Optimal control and homogenization in a mixture of fluids separated by a rapidly oscillating interface

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1 Electronic Journal of Differential Equations, Vol. 00(00), No. 7, pp. 3. ISSN: URL: or ftp ejde.math.swt.edu (login: ftp) Optimal control and homogenization in a mixture of fluids separated by a rapidly oscillating interface Hakima Zoubairi Abstract We study the limiting behaviour of the solution to optimal control problems in a mathematical mixture of two homogeneous viscous fluids. These fluids are separated by a rapidly oscillating periodic interface with constant amplitude. We show that the limit of the optimal control is the optimal control for the limiting problem Introduction The aim of this paper is to study the optimal control problem in a mixture of fluids. More precisely, we consider a mixture of two viscous, homogeneous and incompressible fluids occupying sub-domains of a bounded domain R n+ (n = or ). These fluids are separated by a given interface whose form is determined using a rapidly oscillating function of period ε > 0 and constant amplitude h > 0. We assume that the velocity and pressure of both fluids satisfy the Stokes equations. On the interface, we assume that the velocity is continuous and that the normal forces that the fluids exert each other are equal in magnitude and opposite in direction (hence, surface tension effects are neglected). We associate an optimal control problem to these equations and our aim is to study the limiting behaviour of the solutions when the oscillating period ε tends to 0. To do so, we use some homogenization tools (see Bensoussan-Lions- Papanicolaou 4] and Sanchez-Palencia 5]) and Murat s compactness result ]. This work is based on the mathematical framework of Baffico & Conca 3] for the Stokes problem, of Brizzi 5] for the transmission problem, and of Baffico & Conca, ] for the transmission problem in elasticity. The plan of this paper is as follows. In Section, we present the domain with the rapidly oscillating interface, the Stokes problem posed in this domain Mathematics Subject Classifications: 35B7, 35B37, 49J0, 76D07. Key words: Optimal control, homogenization, Stokes equations. c 00 Southwest Texas State University. Submitted January 3, 00. Published March 5, 00.

2 Optimal control and homogenization EJDE 00/7 and the definition of the associated optimal control problem. In Section 3, we introduce the adjoint problem and present related convergence results. In Section 4, we prove the results announced in the previous section. In Section 5, we sketch the proof of the convergence results concerning the optimal control. In Section 6, we present the case R. Setting of the problem For n = or, let =]0, n and =]0, L i n with L i > 0, i =,..., n. Let h : Ȳ R, h 0 be a smooth function such that i) h = h where h = max{h(y) : y Ȳ } and h > 0. ii) Exists y 0 such that h(y 0 ) = 0 and y h(y 0 ) = 0. For z o R +, define R n+ by = ] z 0, z 0 and Γ the boundary by Γ = { z 0 } ] z 0, z 0 {z 0 }. To define the reference cell, we introduce the sub-domains: which are separated by the interface = {(y, z) R : h(y) < z < z 0 } = {(y, z) R : z 0 < z < h(y)}, Γ = {(y, z) R : h(y) = z}. So that, we have the decomposition of the reference cell Λ (as in figure ) Λ = Γ = ( ] z 0, z 0 ) When we intersect Λ with the hyperplane {Z = z}(0 < z < h ) we obtain {z} and the following decomposition for (as in figure ) where (z) = {y : h(y) > z}, = (z) γ(z) O(z), γ(z) = {y : h(y) = z}. O(z) = {y : h(y) < z}, Let ε > 0 be a small positive parameter. We extend h by -periodicity to R n, we restrict this function to (this function is still denoted by h). Let h ε (x) = h( x ε ) x. Now we introduce ε = {(x, z) R : h ε (x) < z < z 0 }

3 EJDE 00/7 Hakima Zoubairi 3 Figure : The reference cell Λ ε = {(x, z) R : z 0 < z < h ε (x)} and the rapidly oscillating interface is therefore defined by Γ ε = {(x, z) R : h ε (x) = z}. So that, we obtain the following decomposition of (see figure ): = ε Γ ε ε. Finally, as in figure, we set = ]h, z 0, = ]0, h, = ] z 0, 0. We notice that =. The Stokes Problem Let the viscosity of the problem defined by µ ε = µ χ ε + µ χ ε, where µ, µ > 0, µ µ, and χ ε i correspond to the characteristic functions of ε i (i =, ). We denote by v = (v, v n+ ), a vector of R n+. Throughout this paper, C denotes various real positive constants independent of ε. We also denote by the n-dimensional Lebesgue measure and by (e k ) k n the canonical basis of R n (y k is the k-th component of y R n in this basis).

4 4 Optimal control and homogenization EJDE 00/7 Figure : Decomposition of when n = (left) and when n = (rigut) Figure 3: Rapidly oscillating interface (left) and its homogenized version (right)

5 EJDE 00/7 Hakima Zoubairi 5 We define the optimal control as follows. Let U ad L () n+ be a closed non empty convex subset. Let N > 0 be a given constant. For θ U ad the state equation is given by the Stokes problem div(µ ε e( u ε )) = f ε p ε + θ div u ε = 0 in u ε = 0 on. in (.) where ( u ε, p ε ) are respectively the velocity and the pressure of the fluid, θ is the control and f ε is a density of external forces defined by f ε = f χ ε + f χ ε with f i L () n+ (i =, ). The rate-of-strain tensor e( u ε ) is e( u ε ) = ( u ε + t u ε). The cost function is J ε ( θ) = e( u ε ) : e( u ε )dx + N θ dx. (.) The optimal control θ ε is the function in U ad which minimizes J ε ( θ) for θ U ad, in other words J ε ( θ ) ε = min J ε ( θ). (.3) θ U ad This problem is standard and admits an unique optimal solution θ ε U ad (see Lions 0]). Our aim is to study the limiting behaviour of θ ε as ε 0. In particular, it can be shown that (for a subsequence) θ ε θ weakly in L () n+. Our objective is to characterize θ as the optimal control of a similar problem with limiting tensors A and B and to identify these tensors. The homogenization of the problem (.) is thanks to Baffico & Conca 3]. About the optimal control, Saint Jean Paulin & Zoubairi ] studied the problem of a mixture of two fluids periodically distributed one in the other. Also this type of problem has been studied by Kesavan & Vanninathan 9] in the periodic case for a problem which the state equation is a second order elliptic problem with rapidly oscillating coefficients and by Kesavan & Saint Jean Paulin 7] and 8] in the general case (with H-convergence). In this paper, we adapt these methods to the Stokes problem following the technique used by Baffico & Conca ] and 3], by Kesavan & Saint Jean Paulin 7] and 8], and by Saint Jean Paulin & Zoubairi ]. Definition We define L 0() by L 0() = { g L () : g(y)dy = 0 }.

6 6 Optimal control and homogenization EJDE 00/7 The problem (.) (.3) can be reduced to a system of equations by introducing the adjoint state ( v ε, p ε ) H () n+ L 0(). Thus we get div(µ ε e( u ε )) = f ε p ε + θ div(µ ε e( v ε ) e( u ε )) = p ε div u ε = div v ε = 0 in u ε = v ε = 0 on, in in (.4) the optimal control θ ε is characterized by the variationnal inequality θ ε U ad and ( v ε + Nθ ).( ε θ θ )dx ε 0 θ U ad. 3 Convergence results The homogenized adjoint problem Let µ = µ(y, z) (where y and z ]0, h ), be the variable viscosity given by µ(y, z) = µ χ O(z) (y) + µ χ (z)(y). Let us introduce some functions as solution of the Stokes problem defined on. These functions, introduced by Baffico & Conca 3], are associated to the state equation. Let k, l n, and let (χ kl, r kl ) be the solution of div y (µe y ( χ kl + P kl )) = y r kl in div y χ kl = 0 in χ kl, r kl -periodic, (3.) where P kl = ( yk e l +y l e k ). We define M kl the n n matrix by M kl = e y (P kl ). We notice that M kl ] ij = (δ ikδ jl + δ il δ jk ) for all i, j, k, l n. We also define the n + n + matrix E kl ] ij = (δ ikδ jl + δ il δ jk ) i, j, k, l n +. For each z ]0, h, problem (3.) admits an unique solution in ( H ( )n /R ) L 0( ) (see Sanchez-Palencia 5] or Baffico & Conca 6]). Now, we consider the periodic problem div y (µ y ( ϕ k + y k )) = 0 ϕ k -periodic. in (3.) For each z ]0, h fixed, problem (3.) has an unique solution in H ( ) up to an additive constant.

7 EJDE 00/7 Hakima Zoubairi 7 Let A(z) be the fourth-order tensor whose coefficients are defined by µ E kl ] ij h < z < z 0 a ijkl = ã ijkl 0 < z < h i, j, k, l n + (3.3) µ E kl ] ij z 0 < z < h with ã ijkl = µ e y ( χ kl + P kl ) ] ij dy µ ( ϕk +y k ) µ ( ϕl +y l ) µ ( ϕk +y k ) µ ( ϕl +y l ) i, j, k, l n y i dy i, k n j, l = n + y i dy i, l n j, k = n + y j dy j, k n i, l = n + y j dy j, l n i, k = n + µdy i, j, k, l = n + 0 otherwise. (3.4) This tensor (introduced by Baffico & Conca 3]) is the homogenized tensor associated to the state equation. By the same way, we introduce other test functions which will be associated to the adjoint state. Let (ψ kl, r kl ) be the solution of div y (µe y (ψ kl ) + e y ( χ kl + P kl )) = y r kl in div y ψ kl = 0 in ψ kl, r kl -periodic, (3.5) For each z ]0, h, problem (3.5) has an unique solution in ( H ( )n /R ) L 0( ). As we did for the problem (3.), we introduce the scalar problem div y (µ y ψ k + y ( ϕ k + y k )) = 0 ψ k -periodic. in (3.6) For z ]0, h fixed, this problem admits an unique solution in H ( ) up to an additive constant. Let B(z) be the fourth order tensor which coefficients are given by E kl ] ij h < z < z 0 b ijkl = b ijkl 0 < z < h i, j, k, l n + (3.7) E kl ] ij z 0 < z < h

8 8 Optimal control and homogenization EJDE 00/7 with ( µey (ψ kl )] ij + e y ( χ kl + P kl ) )] ij dy i, j, k, l n ( 4 µ ψ k y i + ( ϕk +y k )) y i dy i, k n j, l = n + ( 4 µ ψ l y i + ( ϕl +y l )) y i dy i, l n j, k = n + ( b ijkl = 4 µ ψ k y j + ( ϕk +y k )) y j dy j, k n i, l = n + ( 4 µ ψ l y j + ( ϕl +y l )) y j dy j, l n i, k = n + i, j, k, l = n + 0 otherwise. (3.8) Now we give a result concerning some properties of tensor A. Proposition 3. (Baffico & Conca 3]) The coefficients of A in (3.3) satisfy: a) a ijkl (z) = a klij (z) = a ijlk (z) i, j, k, l n +, z ] z 0, z 0 b) there exists α > 0 such that for all ξ, n + n + symmetric matrix, A(z)ξ : ξ αξ : ξ z ] z 0, z 0. Now we give a result concerning some symmetry and ellipticity properties of the tensor B. Proposition 3. The coefficients of B (voir (3.7)) are such that: a) b ijkl (z) = b klij (z) = b ijlk (z) for i, j, k, l n +, for all z ] z 0, z 0 b) There exists β > 0 such that for all ξ, the n + n + symmetric matrix, B(z)ξ : ξ βξ : ξ z ] z 0, z 0. Proof. Throughout this proof, we adopt the convention of summation over repeated indices. To prove a), we first study the coefficients of tensor B with indexes i, j, k, l n. The symmetry of these coefficients is evident when z ]h, z 0 and z ] z 0, h. Let study the case where z ]0, h. In this case (cf (3.8)) b ijkl = b ( ijkl = µ ey (ψ kl ) ] ij + e y ( χ kl + P kl ) ] ) ij dy. (3.9) Following the ideas in 7,, 4], we transform the above expression to obtain a symmetric form. Let ( kl, r kl 3 ) be the solution of div y (µe y ( kl + P kl )) = y r kl 3 in div y kl = 0 in kl, r kl -periodic, (3.0)

9 EJDE 00/7 Hakima Zoubairi 9 Introducing kl the solution of the previous local problem, the coefficients (3.9) can be rewritten as b ijkl = ey ( kl + P kl ) ] ij dy + ( µ ey (ψ kl ) ] ij e y (χ kl kl ) ] ) ij dy. (3.) The first term of the second integral of the right-hand side of this equation is evaluated as follows (using the fact that e y (ψ kl ) ] ij = e y(ψ kl ) ] ji ) µ e y (ψ kl ) ] ij dy = = = µ e y (ψ kl ) ] βm δ βiδ mj dy µ e y (ψ kl ) ] βm (δ βiδ mj + δ βj δ mi )dy µ e y (ψ kl ) ] βm ey (P ij ) ] βm dy. Using successively (3.),(3.5) and (3.0), we have µ e y (ψ kl ) ] ey (P ij ) ] βm βm dy = µ e y (ψ kl ) ] βm ey (χ ij ) ] dy βm = ey (χ kl P kl ) ] ey (χ ij ) ] βm dy βm = ey (χ kl kl ) ] ey (χ ij ) ] dy. βm βm Moreover using (3.0), we can rewrite the last integral as ey (χ kl kl ) ] ey (χ ij ) ] dy βm βm = ey (χ kl kl ) ] ey (χ ij ij ) ] βm βm dy + ey (χ kl kl ) ] ey ( ij ) ] dy βm βm = ey (χ kl kl ) ] ey (χ ij ij ) ] βm βm dy + ey (χ kl kl ) ] ey (P ij ) ] dy βm βm = ey (χ kl kl ) ] ey (χ ij ij ) ] dy + ey (χ kl kl ) ] dy βm βm ij Thus the second integral of the right-hand side of (3.) can be rewritten as ( µ ey (ψ kl ) ] ij e y (χ kl kl ) ] ij) dy ] = ey (χ kl kl ) ] ey (χ ij ij ) ] βm βm dy.

10 0 Optimal control and homogenization EJDE 00/7 Let us now consider the first integral in (3.). Multiplying the first equation of (3.0) by ij and integrating by parts we have ey ( kl + P kl ) ] ey ( ij ) ] dy = 0. βm βm Using the fact that e y (P ij ) ] βm = ( δβi δ mj + δ βj δ mi ), we obtain = = ey ( kl + P kl ) ] βm ey ( kl + P kl ) ] βm ey ( kl + P kl ) ] ij dy. ey ( ij + P ij ) ] βm dy ey (P ij ) ] βm dy Then using definition (3.), we derive b ijkl = e y ( kl +P kl ) : e y ( ij +P ij )dy+ e y (χ kl kl ) : e y (χ ij ij )dy. (3.) It is immediate from the above form that the coefficients of B satisfy b ijkl = b klij. On the other hand, since e y (P kl ) = e y (P lk ) then by uniqueness of problem (3.), we have χ kl = χ lk (up to an additive constant) and then b ijkl = b ijlk. We now study the coefficients b ijkl with i = k = n+ and j, l n. From the definition of B (cf (3.7)), these coefficients are symmetric when z ]h, z 0 and z ] z 0, h. To prove the symmetry when z ]0, h, we proceed as we did before. These coefficients are as follows b n+jn+l = ( ( ϕ l + y l ) + µ ψl ) dy. 4 y j y j (3.3) Let τ k be the solution of y ( τ k + y k ) = 0 τ k -périodic. in (3.4) The expression (3.3) can be rewritten as follows (using τ l ): b n+jn+l = 4 ( τ l + y l ) dy + ( ψ l µ (ϕl τ l )) dy. (3.5) y j 4 y j y j Using exactly the same technique used above, we obtain (using (3.), (3.6) and (3.4)) µ ψl = y j (ϕ l τ l ) (ϕ j τ j ) dy + y k y k (ϕ l τ l ) dy. y j

11 EJDE 00/7 Hakima Zoubairi Therefore, ( µ ψ l (ϕl τ l )) dy = y j y j (ϕ l τ l ) (ϕ j τ j ) dy. y k y k Concerning the first integral in (3.5), we consider τ l the solution of (3.4): multiplying the first equation by τ j and integrating by parts, we have ( τ l + y l ) τ j dy = 0, y k y k so we derive ( τ l + y l ) ( τ j + y j ) dy = y k y k ( τ l + y l ) dy. y j Finally, we obtain the following expression b n+jn+l = ( τ l +y l ). ( τ j +y j )dy + (ϕ l τ l ). (ϕ j τ j )dy. 8 8 (3.6) From the form of (3.6), it is evident that b n+jn+l = b n+ln+j. By construction of B, we also have b n+ln+j = b n+ljn+. For the other nonzero terms, the same method can be used to obtain b in+kn+ = b in+n+l = b n+jkn+ = b kn+in+ = b in+ln+ = b n+jn+k = b in+n+k b n+lin+ b kn+n+j. To prove part b), we first notice that the coerciveness of B when z ]h, z 0 and z ] z 0, h is evident. When z ]0, h, from the form of b ijkl i, j, k, l n (see (3.)) and the form of b n+jn+l j, l n (see (3.6), we have that B is elliptic. Now we introduce the homogenized problem. ( H () n+ L 0() ) and be the solution of Let ( u, p) and ( v, p ) be in div ( Ae( u) ) = f p + θ div ( Ae( v) Be( u) ) = p div u = div v = 0 in u = v = 0 on, in in (3.7) where f is the weak limit of f ε in L () n+ and we precise it later, (4.0).

12 Optimal control and homogenization EJDE 00/7 Remark 3.3 Since Propositions 3. and 3. hold, problem (3.7) admits an unique solution. Now we state the main result of this paper, and we will prove it in the next section. Theorem 3.3 Under some regularity hypotheses concerning the solutions of (3.), (3.), (3.5), and (3.6) (detailed in Section 4) the solutions ( u ε, p ε ) and ( v ε, p ε ) of (.4) are such that u ε u weakly in H () n+, v ε v weakly in H () n+, p ε p weakly in L 0(), p ε p weakly in L 0(), where ( u, p) and ( v, p ) are the unique solutions of (3.7). 4 Proof of the convergence result A priori estimates Let ξ ε = µ ε e( u ε ), (4.) q ε = µ ε e( v ε ) e( u ε ). (4.) Proposition 4. The sequences ( u ε, p ε ), ( v ε, p ε ), ξ ε and q ε are such that (up to subsequences) u ε u weakly in H () n+ v ε v weakly in H () n+ p ε p weakly in L 0() p ε p weakly in L 0() ξ ε ξ weakly in L () n+ n+ q ε q weakly in L () n+ n+. Proof. Using u ε as a test function in the first equation of (.4), we can easily see that there exists a constant C > 0 independent of ε such that u ε H () n+ C; (4.3) therefore, we have for a subsequence (still denoted by ε) u ε u weakly in H () n+. (4.4) Similarly, multiplying the second equation of (.4) by v ε, integrating by parts and using (4.3), we obtain so we have (for a subsequence) v ε H () n+ C, v ε v weakly in H () n+. (4.5) Now since div (µ ε e( u ε )) H () n+ is bounded, we have pε H () n+ C, this implies (see Temam 6]) p ε L 0 () C, we derive (for a subsequence), p ε p weakly in L 0().

13 EJDE 00/7 Hakima Zoubairi 3 Also by similar arguments, we get provides from (4.3) and we derive by ex- The boundedness of ξ ε L () (n+) traction of subsequences p ε p weakly in L 0(). ξ ε ξ weakly in L () (n+). Similarly, the boundedness of q ε L () (n+) we can extract a subsequence such that provides from (4.3) and (4..6), so q ε q weakly in L () (n+). (4.6) Since ( u ε, p ε ) and ( v ε, p ε ) are solutions of (.4) and since Proposition 4. holds, we obtain that ( u, p), ( v, p ), ξ and q satisfy in the distribution sense div ( ξ ) = f p + θ div ( q ) = p in div u = div v = 0 u = v = 0 on, in in (4.7) where f is the weak limit of f ε in L () n+. This limit can be identified explicitly (cf 3] or 5]). Indeed, the characteristic functions χ ε (i =, ) are such that χ ε i ρ and χ ε ( ρ) weakly in L (), (4.8) where Then we have in O(z) ρ(x, z) = in 0 in. (4.9) f ε f = f ρ + f ( ρ) weakly in L () n+. (4.0) Proposition 4. (Baffico & Conca 3]) Under the hypotheses (4.), (4.5) and (4.8) (introduced in the next subsections), ξ = Ae( u), where A is defined by (3.4). To prove Theorem 3.3, we have to show that q, u and v are related by q = Ae( v) Be( u). (4.) Using the same method as Baffico & Conca3], we show that the identification of q is carried out in, and independently. In and, this identification will pose no particular problem. In, following the ideas of Baffico & Conca (3]), there is three steps: we first identify the components q] ij of q for i, j n, and then q] n+j for j n and finally we identify q] n+n+. To do so, we use some suitable test functions and the energy method ( cf Bensoussan, Lions & Papanicolaou 4] or Sanchez-Palencia 5]).

14 4 Optimal control and homogenization EJDE 00/7 Identification of q] ij i, j n in In what follows, we construct at first the test functions which allow us the identification of q] ij, then we introduce the regularity conditions that these functions must satisfy and finally we establish the identification. Let w kl = χ kl + P kl and σ kl = µe y (w kl ) where (χ kl, r kl ) be the solution of (3.). We assume that (χ kl, r kl ), as function of (y, z) ]0, h, satisfies the regularity hypothesis a) χ kl L loc (0, h, H ( ) n ) (L loc (]0, h R n )) n b) ( (χ kl ) ) i L loc (0, h, L ( ) n ) L loc (]0, h R) i, j n (4.) We define the following functions by extension by -periodicity to R n+ and by restriction to : w ε,kl (x, z) = εw kl ( x ε, z) r ε,kl (x, z) = r kl ( x ε, z) (4.3) It is easy to check that σ ε,kl (x, z) = σ kl ( x ε, z) div x (σ ε,kl ) = x r ε,kl in div x (w ε,kl ) = div x (P kl ) = δ kl in. We also need the Murat s compactness result ]. (4.4) Lemma 4.3 If the sequence (g n ) n belongs to a bounded subset of W,p () for some p >, and (g n ) n 0 in the following sense i.e., for all φ D() such that φ 0 then for all n > 0 g n, φ 0. Then (g n ) n belongs to a compact subset of H (). If we suppose that r ε,kl satisfy a) r ε,kl L p loc () for some p >, locally bounded ( b) r ε,kl) (4.5) 0 in distribution sense, Then using Lemma 4.3 and hypothesis (4.), we have the following result. Proposition 4.4 (Baffico & Conca 3]) If (4.) and (4.5) hold. Then for all, we have the following convergence a) w ε,kl P kl weakly in H ( ) n b) ( (w ε,kl ) i ) 0 strongly in L ( ) n, i n c) r ε,kl 0 weakly in L ( ) n ( d) r ε,kl) 0 strongly in H ( ) n, i n e) σ ε,kl σ kl = m (µe y (w kl ) weakly in L ( ) n n. (4.6)

15 EJDE 00/7 Hakima Zoubairi 5 In the same way, we assume that the solution (ψ kl, r kl ) of (3.5) satisfies the following convergence hypothesis: a) ψ kl L loc (0, h, H ( ) n ) (L loc (]0, h R n )) n b) ( (ψ kl ) ) i L loc (0, h, L ( ) n ) L loc (]0, h R) We define i, j n (4.7) so we obtain ψ ε,kl (x, z) = εψ kl ( x, z), and rε,kl (x, z) = r kl ( x, z) (4.8) ε ε ( div µ ε e x (ψ ε,kl ) + e x (w ε,kl ) ) = x r ε,kl x in (4.9) div ψ ε,kl = 0 in x If we suppose that r ε,kl satisfy a) r ε,kl L p loc () for some p >, locally bounded ( b) r ε,kl) (4.0) 0 in distribution sense, then using Lemma 4.3, we derive ( r ε,kl) 0 strongly in H ( ) n. i n We have the following result concerning these functions Proposition 4.5 If (4.) and (4.5) hold. Then for all, we have a) ψ ε,kl 0 weakly in H ( ) n b) ( (ψ ε,kl ) ) i 0 strongly in L ( ) n, i n c) r ε,kl 0 weakly in L ( ) n. (4.) To prove this proposition, we use well-known results concerning the convergence of periodic functions. We shall prove now the principal result of this section Proposition 4.6 If (4.), (4.5), (4.7), and (4.0) hold, then q ε ] kl q] kl weakly in L ( ) (up to a subsequence) for all k, l n where n i,j= { q] kl = n { µe y ( χ ij + P ij )] kl dy } e( v) ] ij i,j= ( µey ( ψ ij )] kl + e y ( χ ij + P ij )] kl ) dy } e( u) ] ij

16 6 Optimal control and homogenization EJDE 00/7 Proof. Let φ D( ) and w ε,kl = (w ε,kl, 0). Multiplying the second equation of (.4) by φ w ε,kl, integrating by parts and using (4.), we obtain p ε.(φ w ε,kl )dxdz = (q ε φ). w ε,kl dxdz + e( u ε ) : w ε,kl φdxdz m µ ε e( v ε ) : w ε,kl φdxdz. Developing the second and third integral of the right-hand side of the above equation, p ε.φ w ε,kl dxdz = (q ε φ). w ε,kl dxdz + e x (u ε ) : e x (w ε,kl )φdxdz n µ ε e x (v ε ) : e x (w ε,kl )φdxdz q ε ( ] n+j (w ε,kl ) ) j φdxdz. j= (4.) Let ψ ε,kl = (ψ ε,kl, 0). Multiplying the first equation of (.4) by φψ ε,kl, integrating by parts and using definition (4.), we obtain (after algebraic developments) ( f ε p ε + θ).φ ψ ε,kl dxdz = (ξ ε φ). ψ ε,kl + µ ε e x (u ε ) : e x (ψ ε,kl )φdxdz n + ξ ε ( ] n+j (ψ ε,kl ) ) j φdxdz. j= Integrating by parts now the second integral of the right-hand side, we have ( f ε p ε + θ).φ ψ ε,kl dxdz = (ξ ε φ). ψ ε,kl ( div µ ε e x (ψ ε,kl ) ).(u ε φ)dxdz x m n (µ ε e x (ψ ε,kl ) x φ).u ε dxdz + ξ ε ( ] n+j (ψ ε,kl ) ) j φdxdz. j= Using the equation that ψ ε,kl satisfy (see (4.9)), ( f ε p ε + θ).φ ψ ε,kl dxdz

17 EJDE 00/7 Hakima Zoubairi 7 = (ξ ε φ). ψ ε,kl u ε ( div ex (w ε,kl ) ) φdxdz + x r ε,kl x.(φu ε )dxdz m ( µ ε e x (ψ ε,kl ) x φ ) n.u ε dxdz + ξ ε ( ] n+j (ψ ε,kl ) ) j φdxdz. j= Integrating again by parts the second integral, ( f ε + θ p ε ).φψ ε,kl dxdz = (ξ ε φ). ψ ε,kl e x (u ε ) : e x (w ε,kl )φdxdz m + x r ε,kl.(φu ε ( )dxdz µ ε e x (ψ ε,kl ) x φ ).u ε dxdz ( ex (w ε,kl ) x φ ) n.u ε dxdz + ξ ε ( ] n+j (ψ ε,kl ) ) j φdxdz. j= Adding (4.) and the above equation, we obtain ( f ε + θ).φ ψ ε,kl dxdz p ε.φψ ε,kl dxdz p ε.φ w ε,kl dxdz m = (q ε φ). w ε,kl dxdz µ ε e x (v ε ) : e x (w ε,kl )φdxdz n q ε ( ] n+j (w ε,kl ) ) j φdxdz + (ξ j= m ε φ). ψ ε,kl + x r ε,kl.(φu ε )dxdz (b ε,kl x φ).u ε dxdz n + ξ ε ( ] n+j (ψ ε,kl ) ) j φdxdz. j= (4.3) where We obtain easily that (using Problem (3.5)) b ε,kl = µ ε e x (ψ ε,kl ) + e x (w ε,kl ). (4.4) div x (b ε,kl ) = x r ε,kl in. We now pass to the limit in (4.3) as ε tends to 0. In order to do so, we need some preliminaries results. By Definition (4.4) and classical arguments concerning the convergence of periodic functions, we conclude that for all, b ε,kl b kl = m (µe y (ψ kl ) + e y (w kl )) weakly in L ( ) n n and div x (b kl ) = 0 in. (4.5)

18 8 Optimal control and homogenization EJDE 00/7 By the convergence (4.6) a), we have Also by (4.) a), we get w ε,kl P kl = (P kl, 0) strongly in L ( ) n+. ψ ε,kl 0 strongly in L ( ) n+. Now passing to the limit in (4.3) taking into account the precedent convergence results, we obtain p.φp kl dxdz = (q φ). P kl dxdz σ kl : e x (v)φdxdz (b kl x φ).udxdz Integrating by parts the right-hand side of the above expression, using the second equation of (4..4) and the expression (4.5), we obtain 0 = q : e( P kl )φdxdz σ kl : e x (v)φdxdz + e x (u) : b kl φdxdz Since e( P kl ) ] = M kl], then we obtain in the distribution sense ij ij q] kl = n σ kl ] ij ex (v) ] n ij b kl ] ex (u) ]. ij ij i,j= i,j= Now since (4.6) e), (4.5) hold and since e x (v) ] = e( v) ] and e ij ij x (u) ] ] = ij e( u) for all i, j n, we get ij q] kl = n { µe y ( χ kl + P kl )] ij dy } e( v) ] ij i,j= n ( µey ( ψ kl )] ij + e y ( χ kl + P kl ) } ] )] ij dy e( u) i,j= { Also since the following symmetry property holds (see Proposition 3.) µe y ( χ kl + P kl )] ij dy = µe y ( χ ij + P ij )] kl dy, and the following s one holds too (see Proposition 3.) ( µ ey ( ψ kl ) + e y ( χ kl + P kl ) ] ) ij dy ij. (4.6) = ( µ ey ( ψ ij ) + e y ( χ ij + P ij ) ] ) kl dy, (4.7)

19 EJDE 00/7 Hakima Zoubairi 9 we conclude that q] kl = n { µe y ( χ ij + P ij )] kl dy } e( v) ] ij i,j= n { ( µey ( ψ ij )] kl + e y ( χ ij + P ij ) } ] )] kl dy e( u) i,j= This completes the proof. ij Identification of q] n+j, j n in Let ϕ k be the solution of Problem (3.). We assume that ϕ k = ϕ k (y, z) satisfy the regularity hypotheses a) ϕ k L loc (0, h, H ( )) L loc (]0, h R n ) b) ϕ k L loc (0, h, L ( )) L loc (]0, h R n ) (4.8) Let us define ζ k = ϕ k + y k and η k = µ y ζ k. We also define the following functions by -periodicity: ζ ε,k (x, z) = εζ k ( x ε, z), ηε,k (x, z) = η k ( x, z) (4.9) ε It is easy to see that div x η ε,k = 0 in. We introduce a supplementary hypothesis concerning η ε,k : a) { ( η ε,k) } j ε>0 L p loc () for some p >, locally bounded ( b) η ε,k ) (4.30) 0 in the distribution sense. j Then we have the following result. Proposition 4.7 (Baffico & Conca 3]) Assume (4.8) and (4.30). for all, we have a) ζ ε,k y k weakly in H ( ) ( b) ) ζ ε,k 0 strongly in L ( ) c) η ε,k η k = m (η k ) weakly in L ( ) n ( d) ) η ε,k j ( ) η k j strongly in H ( ), j n. Then (4.3) In view of (4.3.5) c) and (4.3.3), we get div x η k = 0 in. Similarly we assume that ψ k, the solution of (3.6), satisfies a) ψ k L loc (0, h, H ( )) L loc (]0, h R n ) b) ψ k L loc (0, h, L ( )) L loc (]0, h R n ) (4.3)

20 0 Optimal control and homogenization EJDE 00/7 Let ψ ε,k (x, z) = ψ k ( x ε, z) and dε,k = µ ε x ψ ε,k + x ζ ε,k, (4.33) then using (3.6), (4.9) and (4.33), we get div x d ε,k = 0 in. (4.34) We assume that d ε,k satisfies the regularity conditions: {( a) d ε,k ) j} ε>0 Lp loc () for some p >, locally bounded ( b) d ε,k ) (4.35) j 0 in the distribution sense. We have the following result. Proposition 4.8 Assume hypotheses (4.3.7) and (4.35) hold., we have Then for all a) ψ ε,k 0 weakly in H ( ) b) ( ψ ε,k ) 0 strongly in L ( ) c) d ε,k d k = m (µ y ψ k + y ζ k ) weakly in L ( ) n d) ( d ε,k ) ( j d k ) j H ( ), j n. (4.36) Remark 4.9 From (4.34) and (4.36) c), we have div x d k = 0 in. Proof of Proposition 4.8 Using classical arguments concerning convergence of periodic functions, we can prove the three first assertions. For the last one, we use the compactness Lemma 4.3 (the hypotheses of this lemma hold since we suppose that (4.35) is satisfied). Proposition 4.0 If (4.8), (4.30), (4.3) and (4.35) hold, then up to a subsequence, we have q ε ] n+k q] n+k weakly in L ( ) j n, where q] n+k = n { i= n i= { µ ( ϕi + y i ) dy } e( v) ] y n+i k ( µ ψ i y k + ( ϕi + y i ) y k dy } e( u) ] n+i. Proof. Let φ D( ) and ζ ε,k = (0, ζ ε,k ). Multiplying the second equation of (.4) by φζ ε,k, integrating by parts and using (4.), we obtain p ε. ζ ε,k φdxdz = (q ε φ). ζ ε,k dxdz + e( u ε ) : ζ ε,k φdxdz

21 EJDE 00/7 Hakima Zoubairi µ ε e( v ε ) : ζ ε,k φdxdz. After some elementary computations on the second and third integral of the right-hand side of the above equation, p ε. ζ ε,k φdxdz = (q ε φ). ζ ε,k dxdz µ ε x vn+. ε x ζ ε,k φdxdz n µ ε vε i ζ ε,k φdxdz + x u ε x i= i n+. x ζ ε,k φdxdz n q ε ζ ε,k ] n+n+ φdxdz + n µ ε uε i ζ ε,k φdxdz. x i j= i= (4.37) Let ψ ε,k = (0, ψ ε,k ). Multiplying the first equation of (.4) by φψ ε,k, integrating by parts and using Definition (4.), we get ( f ε p ε + θ). ψ ε,k φdxdz = (ξ ε φ). ψ ε,k + µ ε e( u ε ) : ψ ε,k φdxdz Rewriting the second integral of the right-hand side differently, the above expression becomes ( f ε p ε + θ). ψ ε,k φdxdz = (ξ ε φ). ψ ε,k + µ ε x u ε n+. x ψ ε,k φdxdz (4.38) + n i= (µ ε uε i ) ψε,k x i φdxdz + i= n ξ ε ψ ε,k ] n+n+ φdxdz. Integrating by parts the second integral of the right-hand side of the above equation and using (4.33) and(4.3.0), we derive µ ε x u ε n+. x ψ ε,k φdxdz = u ε n+ div(µ ε x ψ ε,k )φdxdz µ ε u ε x n+ x φ. x ψ ε,k dxdz m = u ε n+ div( x ζ ε,k )φdxdz µ ε u ε x n+ x φ. x ψ ε,k dxdz m Now, integrating by parts the first integral of the right-hand side of the above equation and using (4.33), we get µ ε x u ε n+. x ψ ε,k φdxdz

22 Optimal control and homogenization EJDE 00/7 = x u ε n+. x ζ ε,k φdxdz Therefore the expression (4.38) becomes ( f ε p ε + θ). ψ ε,k φdxdz = (ξ ε φ). ψ ε,k x u ε n+. x ζ ε,k φdxdz n u ε n+(d ε,k. x φ)dxdz + (µ ε uε i i= n + ξ ε ψ ε,k ] n+n+ φdxdz. i= u ε n+(d ε,k. x φ)dxdz ) ψε,k x i φdxdz Adding (4.37) and the above equation, we have (using Definition (4.33)) ( f ε p ε + θ). ψ ε,k φdxdz p ε. ζ ε,k φdxdz m = (q ε φ). ζ ε,k dxdz η ε,k. x vn+φdxdz ε n µ ε qε i ζ ε,k n φdxdz q ε ζ ε,k ] n+n+ x i= i φdxdz (4.39) j= + n u ε i (dε,k ) i φdxdz + (ξ ε φ). ψ ε,k i= n u ε n+(d ε,k. x φ)dxdz + ξ ε ψ ε,k ] n+n+ φdxdz. i= We now pass to the limit in the above equation as ε 0. To do so, we need some preliminary results. From convergence (4.3.5) a), we have ζ ε,k ζ k = (0, y k ) weakly in L ( ) n+. and from (4.3.) a), we get ψ ε,k 0 weakly in L ( ) n+. Now passing to the limit in (4.3.) taking into account the precedent convergence results, we obtain p. ζ k φdxdz = (q φ). ζ k dxdz η k. x v n+ φdxdz n (η k v i ) i φdxdz u n+ (d k. x φ)dxdz + i= n i= (d k ) i u i φdxdz.

23 EJDE 00/7 Hakima Zoubairi 3 Integrating by parts the above expression, we derive 0 = q : e( n ζ k )φdxdz (η k ) i e( v) ]n+i φdxdz i= n + (d k ) i e( u) ]n+i φdxdz. i= Using the fact that q : e( ζ k ) = q] n+k, we obtain, in the distribution sense, n q] n+k = (η k ] ) i e( v) n+i n (d k ) i e( u) ]n+i. i= Also by Definition (4.3) c) and (4.36) c) of η k and of d k, we have n+k = n { i= n ( ψ k µ i= { Hence, since Proposition 3. holds, µ ( ϕk + y k ) dy = y i and since Proposition 3. holds, ( ψ k µ + ( ϕk + y k )) dy = y i y i i= µ ( ϕk + y k ) dy } e( v) ] y n+i i y i + ( ϕk + y k ) y i ) dy } e( u) ] n+i. (4.40) µ ( ϕi + y i ) y k dy, (4.4) ( µ ψ i y k + ( ϕi + y i ) y k ) dy. (4.4) Finally using (4.4) and (4.4) in (4.40) we obtain the announced result, i.e., q] n+k = n { µ ( ϕi + y i ) dy } e( v) ] i= y n+i k n i= { ( µ ψ i y k + ( ϕi + y i ) y k dy } e( u) ] n+i. Since the matrix q ε is symmetric, this implies that q is symmetric. Hence q] n+k = q] kn+ which completes the proof. Identification of q] n+n+ in The following proposition gives a result concerning the identification of the last component of q. Proposition 4. q ε ] n+n+ q] n+n+ weakly in L ( ) (up to a subsequence), where q] n+n+ = { µdy } e( v) ] e( u) ]. n+n+ n+n+

24 4 Optimal control and homogenization EJDE 00/7 Proof. To prove this result, we proceed as in, ]; We use the method of Brizzi 5]. From (4.) we have q ε ] n+n+ = µ ε vε n+ µ χ ε + µ χ ε, hence uε n+ and µ ε = q ε ] n+n+ = µ P ( (v ε n+) ) + µ P ( (v ε n+) ) P ( (u ε n+) ) ( (u ε P n+) ), where (u ε n+) i = u ε n+ ε i, (vn+) ε i = vn+ ε ε i (i =, ) and P i represent the extension by 0 in \ ε i (i =, ). ( (u ε ) ( n+ It is easy to see that P )i (v ε ) i and n+ )i Pi are bounded in L ( ). Therefore (up to a subsequence), there exists ν i and γ i L ( ) such that ( (u ε P n+) i ) i νi weakly in L ( ), (4.43) ( (v P n+) ε i ) i γi weakly in L ( ). (4.44) We now proceed to identify these limits. Concerning ν i, we have ν = ρ(x, z) u n+, ν = ( ρ(x, z)) u n+ (4.45) where ρ is defined by (4.9) (see Baffico & Conca 3]). In the same way, we can find explicitly γ i : Let φ D(), then H ( ) (χ ε m), φvε n+ H 0 ( ) = χ ε v ε φ n+ dxdz ( (v P n+) ε i ) i φdxdz. (4.46) Using Lemma 4.3 for the sequence { (χ ε m)}, it is shown that this ε>0 sequence satisfies the hypotheses required, 5]. Using (4.5), we can pass to the limit in the left-hand side of (4.46). For the right-hand side, from (4.4), (4.8) and (4.44), in the limit, we have H ( ) ( O(z) ), φvn+ H 0 ( ) = ( O(z) ) φ vn+ dxdz γ φdxdz and developing the duality product in the last equation, we obtain in the distribution sense the identity γ = ρ(x, z) v n+. (4.47)

25 EJDE 00/7 Hakima Zoubairi 5 In the same way, passing to the limit in (χ ε m), φvε n+ and using again the compactness Lemma, we get γ = ( ρ(x, z)) v n+. (4.48) From (4.43) and (4.44), we have q] n+n+ = µ γ + µ γ ν ν. Hence using (4.4.5), (4.47) and (4.48), we conclude q] n+n+ = ( µ ρ(x, z) + µ ( ρ(x, z) )) v n+ u n+, which gives the announced result, that is q] n+n+ = { µdy } e( v) ] n+n+ e( u) ] n+n+. Hence Proposition 4. is proved. Identification of q in and Proposition 4. For i =, q ε i q i weakly in L ( ) n+ (up to a subsequence), where q i = µ i e( v i ) e( u i ). Proof. From (4.), we have q ε i = µ i e( v ε i ) e( u ε i ). From the convergence (4.4), (4.5), and (4.6), we easily obtain the result of Proposition 4.. Conclusion From Propositions 4.6, 4.0, 4., and 4., from the definition of tensors A and B, we conclude that q, u, and v are related by (4.). Therefore, since q satisfies (4.7) by Proposition 4., we have that ( u, p) and ( v, p) are solutions of (3.6). From the properties of the tensors A and B, problem (3.6) admits a unique solution and hence, the whole sequence u ε and v ε converge weakly to u and v respectively. This completes the proof of Theorem Optimal control The following theorem gives a convergence result of the optimal control. Theorem 5. For θ fixed in U ad, we consider ( u, p) as the solution of div(ae( u)) = f p + θ div u = 0 in u = 0 on. in (5.)

26 6 Optimal control and homogenization EJDE 00/7 The cost function is J 0 ( θ) = Be( u) : e( u)dxdz + N θ dxdz. Then the optimal control θ ε of problem (.) (.3) satisfies θ ε θ strongly in L () n+ and θ satisfies the optimality condition J 0 ( θ ) = min θ U ad J 0 ( θ), (5.) Furthermore there is convergence of the minimal cost, i.e., lim J ε( θ ) ε = J 0 ( θ ). (5.3) ε 0 Proof. Step : A priori estimates By the optimal control definition, we have for all θ in U ad N θ ε L () n+ J ε( θ ε ) J ε ( θ) C, so θ ε is bounded in L () n and is such that (for a subsequence) θ ε θ weakly in L () n+. (5.4) We will show later in the proof that the above weakly convergence is in fact a strong convergence (cf 5.4). Let ( u ε, p ε ) and ( v, ε p ε ) the optimal state and the corresponding adjoint state respectively associated to θ. ε By the same arguments as those used in the proof of Theorem 3.3 (the fact that θ is replaced by θ ε poses no problem), we get u ε u weakly in H () n+, v ε v weakly in H () n+, where ( u, p ) and ( v, p ) satisfy p ε p weakly in L 0(), p ε p weakly in L 0() (5.5) div ( Ae( u ) ) = f p + θ in div ( Ae( v ) Be( u ) ) = p dans div u = div v = 0 in u = v = 0 on, (5.6) the optimal control θ is characterized by the variationnal inequality θ U ad and ( v + Nθ ).( θ θ ) 0 θ U ad.

27 EJDE 00/7 Hakima Zoubairi 7 Step : Energy convergence Using integration by parts and the first equation of (.4), we have e( u ε ) : e( u ε )dxdz = div(e( u ε )). u ε dxdz (5.7) = div(µ ε e( v )). u ε ε dxdz p ε. u ε dxdz. Integrating now by parts the right-hand side of the above equation and using the second equation of (.4), e( u ε ) : e( u ε )dxdz = µ ε e( v ) ε : e( u ε )dxdz = µ ε e( u ε ) : e( v )dxdz ε (5.8) = div(µ ε e( u ε )). v dxdz ε = ( f ε p ε + θ ). v ε dxdz. ε Using (4.0),(5.5) and (5.6), we have e( u ε ) : e( u ε )dxdz ( f p + θ ). v dxdz. Using the first equation of (5.6) and integrating by parts in the right-hand side of the above equation, we get (using the symmetry properties of A), ( f p + θ ). v dxdz = div(ae( u )). v dxdz = Ae( u ) : e( v )dxdz (5.9) = e( u ) : Ae( v )dxdz = div(ae( v )). u dxdz. Using now the second equation of (5.7) and integrating by parts in the last integral of (5.9), div(ae( v )). u dxdz = div(be( u )). u dxdz + p. u dxdz = div(be( u )). u dxdz = Be( u ) : e( u )dxdz. (5.0)

28 8 Optimal control and homogenization EJDE 00/7 Finally, by (5.8)-(5.0), we have the energy convergence e( u ε ) : e( u ε )dxdz Be( u ) : e( u )dxdz. (5.) Step 3 Taking into account the definition of optimal control (5.), we have θ U ad J ε ( θ) J ε ( θ ε ). Now passing to the limit in inequality (4.38) and using (4.37), we get J 0 ( θ) Be( u ) : e( u )dxdz + N lim sup θ ε dxdz. (5.) ε 0 Thus taking θ = θ in the above equation, we have lim sup θ ε dxdz θ dxdz. ε 0 By (5.4) and the above equation, we obtain θ ε dxdz = lim ε 0 θ dxdz. (5.3) From (5.) and the above equation, we have (5.). Now from (5.) and the above equation, we get (5.3). Finally from (5.4) and (5.3), we derive θ ε θ strongly in L () n+. (5.4) This completes the proof. 6 The case R When R, it is possible to find explicit functions which are solutions of (3.), (3.), (3.5), and (3.6) and satisfies hypotheses (4.), (4.5), (4.7), (4.0), (4.8), (4.30), (4.3) and (4.35). In this case =]0, is as in figure (left) and Problems (3.) and (3.) become d ( d µ dy dy ( χ + y)) = dr in ]0, dy dχ (6.) = 0 in ]0, dy χ(0) = χ(), r (0) = r () and d ( d µ dy dy ( ϕ + y)) = 0 in ]0, ϕ(0) = ϕ(). We have the following result (see Baffico & Conca 3]): (6.)

29 EJDE 00/7 Hakima Zoubairi 9 Proposition 6. For z 0, h fixed. Let χ = 0, r = µ and ( C µ )y if y ]0, a(z) ϕ(y) = ( C µ )y + a( C µ C µ ) if y ]a(z), b(z) ( C µ )y + (a b)( C µ C µ ) if y ]b(z), with a = a(z), b = b(z), and C = C(z) = ( µ dy). Then (χ, r ) and ϕ are the unique solution (up to an additive constant) of Problems (6.) and (6.). Remark 6. We can calculate the value of C(z) (see, 3]), we get C(z) = µ µ (µ µ )(b(z) a(z)) + µ. Studying the regularity of the solutions of Problems (6.) and (6.) and supposing that µ < µ, hypotheses (4.), (4.5), (4.8) and (4.30) are satisfied (cf 3]). and Let study now Problems (3.5) and (3.6), they becomes d ( dλ µ dy dy + d dy ( χ + y)) = dr dy λ = 0 in ]0, dy λ(0) = λ(), r (0) = r () in ]0, d ( dψ µ dy dy + d dy ( ϕ + y)) = 0 in ]0, ψ(0) = ψ(). (6.3) (6.4) Proposition 6.3 For z fixed in 0, h, Let λ = c, r = c, where c and c are constants, and let µ (K C µ )y if y ]0, a(z) µ ψ(y) = (K C µ )y + a ( ( K µ K µ ) ( C C ) ) if y ]a(z), b(z) µ µ µ (K C µ )y +(a b) ( ( K µ K µ ) ( C C ) ) if y ]b(z), µ µ with K = K(z) = C ( (µ + µ ) C ) µ µ and the constant C defined in Proposition 6.. Then (λ, r ) and ψ are the unique solutions (up to an additive constant) of Problems (6.3) and (6.4).

30 30 Optimal control and homogenization EJDE 00/7 Remark 6.4 The hypotheses (4.7), (4.0), (4.3) and (4.35) are satisfied as shown next. From Proposition 6.3, it is easy to check that λ = c verifies hypothesis (4.7). It is no longer necessary to suppose hypothesis (4.0), since we can pass to the limit in rε, u ε n+ using the explicit form of r and r. ε By the form of ψ (see Proposition 6.3), and since the function h satisfies i) and ii) (see Section ), we show that ψ = ψ(y, z) H loc ( 0, h ) (using the implicit function theorem). Therefore, Hypothesis (4.3.7) is fulfilled. Concerning Hypothesis (4.35), since we can calculate explicitly ψ ϕ y and y, the expression (4.33) becomes d ε = K(z) where K is defined in Proposition 6.3. Part a) of (4.35) follows from regularity of K(z) which follows from the regularity of C(z). i.e Concerning part b), we must show that dε 0 in the distributions sense, dε, φ 0. φ D(), φ 0 Indeed, from the form of K and after elementary computations, we get dε, φ = C(z)( µ + µ ) C(z) µ µ, φ. From 3], if µ < µ, we have C(z), φ 0, and since µ, µ > 0, hence Also C(z) µ µ, φ 0; therefore, dε C(z) ( µ + µ ), φ 0. 0 in the distribution sense. Remark 6.5 Propositions 6. and 6.3 allow us to compute explicitly the coefficients of the tensors A and B. For tensor A (cf 3]), we have µ if h < z < z 0 a = a = µ if 0 < z < h µ if z 0 < z < 0 µ if h < z < z 0 a = a = a = a µ + if 0 < z < h µ if z 0 < z < 0 where µ = µdy, µ+ = ( µ dy) = C(z) and the rest of the coefficients of A are 0. For tensor B, we derive b = b = z ] z 0, z 0

31 EJDE 00/7 Hakima Zoubairi 3 if h < z < z 0 K(z) b = b = b = b 4 if 0 < z < h if z 0 < z < 0 and the rest of the coefficients of B are 0. References ] Baffico L. and Conca C., Homogenization of a transmission problem in D elasticity. in 5 Computational sciences for the st century (M.-O. Bristeau et al., Eds.), , Wiley, New-ork, 997. ] Baffico L. and Conca C., Homogenization of a transmission problem in solid mechanics. J. Math. Anal. and Appl. 33, , ] Baffico L. and Conca C., A mixing procedure of two viscous fluids using some homogenization tools, Computt. Methods Appl. Mech. Engrg., 90 (3 & 33), , 00 4] Bensoussan A., Lions J.L. and Papanicolaou G., Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, ] Brizzi R., Transmission problem and boundary homogenization, Rev. Mat. Apl., 5, -6, ] Conca C., On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl., 64, 3-75, ] Kesavan S. and Saint Jean Paulin J., Homogenization of an optimal control problem, SIAM J.Cont.Optim., 35, , ] Kesavan S. and Saint Jean Paulin J., Optimal Control on Perforated Domains, J. Math. Anal. Appl., 9, No., , ] Kesavan S. and Vanninathan M., L homogénéisation d un problème de contrôle optimal, C.R.A.S, Paris, Sér. A, 85, , ] Lions J.L., Sur le contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris, 968. ] Murat F., L injection du cône positif de H dans W,q, est compacte pour tout q <, J. Math. Pures Appl., 60, 309-3, 98. ] Saint Jean Paulin J. and Zoubairi H., Optimal control and homogenization in a mixture of fluids, Ricerche di Matematica, (to appear). 3] Saint Jean Paulin J. and Zoubairi H., Optimal control and strange term for a Stokes problem in perforated domains, Portugaliae Matematica, (to appear).

32 3 Optimal control and homogenization EJDE 00/7 4] Saint Jean Paulin J. and Zoubairi H., Etude du contrôle optimal pour un problème de torsion élastique, Portugaliae Matematica, (submitted). 5] Sánchez-Palencia E., Nonhomogeneous media and vibrating theory, Lecture notes in Physics, Vol 7, Springer Verlag, Berlin, ] Temam R., Navier Stokes equations, North Holland, Amsterdam, 977. Hakima Zoubairi Département de Mathématiques Ile du Saulcy ISGMP, Batiment A F Metz, France. zoubairi@poncelet.univ-metz.fr