OPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS

Transcription

1 PORTUGALIAE MATHEMATICA Vol. 59 Fasc Nova Série OPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS J. Saint Jean Paulin and H. Zoubairi Abstract: We study a problem of optimal control for Stokes equations in perforated domains with Dirichlet conditions on the boundary of holes. We consider different sizes of holes. 1 Introduction The aim of this paper is to study an optimal control problem for Stokes equations in perforated domains with Dirichlet conditions on the boundary of holes. Let be a bounded connected open set in R n (n 2 ) with Lipschitz boundary. Let ε be a sequence of positive real numbers which tends to zero. We cover the set with a regular mesh of size 2ε, each cell is a cube Pi ε, i = 1,..., N(ε), similar to [ ε, ε] n. We make a hole Ti ε at the center of each cube Pi ε, included in. We define the holes as follows: each hole Ti ε is equal to a ε T where T is a given closed set independent of ε, and a ε is the size of the hole (0 < a ε < ε). Then the perforated domain ε is defined by ε = \ Ti ε. There are different possible sizes of the holes which can be considered ( critical, smaller and larger holes). So we define a ratio σ ε between the current size of the holes and the critical one: ( 1/2 (1.1) σ ε = (ε n /a n 2 ε ) 1/2 for n 3, σ ε = ε log(a ε /ε)) for n = 2. Received: August 4, 2000; Revised: January 6, AMS Subject Classification: 35B27; 49J20; 76D07. Keywords: Optimal control; Homogenization; Stokes equations; Perforated domains.

2 162 J. SAINT JEAN PAULIN and H. ZOUBAIRI If the limit of σ ε as ε tends to zero, is positive and finite then the size of the holes is called critical. If the lim σ ε = +, the size of holes is smaller and if lim σ ε = 0, the holes are larger (cf. Cioranescu and Murat [2] and Allaire [1]). Throughout all the sequel, we use the convention of summation over repeated indices. We denote by the extension by zero onto the holes. Let B = (b ij ) be a symmetric matrix such that (1.2) α m ξ i ξ i b ij (x) ξ i ξ j α M ξ i ξ i a.e. in and b ij L (), where α m and α M are constants such that α M > α m > 0. For ε > 0 fixed, we define the optimal control problem as follows. Let Uad ε L2 ( ε ) n be a closed convex set. Let f L 2 () n be a given function and let N > 0 be a given constant. For θ ε Uad ε, we define the state equation of the Stokes problem by (1.3) p ε u ε = f + θ ε in ε, div u ε = 0 in ε, u ε = 0 on ε. where u ε, p ε are respectively the velocity, the pressure of the fluid and θ ε is the control. The cost functional is then given by (1.4) J ε (θ ε ) = 1 B u ε u ε dx + N θε 2 dx. 2 ε 2 ε The second integral corresponds to the cost of the control whereas the first one corresponds to the energy of the fluid. The matrix B is used in order to generalize the usual energy (we obtain this energy when the matrix B is equal to identity). The optimal control θε is the function in Uad ε which minimizes J ε(θ ε ) for θ ε Uad ε, i.e. (1.5) θε Uad ε and J ε (θε) = min J ε (θ ε ). θ ε Uad ε This problem admits a unique optimal solution θε (see Lions [6]). The problem (1.3) (1.5) can be reduced to a system of equations by introducing the adjoint state (v ε, p ε) of (u ε, p ε ). Thus we get p ε + v ε = div(b u ε ) in ε, (1.6) div v ε = 0 in ε, v ε = 0 on ε. where (v, p ε) (H 1 () n L 2 0 ()).

3 OPTIMAL CONTROL IN PERFORATED DOMAINS 163 The optimal control θ ε is characterized by the variational inequality (1.7) θ ε U ε ad and ε (v ε + N θ ε) (θ ε θ ε) dx 0 θ ε U ε ad. Our aim is to study the limiting behaviour of the optimal control θε as ε 0. In fact, it can be shown that (up to a subsequence) θ ε θ0 weakly in L2 () n. Our objective is to characterize θ0 as the optimal control of a similar problem set in the non-perforated domain. The type of optimal control problem which we consider, was studied by Kesavan and Vanninathan [5], Kesavan and Saint Jean Paulin [3] in non-perforated domains and by Kesavan and Saint Jean Paulin [4] in perforated domains. They studied in [4] the Laplace problem with Neumann conditions on the boundary. Also Rajesh [7] considered the optimal control problem for the Dirichlet problem in perforated domains and he obtained a strange term in the limit. This paper is organized as follows. In Section 2, we recall some hypotheses (H1) (H6) in perforated domains concerning the holes (see Allaire [1]) and the main results of the homogenization of Stokes equations. In Section 3, we consider the critical case and we homogenize the adjoint problem and establish convergence results of energies which appear in the cost functional. In Section 4, we obtain the limiting optimal control problem. In Section 5, we study the optimal problem for smaller sizes of holes (for which lim σ ε = + ). Notation. Throughout this paper, C denotes various real positive constants independent of ε. The duality products between H 1 0 () and H 1 (), and between (H 1 0 ())n and (H 1 ()) n, are each denoted by,. We denote by (e k ) 1 k n the canonical basis of R n. Definition 1.1. (1.8) L 2 0() = We define the set L 2 0 () by { f L 2 () } f(x) dx = 0. 2 Hypotheses on the perforations and preliminary results We make on the holes the same assumptions as Allaire [1], so there exist functions (ω ε k, rε k, µ k) and a linear mapping R ε such that

4 164 J. SAINT JEAN PAULIN and H. ZOUBAIRI (H1) ω ε k H1 () n, r ε k L2 (), (H2) div ω ε k = 0 in and ωε k = 0 in T ε i, (H3) ω ε k e k weakly in H 1 () n and r ε k 0 weakly in L2 0 (), (H4) µ k W 1, () n, (H5) v ε and v such that v ε v weakly in H 1 () n, φ D(), rk ε ωk ε, φ v ε µ k, φv, v ε = 0 in T ε i and R ε L(H0 1()n, H0 1( ε) n ), (H6) If u H0 1( ε) n then R ε ũ = u in ε, If div u = 0 in then div(r ε u) = 0 in ε, R ε u H 1 0 ( ε) n c u H0 1. ()n Example 2.1. The assumptions (H1) (H6) are satisfied in the particular case where each hole Ti ε is a ball of radius a ε where a ε = C 0 ε n/n 2 for n 3 and a ε = e C 0/ε 2 for n = 2 with C 0 > 0 and in a such geometry we can compute explicitly the functions ωk ε, rε k and µ k which satisfy (H1)- -(H6) (see [1]). In this case, the diameter of the holes is such that a ε << ε. Note also that, the case where the diameter of the holes a ε is of the same order as ε corresponds to the classical homogenization. Assumptions (H1) (H6) hold throughout the paper. We define the matrix M (W 1, ()) n n by (see [1]) (2.1) Me k = µ k. This matrix is symmetric and under the above assumptions, we have the following result which is due to Allaire [1]. The extension ũ ε of the velocity u ε and the extension P ε p ε of the pressure p ε (defined by Allaire [1]) satisfy Theorem 2.2 (Allaire [1]). Depending on the size of the holes, there are three different limit flow regimes for the solution of (1.3):

5 OPTIMAL CONTROL IN PERFORATED DOMAINS 165 (i) If lim σ ε = + then (ũ ε, P ε p ε ) converges strongly to (u, p) in H0 1()n L 2 0 (), where (u, p) is the unique solution of the Stokes problem p u = f + θ in, (2.2) div u = 0 in, u = 0 on. (ii) If lim σ ε = σ >0 then there exist a measure µ k and a matrix M such that Me k =µ k such that (ũ ε, P ε p ε ) converges weakly to (u, p) in H0 1()n L 2 0 (), where (u, p) is the unique solution of the Brinkman-type law p u + Mu = f + θ in, (2.3) div u = 0 in, u = 0 on. Remark 2.3. Under hypotheses similar to (H1) (H6) (with a scaling depending of σ ε ), if lim σ ε = 0 then there exist a matrix M 0 such that (ũ ε /σ 2 ε, P ε p ε ) converges strongly to (u, p) in L 2 () n L 2 0 (), where (u, p) is the unique solution of Darcy s law (2.4) u = M0 1 (f p + θ) in, div u = 0 in, u. n = 0 on. with n the exterior normal vector to (see Allaire [1] for more details concerning these hypotheses and the matrix M 0 ). 3 Homogenization and convergence of some energies In this section and in Section 4, we assume that (3.1) lim σ ε = σ > 0. Following the approach of Kesavan and Saint Jean Paulin [4], we introduce the adjoint state variable and pass to the limit in the resulting system. Assuming (3.1), there exists a sequence (ωk ε, rε k ) satisfying (H1) (H6). We show that there exists n distributions µ k B (k = 1,..., n) and a matrix M B

6 166 J. SAINT JEAN PAULIN and H. ZOUBAIRI defined below by (3.8) such that, given any f L 2 () n, if (u ε, p ε ) solves the Stokes problem (1.3), then (up to a subsequence), we have the following convergence of energies. (3.2) (3.3) ε B u ε u ε dx B u u dx + M B u, u B u ε u ε dx B u u + t (M B u) u in D (), where (u, p) solves the problem (2.3). This type of results was shown by Rajesh [7] for the Dirichlet problem for the Laplace operator. We introduce some auxiliary test functions which are used to homogenize the adjoint problem (1.6). Lemma 3.1. Assume (3.1) and let (ψk ε, sε k ) H1 0 ( ε) n L 2 0 ( ε) be the solution of the auxiliary system (3.4) s ε k + ψε k = div(t B ωk ε) in ε, div ψk ε = 0 in ε, ψk ε = 0 on ε. Then there exist ψ k and s k such that (for a subsequence) (3.5) (3.6) ψ ε k ψ k weakly in H 1 0 () n, P ε (s ε k) s k weakly in L 2 0(). Proof: Multiplying the first equation of (3.4) by ψk ε, integrating by parts and taking into account the boundedness of ωk ε in H1 () n, we have the announced result. Definition 3.2. Let us define the distributions µ k B D (), k = 1,..., n by (3.7) µ k B = M ψ k + ( s k + ψ k ), and the matrix M B (W 1, ) n n by (3.8) M B e k = µ k B.

7 OPTIMAL CONTROL IN PERFORATED DOMAINS 167 Proposition 3.3. Let f L 2 (). Define M by (2.1) and M B by (3.8). Assume that (3.1) holds and that θ ε is such that θ ε is bounded in L 2 () n. Let (u ε, p ε ) and (v ε, p ε) in (H0 1( ε) n L 2 0 ( ε)) 2 be the solution of the system p ε u ε = f + θ ε in ε, p ε + v ε = div(b u ε ) in ε, (3.9) div u ε = div v ε = 0 in ε, u ε = v ε = 0 on ε. Then, up to subsequences θ ε θ weakly in L 2 () n, (3.10) ũ ε u weakly in H0 1 ()n, ṽ ε v weakly in H0 1()n and (3.11) { P ε p ε p weakly in L 2 0 (), P ε p ε p weakly in L 2 0 (), where the limits (u, p) and (v, p ) are solution of the Brinkman type system p u + Mu = f + θ in, p + v Mv = div(b u) t M B u in, (3.12) div u = div v = 0 in, u = v = 0 on. Proof: Step 1: A priori estimates Since θ ε is bounded in L 2 (), it is clear that ũ ε and ṽ ε are uniformly bounded in H0 1()n and, also {P ε p ε } and {P ε p ε} are uniformly bounded in L 2 0 (). Hence we can extract a subsequence (again indexed by ε for convenience) such that (3.10) and (3.11) holds. The homogenization of the state equation (1.3) is known (see Theorem 2.1 (ii)).

8 168 J. SAINT JEAN PAULIN and H. ZOUBAIRI Step 2: Energy method To pass to the limit in the second equation in (3.9), we use the test functions (ωk ε, rε k ) defined in (H1) (H6) and the auxiliary functions (ψε k, sε k ) defined by (3.4). Let φ D(). Multiplying the second equation in (3.9) by φωk ε and integrating by parts and using assumption (H2), we get p ε φ ωk ε dx = z ε φ ωk ε dx v ε ωk ε φ dx (3.14) ε ε ε + B u ε ωk ε φ dx, ε where (3.15) z ε = v ε B u ε. Similarly, multiplying the first equation in (3.9) by φ ψk ε, integrating by parts and taking into account the definition of ψk ε (see equation (3.4)), we obtain (f + θ ε ) φ ψk ε dx + ε p ε φ ψk ε dx = ε (3.16) = u ε φ ψk ε dx u ε φ s ε k dx ε ε u ε φ ψk ε dx ε ( t B ωk) ε u ε φ dx ε ( t B ωk) ε u ε φ dx. ε Adding (3.14) and (3.16) and transforming all the integrals over ε into integrals over, we get (f + θ ε ) φ ψ k ε dx + P ε p ε φ ψ k ε dx + P ε p ε φ ωk ε dx = (3.17) where = z ε φ ωk ε dx b ε k ũε φ dx (3.18) b ε k = t B ω ε k + ψ ε k. Since div v ε = 0 in ε, we get (3.19) rk ε φ div ṽ ε dx = 0. ṽ ε ωk ε φ dx + ũ ε φ P ε s ε k dx, ũ ε φ ψ ε k dx

9 OPTIMAL CONTROL IN PERFORATED DOMAINS 169 Adding (3.17) and (3.19) and integrating by parts, we get (f + θ ε ) φ ψ k ε dx + P ε p ε φ ψ k ε dx + P ε p ε φ ωk ε dx = (3.20) = Step 3: Passing to the limit z ε φ ωk ε dx + ωk ε rk, ε φ ṽ ε rk ε φ ṽ ε dx + ũ ε φ ψ k ε dx b ε k ũε φ dx ũ ε φ P ε s ε k dx. + ṽ ε ωk ε φ dx We now pass to the limit in (3.20) as ε tends to 0. In order to do so, we need some preliminary results. Using (H3), we have (3.21) ω ε k 0 weakly in L 2 () n n. By the definition (3.18) and using the convergences (3.5) and (3.21), we can extract a subsequence such that (3.22) b ε k ψ k weakly in L 2 () n n. Also by the definition (3.15) and using the convergence (3.10), we get (up to subsequences) (3.23) z ε z = v B u weakly in L 2 () n n. Now passing to the limit in (3.20), taking into account the convergences in (H3), (H5), (3.5), (3.6), (3.10), (3.11) and (3.21) (3.23), we get, (up to subsequences) (3.24) (f + θ) φ ψ k dx + = p φ ψ k dx + p φ e k dx = z φ e k dx µ k, φ v + u φ ψ k dx ψ k u φ dx u φ s k dx. Therefore, integrating by parts the right-hand side of (3.24) and using Theorem 2.1 (ii), we have Mu φ ψ k dx p φ e k dx = (3.25) = (div z) φ e k dx µ k, φv + ψ k u φ dx + s k u φ dx.

10 170 J. SAINT JEAN PAULIN and H. ZOUBAIRI Since the above relation holds for all φ D() and since M is symmetric, we have (3.26) p + div z Mv = t M B u i.e. (u, p) and (v, p ) satisfy (3.12). Since M is symmetric and positive definite, the solutions (u, p) and (v, p ) of (3.12) are unique, and therefore, it follows that the whole sequences (u ε, P ε p ε ) and (v ε, P ε p ε) converge. This completes the proof of the proposition. Now, we treat the convergence of the energies B u ε u ε dx. This type of ε convergence have been studied by Rajesh [7] for the Dirichlet problem. He has shown in [7] that a strange term for the energy appears in the limit using ideas of [2]. Similarly, we show a same type of result i.e. a strange term in the limiting energy for Stokes problem following ideas of [1] and [7]. Theorem 3.4. Let f L 2 () n and (u ε, p ε ) be the solution of the Stokes problem (1.3). Let M B given by (3.8). Then (3.27) and ε B u ε u ε dx B u u dx + M B u, u (3.28) B ũ ε ũ ε B u u + t (M B u) u in D (). Proof: Using the fact that (u ε, p ε ) and (v ε, p ε) are solution of (3.9), we have (3.29) B u ε u ε dx = ( v ε B u ε ) u ε dx + v ε u ε dx ε ε ε = p ε u ε dx + v ε u ε dx ε ε = v ε u ε dx ε = v ε (f + θ ε ) dx = ṽ ε (f + θ ε ) dx. ε Therefore, integrating by parts and using the homogenization results of Propo-

11 OPTIMAL CONTROL IN PERFORATED DOMAINS 171 sition 3.3 and the fact that M is symmetric, we obtain lim B u ε u ε dx = v(f + θ) dx ε = v( p u + Mu) ε = v u dx + Mv u dx (3.30) = v + Mv, u = p div(b u) + t M B u, u = B u u dx + M B u, u, which proves (3.27). Let φ D(). Set z ε defined by (3.15), integrating by parts and using the problem (3.9), we have B u ε u ε φ dx = v ε u ε φ dx z ε u ε φ dx ε ε ε (3.31) = v ε (f + θ p ε ) φ dx v ε u ε φ dx ε ε p ε u ε φ dx + z ε u ε φ dx. ε ε Using the same arguments as in the proof of Proposition 3.3 and using system (3.12), we derive lim B u ε u ε φ dx = ε (3.32) = v(f +θ p) φ dx v u φ dx p u φ dx + z u φ dx = Mu, v φ + u v φ dx z u φ dx + t M B u Mv, φ u. Therefore, using the fact that M is symmetric, we have lim B u ε u ε φ dx = ( v z) u φ dx + t M B u, φ u (3.33) ε = B u u φ dx + t (M B u) u, φ. This holds for all φ D(). This proves (3.28) and completes the proof.

12 172 J. SAINT JEAN PAULIN and H. ZOUBAIRI Now we give some properties concerning the functions (µ k B ) 1 k n. Theorem 3.5. Let µ k B be as defined in (3.7). Then (3.34) µ k B e i = lim B ω ε i ω ε k in D (). Proof: Let φ D(). Using the problem (3.4), the expression (3.19) and integrating by parts, we have B ωi ε ωk ε φ dx = ( ψk ε + t B ωk) ε ωi ε φ dx ψk ε ωi ε φ dx ε ε ε = s k ε ωi ε φ dx ( ψk ε + t B ωk) ε ωi ε φ dx ε ε (3.35) + ψk ε ωi ε φ dx + ψk ε ωi ε φ dx ε ε = s ε k ωi ε φ dx ( ψk ε + t B ωk) ε ωi ε φ dx ε ε ri ε ωi ε, ψ k ε φ ri ε ψk ε φ dx. ε Passing to the limit (using the convergences (H3), (H5), (3.5) and (3.6)), we get lim B ωi ε ωk ε φ dx = s k e i φ dx ψ k e i φ dx µ i, ψ k φ ε = s k e i φ dx + ψ k e i φ dx µ i, ψ k φ (3.36) = s k + ψ k Mψ k, φ e i This proves (3.34). = µ k B, φ e i = µ k B e i, φ. Corollary 3.6. If B is symmetric positive definite, then µ k B measure and M B is symmetric. is a positive Proof: This is a consequence of Theorem 3.5. In the next section, we return to the control problem we started with.

13 OPTIMAL CONTROL IN PERFORATED DOMAINS Optimal control We denote by χ ε the characteristic function of ε. We now consider the optimal control problem (1.3) (1.5) where the convex set Uad ε L2 ( ε ) is one of the following ones (see [3] and [4]). (4.1) (4.2) (4.3) Uad ε = L 2 ( ε ) n, { Uad ε = θ L 2 ( ε ) n θ } χ ε ψ a.e. in, { Uad ε = θ L 2 ( ε ) n χ ε ψ 1 θ } χ ε ψ 2 a.e. in, where ψ, ψ 1 and ψ 2 are given functions in L 2 () n. Now, since θε is optimal we have N (4.4) (θ 2 ε) 2 dx J ε (θε) J ε (Θ ε ) Θ ε Uad ε. ε This relation holds in particular with the following choice of Θ ε χ ε in the case of (4.1), (4.5) Θ ε = χ ε ψ in the case of (4.2), χ ε ψ 2 in the case of (4.3). In each of the three cases above, we have Lemma 4.1. The optimal control satisfies (up to a subsequence) (4.6) θ ε θ 0 weakly in L 2 () n. Proof: Using (4.5), we have that J ε (Θ ε ) is bounded in L 2 ( ε ) n, so we derive from (4.4) the announced result. Lemma 4.2. The characteristic function χ ε of ε satisfies (4.7) χ ε 1 weakly in L (). Proof: We have, up to a subsequence χ ε χ 0 weakly in L (). Since χ ε ω ε k = ωε k, thus passing to the limit and by uniqueness, we obtain χ 0 =1.

14 174 J. SAINT JEAN PAULIN and H. ZOUBAIRI We proceed to characterize the limiting optimal control problem. We define the set U ad L 2 () as (4.8) (4.9) (4.10) U ad = L 2 () n, { } U ad = θ L 2 () n θ ψ a.e. in, { } U ad = θ L 2 () n ψ 1 θ ψ 2 a.e. in, corresponding to the cases (4.1), (4.2) and (4.3) respectivly. We have the following convergence result of optimal control. Theorem 4.3. Let M B given by (3.8). For θ U ad, let (u, p) H0 1()n L 2 0 () be the solution of (2.2). Let J 0 be the cost functional defined by (4.11) J 0 (θ) = 1 B u u dx M B u, u + N θ 2 dx. 2 Then θ0 satisfy the condition of optimality (4.12) θ 0 U ad and J 0 (θ 0) = min θ U ad J 0 (θ). Further we have the convergence of the minimal costs, i.e. (4.13) lim J ε (θ ε) = J 0 (θ 0). Proof: Step 1: It is clear from the definition of U ad, that if θ U ad then χ ε θ Uad ε. Further, since θ ε θ0 weakly in L2 () n and U ad is a closed convex set, we have θ0 U ad. Step 2: Let (u ε, p ε) be the solution of the state equation (1.3) corresponding to θ ε = θε. Using the convergence (4.6) of Lemma 4.1, we get { ũ ε u weakly in H0 1()n, (4.14) P ε p ε p weakly in L 2 0 (), where (u, p ) is solution of (2.2) with θ = θ in the right-hand side. Step 3: Let (w ε, q ε ) H0 1( ε) n L 2 0 ( ε) be the solution of the state equation (1.3) with the control χ ε θ, θ U ad, that is q ε w ε = f + χ ε θ in ε, (4.15) div w ε = 0 in ε, w ε = 0 on ε.

15 OPTIMAL CONTROL IN PERFORATED DOMAINS 175 Since χ ε θ θ weakly in L 2 () n, it follows that w ε w weakly in H0 1()n and P ε (q ε ) q weakly in L 2 0 () where (w, q) satisfy the following Brinkmann-type problem q w + Mw = f + θ in, (4.16) div w = 0 in, w = 0 on, (see Proposition 3.3). Further, using Theorem 3.4 for θ fixed, we have (4.17) B w ε w ε dx B w w dx + M B w, w. ε Thus (4.18) J ε (χ ε θ) J 0 (θ). Once again, using Theorem 3.4 but now for θ ε = θε, we get (4.19) B u ε u ε dx B u u dx + M B u, u. ε Step 4: Passing to the limit in the inequality (4.20) J ε (χ ε θ) J ε (θ ε). and using Lemma 4.2, we get (4.21) J 0 (θ) 1 B u u dx + lim sup 2 Thus taking θ = θ in the above inequality, we have (4.22) lim sup (θε) 2 dx (θ ) 2 dx. ε Moreover since θ ε θ0 weakly in L2 (), we get (4.23) lim inf (θε) 2 dx (θ ) 2 dx. ε Thus using (4.22) and (4.23), we derive (4.24) lim (θ ε) 2 dx = ε N 2 (θ ) 2 dx. We deduce (4.13). Now (4.12) follows from (4.21) and (4.24). ε (θ ε) 2 dx M B u, u.

16 176 J. SAINT JEAN PAULIN and H. ZOUBAIRI 5 Case of smaller holes We now assume that the size of the holes is smaller than the critical size, i.e. (5.1) lim σ ε = +, in other words, (5.2) a ε << ε n/n 2 for n 3, a ε = exp 1/Cε and C ε << ε 2 for n = 2. Since the size of the holes satisfies (5.1), the hypothese (H3) is replaced by (see [1]) (5.3) ω ε k e k strongly in H 1 () n and r ε k 0 strongly in L 2 0(). Remark 5.1. Hypothese (5.3) is stronger than Hypothese (H3). We have the following result Proposition 5.2. Let the size of the holes satisfy (5.1). Assume that (H1), (H2), (H4) (H6) and (5.3) hold. Let (u ε, p ε ) and (v ε, p ε) be the unique solution of (3.9). Then up to subsequences θ ε θ weakly in L 2 () n, (5.4) ũ ε u strongly in H0 1 ()n, ṽ ε v strongly in H0 1()n and (5.5) { P ε (p ε ) p strongly in L 2 0 (), P ε (p ε) p strongly in L 2 0 (), where (u, p) and (v, p ) are solution of the Stokes problem p u = f + θ in, p + v = div(b u) in, (5.6) div u = div v = 0 in, u = v = 0 on. Proof: To prove this result, we use the same arguments as in the proof of Proposition 3.3.

17 OPTIMAL CONTROL IN PERFORATED DOMAINS 177 Step 1: The fact that M = 0 was established by Allaire [1] and also the following convergence result (5.7) { ũε u strongly in H 1 0 ()n, P ε (p ε ) p strongly in L 2 0 (). Following Allaire [1], we show now that µ k B, defined by (3.7), is equal to zero. Since Hypotheses (H1), (H2), (H4) (H6) and (5.3) are satisfied, all the results of Proposition 3.3 hold. But from (5.3) and Theorem 3.6, we deduce that µ k B = 0 and hence M B = 0. This proves that (u, p) and (v, p ) satisfy the Stokes equations (5.6). We complete the proof by showing the strong convergence of ṽ ε in H 1 0 ()n and of P ε (p ε) in L 2 0 (). Step 2: Using the convergences (3.10) and (3.11) of ṽ ε and P ε (p ε) respectively and defining v by (5.6), we have the following convergence (using classical arguments) { ṽε v strongly in H0 1()n, (5.8) This ends the proof. P ε (p ε) p strongly in L 2 0 (). We now give a convergence result of the optimal control. Let U ε ad L2 ( ε ) n given by (4.1) (4.3) and U ad L 2 () n by (4.8) (4.10). We have the following result Theorem 5.4. Let θ ε be the optimal control for the Stokes problem (1.3) and let the cost functional be given by (1.4). Then (5.9) θ ε θ 0 weakly in L 2 () n and θ 0 (5.10) is the optimal control for the problem p u = f + θ in, div u = 0 in, u = 0 on, with the following cost functional (5.11) J 0 (θ) = 1 B u u dx + N 2 2 θ 2 dx.

18 178 J. SAINT JEAN PAULIN and H. ZOUBAIRI Proof: By Lemma 4.1, we have (5.9). Now using Theorem 4.3 with the matrices M and M B equal to zero, we have immediately the results. This completes the proof. Remark 5.5. In the case where σ ε 0 (i.e. when the holes are larger), we have that ũ ε 0 strongly in H 1 () n, hence ũ ε 0 strongly in L 2 () n n. Then it is obvious that we have the following convergence of energy B u ε u ε dx = B ũ ε ũ ε dx 0. ε Unfortunately, we could not succeed to conclude concerning the optimal control problem in this case. REFERENCES [1] Allaire, G. Homogenization of the Navier Stokes Equations in Open Sets Perforated with Tiny Holes, Arch. Ration. Mech. Anal, 113(3) (1991), [2] Cioranescu, D. and Murat, F. Un terme étrange venu d ailleurs, in Nonlinear Partial Differential Equations and their Applications, (H. Brezis and J.-L. Lions, Eds.), Collège de France Seminar, Vols. 2 & 3, Research Notes in Mathematics 60 & 70, Pitman, London, [3] Kesavan, S. and Saint Jean Paulin, J. Homogenization of an Optimal Control Problem, SIAM J. Contr. Optim., 35 (1997), [4] Kesavan, S. and Saint Jean Paulin, J. Optimal Control on Perforated Domains, J. Math. Anal. Appl., 229(2) (1999), [5] Kesavan, S. and Vanninathan, M. L homogénéisation d un problème de contrôle optimal, C.R.A.S, Paris, sér. A, 285 (1977), [6] Lions, J.L. Sur le Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, [7] Rajesh, M. Convergence of some Energies for the Dirichlet Problem in Perforated Domains (preprint). J. Saint Jean Paulin and H. Zoubairi, Département de Mathématiques, Université de Metz, Ile du Saulcy F-57045, Metz FRANCE zoubairi@poncelet.sciences.univ-metz.fr