v, w) = 3 v j u i w j dx. Ω x i i,j=1

13 Theorem. There exists a constant δ > 0 such that if the following inequalities hold: u 0 2 L 2 (Ω) + α u 0 2 H 1 (Ω) < δ u 0 2 V 2 < δ, ( ) 1/2 ( v(t) 2 L 2 (Ω) + curl(v(t)) 2 L 2 (Ω) ) < δ, 0 ( ) 1/2 ( f(t) 2 L 2 (Ω) + curlf(t) 2 L 2 (Ω) ) < δ, 0 then, the above variational formulation has a unique solution for each t 0. Moreover u L (R +, V 2 ), u L (R +, V)

14 Define: W 1 = {w L (0, T; V 2 ) : w L (0, T; V)}, W 2 = L 2 (0, T, H(curl; Ω)) L (0, T, L 2 (Ω)), W = W 1 W 2. And the admissible controls: ( ) 1/2 U = v W 2 : ( v(t) 2 L 2 (Ω) + curl(v(t)) 2 L 2 (Ω) ) < δ 0

15 Define: W 1 = {w L (0, T; V 2 ) : w L (0, T; V)}, W 2 = L 2 (0, T, H(curl; Ω)) L (0, T, L 2 (Ω)), W = W 1 W 2. And the admissible controls: ( ) 1/2 U = v W 2 : ( v(t) 2 L 2 (Ω) + curl(v(t)) 2 L 2 (Ω) ) < δ 0

16 If we define: M(w, v) = (ψ 1, ψ 2 ) by t (w αaw) νaw + P(curl(w α w) w) f v = ψ 1 w(0) u 0 = ψ 2, (1) where P : L 2 (Ω) V(Ω) is the orthogonal projection and A is the Stokes operator.

17 The optimal control problem is the following: Find u and v such that J(u, v) = inf J(w, ṽ) (w,ṽ) G Where G is the non-empty set: G = {(w, ṽ) W : ṽ U, M(w, ṽ) = 0}.

18 The optimal control problem is the following: Find u and v such that J(u, v) = inf J(w, ṽ) (w,ṽ) G Where G is the non-empty set: G = {(w, ṽ) W : ṽ U, M(w, ṽ) = 0}.

19 Let us see that there exists at least one solution of this problem. Let (w n, ṽ n ) n 1 be a minimizing sequence in G, i.e., lim J(w n, ṽ n ) = inf J(w, ṽ). n (w,ṽ) G

20 Let us see that there exists at least one solution of this problem. Let (w n, ṽ n ) n 1 be a minimizing sequence in G, i.e., lim J(w n, ṽ n ) = inf J(w, ṽ). n (w,ṽ) G

21 Since (w n, ṽ n ) n 1 is uniformly bounded, there exists a subsequence (w nk, ṽ nk ) k 1 that converge to (u, v). By convexity of J, we have that lim inf k J(w n k, ṽ nk ) J(u, v).

22 Since (w n, ṽ n ) n 1 is uniformly bounded, there exists a subsequence (w nk, ṽ nk ) k 1 that converge to (u, v). By convexity of J, we have that lim inf k J(w n k, ṽ nk ) J(u, v).

23 Since M(w nk, ṽ nk ) = 0 for every k N, we have that M(u, v) = 0, then (u, v) G. This conclude the proof.

24 Since M(w nk, ṽ nk ) = 0 for every k N, we have that M(u, v) = 0, then (u, v) G. This conclude the proof.

25 A set K is a cone with vertex a if for every λ > 0, λ(k a) K a Figure, cone with vertex (0, 0): y xx Figure:

26 Consider E X and F X. We define: E = {f X : f(x) 0, x E} F = {x X : f(x) 0, f F} If K X is a closed convex cone, then (K ) = K.

27 Consider E X and F X. We define: E = {f X : f(x) 0, x E} F = {x X : f(x) 0, f F} If K X is a closed convex cone, then (K ) = K.

28 (Dubovitskii-Milyutin Formalism) Let K 1, K 2,..., K n, K n+1 be convex cones in the real normed space X with K 1, K 2,..., K n open. Then n+1 i=1 K i = if and only if there exists f i K, not i all zero, such that f f n + f n+1 = 0.

29 The cone of decreasing directions of J at (u, v) is given by: DC(J; u, v) = {(w, v) W : DJ(u, v)(w, v) < 0} The corresponding dual cone is: [DC(J; u, v)] = { λdj(u, v) : λ 0}

30 The cone of tangent directions of G at (u, v), is: TC(M; u, v) = {(w, v) W : DM(u, v)(w, v) = 0} The cone of feasible directions of U at (u, v) is given by: FC(U; u, v) = {(w, v) W 1 U : ε > 0, such that (u, v) + λ(w, v) W 1 U, λ ]0, ε]}

31 The cone of tangent directions of G at (u, v), is: TC(M; u, v) = {(w, v) W : DM(u, v)(w, v) = 0} The cone of feasible directions of U at (u, v) is given by: FC(U; u, v) = {(w, v) W 1 U : ε > 0, such that (u, v) + λ(w, v) W 1 U, λ ]0, ε]}

32 Or, FC(U; u, v) = W 1 {λ(v v) : v U, λ > 0} And its dual cone: [FC(U; u, v)] = {(0, h) : h W 2 }

33 Or, FC(U; u, v) = W 1 {λ(v v) : v U, λ > 0} And its dual cone: [FC(U; u, v)] = {(0, h) : h W 2 }

34 It is known that DC(J; u, v) TC(M; u, v) FC(U; u, v) = Therefore, there exists f 1 [DC(J; u, v)], f 2 [FC(U; u, v)] y f 3 [TC(M; u, v)] not all zero: f 1 + f 2 + f 3 = 0.

35 It is known that DC(J; u, v) TC(M; u, v) FC(U; u, v) = Therefore, there exists f 1 [DC(J; u, v)], f 2 [FC(U; u, v)] y f 3 [TC(M; u, v)] not all zero: f 1 + f 2 + f 3 = 0.

36 Let us recall that M(u, v) = (M 1 (u, v), M 2 (u, v)), define: M 1 (u, v) = (u Au) νau + P(curl(u u) u) f v t M 2 (u, v) = u(0) u 0 And its derivative: DM 1 (u, v)(w, v) = (w Aw) νaw + P(curl(u u) w t + curl(w w) u) v DM 2 (u, v)(w, v) = w(0)

37 Let us recall that M(u, v) = (M 1 (u, v), M 2 (u, v)), define: M 1 (u, v) = (u Au) νau + P(curl(u u) u) f v t M 2 (u, v) = u(0) u 0 And its derivative: DM 1 (u, v)(w, v) = (w Aw) νaw + P(curl(u u) w t + curl(w w) u) v DM 2 (u, v)(w, v) = w(0)

38 Thus, DM(u, v)(w, v) = 0. It is equivalent to: (w Aw) νaw+ t +P(curl(u u) w + curl(w w) u) = v w(0) = 0

39 Thus, DM(u, v)(w, v) = 0. It is equivalent to: (w Aw) νaw+ t +P(curl(u u) w + curl(w w) u) = v w(0) = 0

40 Therefore, DM(u, v)(w, v) = 0, implies that f 3 (w, v) = 0. Then, (f 1 + f 2 )(w, v) = 0

41 Now f 1 (w, v) = λdj(u, v)(w, v), for some λ 0. But, DJ(u, v)(w, v) = T 0 ω T (u u 1 )wdxdt + 0 v vdxdt Ω

42 Now f 1 (w, v) = λdj(u, v)(w, v), for some λ 0. But, DJ(u, v)(w, v) = T 0 ω T (u u 1 )wdxdt + 0 v vdxdt Ω

43 Now, f 2 (w, v) = h( v). and therefore, by assuming λ = 1: h( v) = T 0 ω T (u u 1 )wdxdt + 0 v vdxdt Ω

44 Now, f 2 (w, v) = h( v). and therefore, by assuming λ = 1: h( v) = T 0 ω T (u u 1 )wdxdt + 0 v vdxdt Ω

45 There exists ξ such that T 0 Ω T ξ vdxdt = (u u 1 )wdxdt 0 ω It follows from the theorem of Riesz. With the right-hand side of the above equality.

46 There exists ξ such that T 0 Ω T ξ vdxdt = (u u 1 )wdxdt 0 ω It follows from the theorem of Riesz. With the right-hand side of the above equality.

47 It is solution of the adjoint problem: (ξ αaξ) νaξ P(curl(u α u) ξ t +curl[(u ξ) α (u ξ)]) +P(χ ω (u u 1 )) = 0, in Ω ]0, T[ ξ = 0, on Ω ]0, T[ ξ(t) = 0, in Ω

48 So: h( v) = T 0 Ω ( ξ + v) vdxdt. Finally: T 0 Ω ( ξ + v)( v v)dxdt 0.

49 So: h( v) = T 0 Ω ( ξ + v) vdxdt. Finally: T 0 Ω ( ξ + v)( v v)dxdt 0.

50 J. E. Dunn and R.L. Fosdick Thermodynamics, stability and boundedness of fluids of complexity two and fluids of second grade. Arch. Rat. Mech. Anal. 56, 191 (1974). J. L. Boldrini, E. Fernández-Cara and M.A. Rojas-Medar, An optimal control problem for a generalized Boussinesq model: the time dependent case, Rev. Mat. Complut. 20 (2007), no. 2, D. Cioranescu and V. Girault, Weak and classical solutions of a family of second grade fluids, Int. J. Non-Linear Mechanics 32 (1997),

51 D. Cioranescu and E. H. Ouazar, Existence and uniqueness for fluids of second grade. In Nonlinear Partial Differential Equations 109, Collége de France Seminar, Pitman (1984). R.L. Fosdick and K. R. Rajapogal, Anomalous features in the model of second grade fluids. Arch. Rat. Mech. Anal. 70, 1 (1979).

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