Optimality conditions for nonconvex multistate control problems in the coefficients

OPTIMALITY CONDITIONS FOR NONCONVEX MULTISTATE CONTROL PROBLEMS IN THE COEFFICIENTS∗ JUAN CASADO-D´ IAZ†, JULIO COUCE-CALVO†,AND JOS´ E D. MART´ IN-G´ OMEZ† SIAM J. CONTROL OPTIM.c 2004 Society for Industrial and Applied Mathematics Vol. 43, No. 1, pp. 216–239 Abstract. The purpose of this paper is to attain some optimality conditions for the identification of a diffusion matrix (material) under several restrictions. Assuming that the set of such diffusion matrices is closed for the H-convergence, we give a method to obtain admissible directions which applies to a not-necessarily convex control set. Our results permit obtaining the diffusion matrix from the state functions. Key words. control in the coefficients, admissible directions, nonconvex control set AMS subject classification. 49K20 DOI. 10.1137/S0363012902411714 1. Introduction. The problem we consider in the present paper is related to the choice of an optimal material under several conditions, or the identification of a material from a finite number of observations. In a mathematical setting, we have the model problem min A∈M(Ω) J(y1,... ,y k),(1.1) where yi=yi(A), 1 ≤i≤k, are the solutions of the equations −div A∇yi=fiin D(Ω), yi∈H1 0(Ω),1≤i≤k. (1.2) Here Ω is a bounded open set of RN,Jis a smooth objective functional in H1 0(Ω)k, f1,... ,f kare kfixed elements of H−1(Ω), and M(Ω) is a given set of measurable functions with values in the space of symmetric matrices of order N. The elements of M(Ω) are uniformly elliptic and bounded. Clearly, other generalizations can be considered: Jdepending on A, other boundary conditions, etc. A physical example is the identification of a material. For this purpose, we apply a finite number kof external conditions (in our case they are represented by fi) and in each case we realize a measure of the corresponding state. For example, we give the value ziof the state in a subset ω⊂Ω. Then the problem can be formulated as min k  i=1 ω|yi−zi|2dx, where yiare the solutions of (1.2). Assuming Jis sequential lower semicontinuous for the weak topology of H1 0(Ω)k and M(Ω) is closed for the H-convergence or the G-convergence, because we are ∗Received by the editors July 22, 2002; accepted for publication (in revised form) November 24, 2003; published electronically June 15, 2004. This work has been partially supported by the projects BFM 2002-00672 of the D.G.I. of the “Ministerio de Ciencia y Tecnolog´ıa” of Spain and FQM309 of the “Junta de Andaluc´ıa.” http://www.siam.org/journals/sicon/43-1/41171.html †Depto. de Ecuaciones Diferenciales y An´alisis Num´erico, Fac. de Matem´aticas, Universidad de Sevilla, C. Tarfia s/n, 41012 Sevilla, Spain ([email protected], [email protected], [email protected]). 216 Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 217 working with symmetric matrices and thus the two concepts are equivalent (see, e.g., [22], [19], [18], [4], [25]), it is well known that (1.1) has, at least, a solution (see, e.g., [25], [17]). If M(Ω) does not satisfy this last condition and Jis sequential continuous for the weak topology of H1 0(Ω)k, we can obtain a relaxed problem replacing M(Ω) by its H-closure. Thus, it is natural to assume M(Ω) is H-closed. The calculus of the H-closure of a set is a very difficult problem and there are a lot of works in this field (see, e.g., [23], [14], [13], [8], [25], [3], [16] and the references in them). In this paper we are interested in obtaining some necessary conditions which must satisfy the optimal solution of (1.1). For k= 1, the problem has been studied in [13], [20], [8], [25], [3]. For k>1, there are few results to our knowledge (see [25], [6], [7], [3]). The paper is organized as follows: In section 3 we give a definition of admissible direction (see Definition 3.1). Then we prove that if Ais a solution of (1.1), y1,... ,y kthe corresponding state functions, p1,... ,p kthe solutions of (3.2) (the adjoint states), and Cthe matrix defined by (3.3), we have Ω H:Cdx≤0(1.3) for every admissible direction H. Related results can be found, for example, in [8], [13], [25], [3]. However, in these papers, the admissible directions are of the form H=B−A, with Bin M(Ω), which needs some convexity assumptions. When k≤N−1 (in particular k= 1) and M(Ω) is obtained by homogenization, mixing a finite number of matrices with fixed proportions, a result of Tartar (see [25]) shows that although M(Ω) is not convex, for every ξ1,... ,ξ k∈RN, the set {(Bξ1,... ,Bξ k)/B∈M(Ω)}⊂L∞(Ω)k (1.4) is convex, and thus the directions H=B−Acan still be considered. However, this is not true for k≥N(or, in principle, for other choices of M(Ω) even if k≤N−1). This is the reason we have given a more general definition of admissible direction. In section 4, assuming M(Ω) local (see Definition 4.1) and closed for the H- convergence, we give an original method to find admissible directions following our definition. As a consequence, we obtain the main result of the paper, Theorem 4.5, where we prove that for every A, B ∈M(Ω), l∈{1,... ,N},W⊂RNlinear subspace of dimension l, and every bounded measurable set Tof W, with l-dimensional positive measure, the matrix Hdefined by H(x)ei=(B(x)−A(x)) ei+1 |T|T∇W zˆwi(x, z)d(z) (1.5) is an admissible direction in A. Here e1,... ,e Nis the standard basis of RN,∇W z denotes the gradient with respect to W, and ˆwiis the solution of the partial differential problem given by (4.4). This direction has the difficulty that ˆwi(and then H) cannot be explicitly obtained. However, we think that it can be interesting, for example, to apply a descent method in order to solve numerically problem (1.1), where we can obtain ˆwinumerically. Related to this point, an interesting question, one that we want to study in the future, is the optimal choice of Wand Tto obtain the steepest descent direction. A criterion to determine this direction (see Remark 4.9) can be to calculate the maximum of H:Con the set of matrices Hobtained by (1.5). Although, as we have said above, it is not possible in general to obtain ˆwiexplicitly, we show in Theorem 4.12 that this can be carried out for a particular choice of Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 218 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ T(which probably is not optimal). This permits us to obtain a family of admissible direction, depending on the subspace Wchosen. Essentially, they are of the form B−Aplus a term which has a growth of order two in B−Afor every B∈M(Ω). When the dimension of Wis equal to 1, the corresponding admissible direction comes just from a lamination. In this case the expression of His known and it can be found, for example, in [25] (it can also be obtained from the results in [10]), but to our knowledge its utility to obtain optimality conditions for problem (1.1) has not been exploited. Most of the consequences we obtain in the present paper using Theorem 4.12 use only, in fact, l= 1. However, we show that in some cases (see Remark 4.17) it is better to use a subspace of dimension greater than one. Using Theorem 4.12 we prove in Corollary 4.18 that for every B∈M(Ω), the condition C:(A−B)≥0(1.6) (which is the condition we find if the admissible directions are of the form B−A) is still true on the set where Chas a nonpositive eigenvalue or where Ker(A−B)={0}.In particular (see Corollary 4.20) the condition (1.6) holds a.e. in Ω for every B∈M(Ω) when k≤N−1. When M(Ω) comes from the mixture of a finite number of materials with fixed proportions, this result can also be obtained from the convexity of the set defined by (1.4), but we note that our set M(Ω) is more general. In section 5 we study the case where M(Ω) is invariable by rotations, which is a natural assumption in the applications. Then we show that condition (1.3) implies that Cand Aare mutually diagonalizable a.e. in Ω. Moreover, assuming further hypotheses (in particular if M(Ω) is H-closed and N≥3), we prove in Proposition 5.4 that the eigenvalues of Aand Care mutually ordered. As application of the results stated above, it is possible to obtain, in some situations, the matrix Afrom Cand then to reduce the set of optimality conditions to a nonlinear partial differential system with variables yi,pi,1≤i≤k. The main problem to carrying out this point is that in general, the H-closure of a given set is unknown. In section 6, we apply our results to two examples: The first one is the mixture of two homogeneous isotropic materials, which has also been studied in [3] (see also [25] for k= 1). In this case M(Ω) is convex. In second problem we consider a polycrystal in dimension 2, where M(Ω) is not convex. 2. Notation. For a linear subspace W⊂RN, we define L(W, W ) as the space of the linear applications from Winto Wand by Ls(W, W ) the subspace of the symmetric applications. When W=RNwe write MN=L(RN,RN), Ms N=Ls(RN,RN). The orthogonal projection of RNinto Wis denoted by PW. For a matrix A∈M N, we define AW∈L(W, W )byAW=PWA|W. The orthogonal subspace of Wis denoted by W⊥. For u:W→R, we denote ∇Wu:W→Wthe gradient of uwith respect to W, i.e., ∇Wuis defined by ∇Wuξ=Dξu∀ξ∈W, where Dξuis the derivative of uin the direction ξ. We denote by {e1,... ,e N}the standard basis of RN. The group of the orthogonal matrices in RNof determinant 1 is denoted by ON. The scalar product of two matrices A, B ∈M Nis written A:B. The tensorial product of two vectors ξ,η ∈RNis denoted as ξ⊗η. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 219 For a bounded open set Ω ⊂RN, we denote by M(Ω) a fixed subset of the space L∞(Ω,Ms N) such that there exist α, β > 0 which satisfy B(x)ξξ ≥α|ξ|2,|B(x)ξ|≤β|ξ|∀B∈M(Ω) a.e. x∈Ω.(2.1) For a matrix A∈M(Ω), KA(M(Ω)) and ¯ KA(M(Ω)) are the cones of admissible directions of M(Ω) in A; see Definition 3.1. For T⊂RNand ∈(0,N], we denote by |T|the -dimensional Hausdorff measure of T. The integral of a function u:T→R, with respect to the -dimensional Hausdorff measure, is written T u(z)dz. When =N, we simplify the notation by writing |T|and T u(z)dz. We use the subindex to mean periodicity. For example, for a cube Y⊂RN, H1 (Y) is the space of functions of H1 loc(RN) which are Y-periodic. 3. Optimality conditions. In this section we introduce the definition of admissible direction. Using it, we obtain the first optimality result for the control problem (1.1). Definition 3.1. For A∈M(Ω), let us define the cone of admissible directions ¯ KA(M(Ω)) as the closure in the weak-∗topology of L∞(Ω,Ms N)of the set KA(M(Ω)), where KA(M(Ω)) is the set of H∈L∞(Ω,Ms N)such that there exist a constant c>0 and Aε∈M(Ω),ε>0, such that ⎧ ⎨ ⎩ Aε−AL∞(Ω,Ms N)≤cε, lim ε→0 Aε−A ε=Ha.e. in Ω. (3.1) Theorem 3.2. We consider J:H1 0(Ω)k→R,Fr´echet derivable, f1,... ,f k∈ H−1(Ω)k.LetA∈M(Ω) be a solution of (1.1) and y1,... ,y kthe solutions of (1.2). We define the adjoint states p1,... ,p kas the solutions of −div(A∇pi)=∂iJ(y1,... ,y k)in D(Ω), pi∈H1 0(Ω),1≤i≤k, (3.2) and the matrix C∈L1(Ω,Ms N)by C=1 2 k  i=1 (∇yi⊗∇pi+∇pi⊗∇yi).(3.3) Then we have Ω H:Cdx≤0∀H∈¯ KA(M(Ω)).(3.4) Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 220 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ Proof. Let us first prove the result for H∈KA(M(Ω)). For ε>0 small enough, we define y∗ i,ε,1≤i≤k, as the solution of −div((A+εH)∇y∗ i,ε)=fiin D(Ω), y∗ i,ε ∈H1 0(Ω). (3.5) Then it is easy to check that for 1 ≤i≤k,wehave lim ε→0 y∗ i,ε −yi ε=˙yiin H1 0(Ω),(3.6) with ˙yithe solutions of −div(A∇˙yi+H∇yi)=0 inD(Ω), ˙yi∈H1 0(Ω). (3.7) Now, for ε>0, we consider Aε∈M(Ω) in the conditions of (3.1). Then for 1≤i≤k, we define yi,ε as the solutions of −div(Aε∇yi,ε)=fiin D(Ω), yi,ε ∈H1 0(Ω). (3.8) Taking y∗ i,ε −yi,ε as test function in the difference of (3.5) and (3.8), and dividing by ε,weget 1 εΩ (A+εH)∇(y∗ i,ε −yi,ε)∇(y∗ i,ε −yi,ε)dx =Ω Aε−(A+εH) ε∇(yi,ε −y∗ i,ε)∇(y∗ i,ε −yi,ε)dx(3.9) +Ω Aε−(A+εH) ε∇y∗ i,ε∇(y∗ i,ε −yi,ε)dx. By the ellipticity of A+εH (for εsmall enough) and (3.1), we deduce from (3.9) the existence of c>0 such that 1 εy∗ i,ε −yi,ε2 H1 0(Ω) ≤cy∗ i,ε −yi,εH1 0(Ω)µεL2(Ω), with µε=Aε−(A+εH) ε∇y∗ i,ε. From (3.1), (3.6), and the Lebesgue-dominated convergence theorem, we deduce that µεconverges strongly to zero in L2(Ω)N. Thus, lim ε→0 y∗ i,ε −yi,ε ε=0 inH1 0(Ω), which, by (3.6), implies lim ε→0 yi,ε −yi ε=˙yiin H1 0(Ω).(3.10) Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 221 On the other hand, since Ais a solution of (1.1) and Aε∈M(Ω), we have J(yε)−J(y) ε≥0∀ε>0, with yε=(y1,ε,... ,y k,ε) and y=(y1,... ,y k). From (3.10) and the Fr´echet derivability of J,weget k  i=1∂iJ(y),˙yi= lim ε→0 J(yε)−J(y) ε≥0.(3.11) But taking ˙yias the test function in (3.2) and pias the test function in (3.7), we have k  i=1∂iJ(y),˙yi= k  i=1 Ω A∇pi∇˙yidx =− k  i=1 Ω H∇yi∇pidx. This proves Ω H:Cdx= k  i=1 Ω H∇yi∇pidx ≤0∀H∈KA(M(Ω)).(3.12) Now let Hbe in ¯ KA(M(Ω)). For δ>0 we define Gδ=M∈L∞(Ω,Ms N)/Ω C:(M−H)dx <δ . Since Gδis a neighborhood of Hin the weak-∗topology of L∞(Ω,Ms N), there exists Hδin Gδ∩KA(M(Ω)) and then, from (3.12), we get Ω C:Hdx=Ω C:Hδdx +Ω C:(H−Hδ)dx<δ for every δ>0. This proves (3.4). Remark 3.3. The above theorem is still true if the elements of M(Ω) are not necessarily symmetric by changing Ato Atin the definition (3.2) of the functions pi and taking C= k  i=1 ∇pi⊗∇yi. Remark 3.4. If M(Ω) is convex, the condition (3.4) implies Ω A:Cdx= max Ω B:Cdx/B∈M(Ω).(3.13) Remark 3.5. Theorem 3.2 still holds if we take KA(M(Ω)) as the cone of matrices H∈L∞(Ω,Ms N(Ω)) such that for every sequence Φ1 ε,... ,Φk ε,which respectively converges in L2(Ω)Nto Φ1,... ,Φk,there exists Aε∈M(Ω) such that Aε−A εΦi ε→HΦiin L2(Ω)N,i=1,... ,k. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 222 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ The advantage of this definition is the following: If M(Ω) is the set we obtain by mixing rmaterials with proportions fixed, then for every ξ1,... ,ξ N−1∈RN, the set {(Bξ1,... ,Bξ N−1)/B ∈M(Ω)}⊂L∞(Ω)N−1 is convex (see [24], [25]). So, with this definition of KA(M(Ω)), the matrices of the form B−Abelong to KA(M(Ω)) if k≤N−1. Thus, (3.13) still holds in this case. Later we will deduce this result (see Corollary 4.20) for more general choices of M(Ω), using simply Definition 3.1 of admissible directions. 4. Calculus of admissible directions. In the following, let us calculate explicitely some admissible directions by imposing additional hypotheses about M(Ω). Definition 4.1. We say that M(Ω) is local if there exists a multivaluated application F:x∈Ω→F(x)⊂M s Nsuch that M(Ω) = {B∈L∞(Ω,Ms N)/B(x)∈F(x)a.e. x∈Ω}, where Fis measurable in the sense that {x∈Ω/F(x)∩G=∅} is measurable ∀G⊂M s Nopen. As it is proved in [21], the local property is satisfied in several typical examples of M(Ω). A first consequence of assuming M(Ω) is local follows. Proposition 4.2. We assume M(Ω) is local. We consider A∈M(Ω),H1,... , Hm∈KA(M(Ω)),ω1,... ,ω m⊂Ωmeasurable such that |ωi∩ωj|=0if i=j. Then the matrix H=m i=1 Hiχωibelongs to KA(M(Ω)). Proof. By Definition 3.1, for every i∈{1,... ,m}there exists Ai ε∈M(Ω) and c>0 (which can be taken independent of i) such that Ai ε−AL∞(Ω,Ms N)≤cε, lim ε→0 Ai ε−A ε=Hia.e. in Ω. Taking then Aε= m  i=1 Ai εχωi+AχΩ\∪m i=1ωi, which belongs to M(Ω) because M(Ω) is local, we have Aε−AL∞(Ω,Ms N)≤cε, lim ε→0 Aε−A ε=Ha.e. in Ω, and then Hbelongs to KA(M(Ω)). Remark 4.3. It is not difficult to show that the above result remains true if we replace KA(M(Ω)) by ¯ KA(M(Ω)). Using Proposition 4.2, we get the following. Proposition 4.4. In the assumptions of Theorem 3.2,ifM(Ω) is local, we have H:C≤0a.e. in Ω∀H∈¯ KA(M(Ω)).(4.1) Proof. By Proposition 4.2, for every H∈KA(M(Ω)) and every ω⊂Ω measurable, the matrix Hχωbelongs to KA(M(Ω)). So, using (3.4), we get ω H:Cdx≤0.(4.2) Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 223 Since ¯ KA(M(Ω)) is the closure of KA(M(Ω)) in the weak-∗topology of L∞(Ω,Ms N), we deduce that (4.2) holds, in fact, for every H∈¯ KA(M(Ω)) and every ω⊂Ω measurable, which implies (4.1). Let us now see how assuming M(Ω) is local permits us to obtain admissible directions. Theorem 4.5. We suppose that M(Ω) is local and closed for the H-convergence. We consider a linear subspace W⊂RNof dimension and a measurable bounded subset T⊂Wsuch that |T|is positive. Then, for every A, B ∈M(Ω), the matrix H∈L∞(Ω,Ms N)defined by H(x)ei=(B(x)−A(x)) ei+1 |T|T∇W zˆwi(x, z)d(z) (4.3) for 1≤i≤Nand a.e. x∈Ωbelongs to KA(M(Ω)).In(4.3), the function ˆwiis defined by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˆwi(x, .)∈H1 loc(W),∇W zˆwi(x, .)∈L2(W, W ), W (A(x)χW\T+B(x)χT)∇W zˆwi(x, .)∇W zˆvd (z) =T (A(x)−B(x))ei∇W zˆvd (z), ∀ˆv∈H1 loc(W),∇W zˆv∈L2(W, W )a.e. x∈Ω. (4.4) Proof. We consider A,B,W, and Tas in the statement of the theorem. For an orthonormal basis {e 1,... ,e  }of W, we denote Y=  i=1 λie i/−1 2<λ i<1 2,1≤i≤⊂W. For ε>0 small enough, we denote Tε=ε1 T⊂Y,˜ Tε=k∈Z(Tε+ i=1 kie i), and we define ˜ Aε:Ω×W→M s Nby ˜ Aε(x, y)=A(x)(1 −χ˜ Tε(y)) + B(x)χ˜ Tε(y). Since M(Ω) is local and closed for the H-convergence, the matrix Aεobtained by taking, for εfixed, the H-limit when δtends to zero of the matrices x→˜ Aε(x, 1 δPW(x)) belongs to M(Ω). Since the matrices ˜ Aεare a tensorial product of functions which only depend on xand functions which only depend on y, it is well known (see, e.g., [5], [2]) that Aεis given by Aε(x)ei=Y ˜ Aε(x, y)(∇W ywi,ε +ei)d(y),(4.5) where wi,ε is the unique solution of ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ wi,ε ∈L2(Ω,H1 (Y)/R), Y ˜ Aε(x, y)(∇W ywi,ε(x, y)+ei)∇W yv(y)d(y)=0 ∀v∈H1 (Y)/Ra.e. x∈Ω. (4.6) Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 224 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ Let us study the asymptotic behavior of Aε. First, we remark that for every v∈ H1 (Y)/Rand a.e. x∈Ω, we have Y ˜ Aε(x, y)ei∇W yv(y)d(y)=Y A(x)ei∇W yv(y)d(y) +Tε (B(x)−A(x))ei∇W yv(y)d(y), but for a.e. x∈Ω, the first term on the right-hand side vanishes. So, wi,ε satisfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Y ˜ Aε(x, y)∇W ywi,ε(x, y)∇W yv(y)d(y) =Tε (A(x)−B(x))ei∇W yv(y)d(y) ∀v∈H1 (Y)/Ra.e. x∈Ω. (4.7) For 1 ≤i≤N, we take wi,ε as the test function in (4.7). Then we get Y ˜ Aε(x, y)∇W ywi,ε(x, y)∇W ywi,ε(x, y)d(y)=Tε (A(x)−B(x))ei∇W ywi,ε(x, y)d(y) for a.e. x∈Ω. Using then (2.1) and the Cauchy–Schwarz inequality, we deduce there exists c>0 such that Y|∇W ywi,ε(x, y)|2d(y)≤c|Tε|=cε(4.8) for ε>0 small enough and a.e. x∈Ω. We define ˆwi,ε :Ω×(ε−1 Y)→Rby ˆwi,ε(x, z)=ε−1 wi,ε(x, ε1 z). From (4.8), we deduce that ∇W zˆwi,ε χε−1 Yis bounded in L∞(Ω,L 2(W, W )). So there exists a subsequence of ε, which we still denote by ε, which converges weak-∗in L∞(Ω,L 2(W, W )). Since the curl of the limit is zero, it is the gradient of a function ˆwi∈L∞(Ω,H1 loc(W)). Once we prove that ˆwiis the solution of (4.4), we conclude that the whole of the sequence converges. We consider ˆv∈D(W) and ε>0 small enough, such that ε1 supp(ˆv)⊂Y, then we define vε∈H1 (Y)by vε(y)=ε1 ˆv(ε−1 y) a.e. y∈Y. Taking vεas the test function in (4.7), using the change of variables z=ε−1 y, and integrating with respect to xin a measurable set ω,weget ωε−1 lY (A(x)χW\T(z)+B(x)χT(z))∇W zˆwi,ε(x, z)∇W zˆv(z)d(z)dx =ωT (A(x)−B(x))ei∇W zˆv(z)d(z)dx. Passing to the limit in this equality and taking into account the arbitrariness of ˆv and ω, and the density of D(RN)/Rin the factor space of functions with gradient in L2(RN) over R(see, e.g., [9]), we show that ˆwiis the solution of (4.4) for 1 ≤i≤N. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 231 Now, if m−m−= 0, then clearly k+≥m−m−. In another case, we consider E= Span({φ− j/1≤j≤k}∪{φ− j,φ + j/1≤j≤k}⊥), which has dimension m−+N−m.If φ∈E⊥, then, as above, statement (i) gives ˜ Cjφφ ≥0, 1 ≤j≤k, and then ˜ Cφφ ≥0. Moreover, ˜ Cφφ =0iff ˜ Cjφφ =0,1≤j≤k, which, by statement (i), implies that φis orthogonal to φ− j,φ+ j,1≤j≤k; i.e., φis in Eand so φ= 0. Thus, taking i=m−+N−min (4.19) and using the compactness of the unitary ball in RN,we get λm−+N−m+1 ≥min φ∈E⊥ |φ|=1 ˜ Cφφ > 0, and thus k+≥m−m−. The other inequalities in (4.18) follow analogously. As a consequence we get the following. Corollary 4.20. In the assumptions of Theorem 3.2,ifM(Ω) is local and closed for the H-convergence and k≤N−1, then condition (4.14) holds. Proof. We apply Theorem 4.19 to ξi=∇yi(x), ηi=∇pi(x), 1 ≤i≤k, a.e. x∈Ω. In this case ˜ C=C(x). Since, clearly, the number m+which appears in this result is less than or equal to k≤N−1, we deduce that the number of positive eigenvalues of Cis less than or equal to N−1, and then there exists at least a nonpositive eigenvalue of Ca.e. in Ω. Corollary 4.18 gives then (4.14). 5. Invariability by rotations. In the applications, it is a natural hypothesis to assume that M(Ω) is invariable by rotations. We show in this section that this assumption implies that the eigenvectors of Aand Cagree. Definition 5.1. We say that M(Ω) is invariable by rotations if for every B∈ M(Ω) and every Q∈L∞(Ω,MN), with Q∈O Na.e. in Ω, the matrix QBQtbelongs to M(Ω). We have the following result. Proposition 5.2. In the assumptions of Theorem 3.2,ifM(Ω) is invariable by rotations, then Aand Care mutually diagonalizable a.e. in Ω. Proof. Let us first prove that given a skew-symmetric matrix Rand a measurable set ω⊂Ω, the function (RA +ARt)χωbelongs to KA(M(Ω)). To this purpose we define G:MN→M N×Rby G(M)=(MMt,det(M)). Since Ker(G(I)) coincides with the space of skew-symmetric matrices, it is known (see, e.g., [1]) that for ε∈R with |ε|small enough, there exists Pε∈M Nsuch that G(Pε)=G(I) or, equivalently, Pε∈O N, and (Pε−I)/ε converges to R. Defining then Aε=PεAPt εχω+AχΩ\ω and using that M(Ω) is invariable by rotations, we deduce that Aεbelongs to M(Ω) and (Aε−A)/ε converges to (RA +ARt)χωin L∞(Ω,Ms N). Thus (RA +ARt)χω belongs to KA(M(Ω)). Using now that the set of skew-symmetric matrices is a vectorial space, condition (3.4), and the arbitrariness of ω, we deduce 2(RA):C=(RA +ARt):C= 0 a.e. in Ω. For i, j ∈{1,... ,N},i=j, we take in the above equation Ras the matrix Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 232 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ defined by Rlk =⎧ ⎪ ⎨ ⎪ ⎩ 1ifl=i, k =j, −1ifl=j, k =i, 0 in another case. Then we get (AC)ij −(CA)ij = 0 a.e. in Ω, i.e., Aand Ccommute, and then they are mutually diagonalizable. Remark 5.3. From Proposition 5.2, assuming that the matrix Cis known and that their eigenvalues are all different, we must look for the optimal solution Aon the set of matrices of M(Ω) which have the same eigenvectors as Ca.e. So it can be interesting to write condition (4.13) assuming that Bis also mutually diagonalizable with C.If we restrict ourselves to the spaces Wwhich are generated by eigenvectors of C,we get the following result: In the conditions of Proposition 5.2, if ci,ai,i∈{1,... ,N}, are, respectively, the eigenvalues of Cand A, then for every B∈M(Ω) mutually diagonalizable with Aand C, with eigenvalues b1,... ,b N,wehave N  i=1 ci(ai−bi) + min 1≤≤Nmin 1≤i1<···<i≤N   j=1 cij(aij−bij)2 bij+(−1)aij≥0.(5.1) We also note that by Remark 4.16, for A,B,Cas above, the condition (5.1) implies in particular (4.15). Assuming stronger hypotheses, we can improve Proposition 5.2. Proposition 5.4 below is related to a theorem due to Lewis [11], which applies to the optimization of a function h:Ms N→Rconvex and invariable by rotations (see also [12], where there is a review of results corresponding to optimization problems on symmetric matrices). Proposition 5.4. In the assumptions of Theorem 3.2, we assume M(Ω) invariable by rotations and at least one of the following hypotheses: (i) M(Ω) is convex. (ii) M(Ω) is H-closed and N≥3. (iii) M(Ω) is H-closed, N=2and k=1. Then there exists Q∈L∞(Ω,MN), with Q∈O Na.e. in Ω, such that QAQt= diag(a1,... ,a N), (5.2) QCQt= diag(c1,... ,c N), and a1≤···≤aN,c1≤···≤cN. Proof. From Proposition 5.2 there exists Q∈L∞(Ω,MN), with Q∈O Na.e. in Ω, such that (5.2) holds. Clearly, we can also assume c1≤···≤cNa.e. in Ω. We consider i, j ∈{1,... ,N},i=j, and we take L∈O N, defined by Lei=ej,Le j=−ei,Le l=el∀l=i, j. Since M(Ω) is invariable by rotations, the matrix B=(LQ)tdiag(a1,... ,a N)LQ belongs to M(Ω). Let us now see that if one of the hypotheses (i), (ii) or (iii) hold, then C:(A−B)≥0 a.e. in Ω.(5.3) Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 233 For this purpose, we define ˜ M(Ω) as the convex hull of {SASt/S∈L∞(Ω,MN),S∈O Na.e. in Ω}, if (i) holds and as the H-closure of this set in cases (ii) and (iii). Then ˜ M(Ω) is contained in M(Ω) and Abelongs to ˜ M(Ω), so Ais also a solution of (1.1) with M(Ω) replaced by ˜ M(Ω). In case (i), ˜ M(Ω) is convex and local, so from (4.1) and B−Ain KA(M(Ω)), we deduce (5.3). In cases (ii) and (iii), ˜ M(Ω) is local and H-closed. Moreover, if (ii) holds, then Ker(B−A)= 0 a.e. in Ω, while if we have (iii) then, from Theorem 4.19, Chas at least a nonpositive eigenvalue. So, in both situations, we deduce (5.3) from (4.17). From (5.3) we get 0≤C:(A−B)=ci(ai−aj)+cj(aj−ai)=(ci−cj)(ai−aj) a.e. in Ω. This finishes the proof of Proposition 5.4. Remark 5.5. As we have seen in the proof of Proposition 5.4, the order relation between the eigenvalues of Cand Ais a consequence of (5.3) with Bdefined as above. So if N= 2 and M(Ω) is H-closed, by Corollary 4.18, the thesis of Proposition 5.4 still holds on the set where Chas a least a nonnegative eigenvalue. Where the two eigenvalues are positive, assuming c1<c 2, the condition (4.15) implies a1<a 2or a1>a 2and c2a2≤c1a1.(5.4) Related to this inequality, we also remark that if in the conditions of Proposition 5.4 (ii) there are two eigenvalues of C,ci,cjsuch that ci≤cj≤0, then besides ai≤aj,wehave|cj|aj≤|ci|ai. 6. Applications. In this section let us show how the condition (3.4) and the consequences we have obtained from it can be used, in some cases, to obtain A∇yi, A∇pi,1≤i≤k, as explicit functions of ∇yi,∇piand then, from (1.2) and (3.2), to reduce the optimality conditions given in Theorem 3.2 to a nonlinear system in ∇yi, ∇pi. The main difficulty in carrying out this idea is that obtaining the H-closure of a subset of L∞(Ω,MN) is a very difficult problem, which has only been solved in some particular cases (see [23], [14], [13], [8], [16], [25]). To simplify the exposition, we have chosen two simple problems where the H-closure is well known. The first consists of the mixture of two homogeneous isotropic materials in dimension two (the problem can also be studied analogously for higher dimensions). This problem has also been studied in [3] and [7]. In this case the set M(Ω) is convex. In the second problem we consider a nonconvex situation corresponding to a polycrystal in dimension two. First problem. We start by recalling the following result which has been proved in [23] and [14]. Theorem 6.1. We assume N=2.For0<α≤βand θ∈L∞(Ω) with 0≤θ≤1 a.e. in Ω, the set Mθ(Ω) of the H-limits of the sequences αI χωn+β(1−χωn)I, such that ωn⊂Ωare measurable sets and satisfy χωnconverges weakly-∗in L∞(Ω) to θ, is characterized as follows: Mθ⊂L∞(Ω,Ms 2)is the set of matrices such that their eigenvalues λ1,λ2satisfy Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 234 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ the following inequalities a.e. in Ω: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ−≤λ1,λ 2≤λ+, 2  i=1 1 λi−α≤1 λ−−α+1 λ+−α, 2  i=1 1 β−λi≤1 β−λ−+1 β−λ+, (6.1) where λ−,λ+are given by λ+=αθ +β(1 −θ),λ −=θ α+1−θ β−1 .(6.2) Let us see how the results obtained in the previous sections permit us to obtain Afrom C, and then from ∇yi,∇pi,1≤i≤k, for some choices of M(Ω) related to Mθ(Ω). In the following, we define ˇ Ω={x∈Ω/(c1(x),c 2(x)) =(0,0)},(6.3) where c1,c2are the eigenvalues of the matrix C. Proposition 6.2. In the assumptions of Theorem 3.2,ifM(Ω) is the set of matrices defined in Theorem 6.1 for a fixed function θand c1,c2,c1≤c2, are the eigenvalues of C, then there exists an associated basis {µ1,µ 2}of eigenvectors of C such that a.e. in Ω, we have Aµi=aiµi,i=1,2,(6.4) where a.e. in ˇ Ω, the functions a1,a2are given by the following: If c2<0and c2 c1≥α α+(β−α)θ⇒⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ a1=α+α(β−α)(1 −θ) 2α+(β−α)θ1+c2 c1, a2=α+α(β−α)(1 −θ) 2α+(β−α)θ1+c1 c2. If c1>0and c2 c1≤β α+(β−α)θ⇒⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ a1=β−β(β−α)θ α+β+(β−α)θ1+c2 c1, a2=β−β(β−α)θ α+β+(β−α)θ1+c1 c2. In another case a1=λ−,a2=λ+, where λ−,λ+are defined by (6.2). Proof. In this case, the set M(Ω) is local, convex, and invariable by rotations. So we can apply Proposition 5.4 and (4.14), which imply that a.e. in Ω, Asatisfies (6.4) for a basis of eigenvectors of C, and a1c1+a2c2is the maximum of λ1c1+λ2c2, with λ1,λ2in the set defined by (6.1). Solving this maximum problem we get the expressions a1and a2given in Proposition 6.2. The above result assumes that θis known, but clearly this is not a realistic situation. Next, we consider two examples where θalso varies. In the first one we impose the condition 1 |Ω|Ω θdx=s,(6.5) Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 235 with s∈(0,1). This means we know the proportion of the materials defined by αand βbut not its local distribution. This usually holds when one material is better than the other but it is also more expensive. In the second situation we consider the case where we do not have any restriction on θ. Proposition 6.3. In the assumptions of Theorem 3.2, if for s∈(0,1) given, M(Ω) is the set of matrices defined in Theorem 6.1, with θ∈L∞(Ω),0≤θ≤1a.e. in Ωand such that (6.5) holds, then the matrix Asatisfies the thesis of Proposition 6.2, where the corresponding function θis such that defining F∈L∞(ˇ Ω) by F(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α(β2−α2)(√−c1−√−c2)2 (2α+(β−α)θ)2if c2<0and c2 c1≥α α+(β−α)θ, −β(β2−α2)(√c1+√c2)2 (α+β+(β−α)θ)2if c1>0and c2 c1≤β α+(β−α)θ, −(β−α)αβ (α+(β−α)θ)2c1+c2in another case, there exists r∈Rwhich satisfies ⎧ ⎨ ⎩ F(x)≤ra.e. in {θ=0}∩ˇ Ω, F(x)=ra.e. in {0<θ<1}∩ˇ Ω, F(x)≥ra.e. in {θ=1}∩ˇ Ω. (6.6) Proof. We remark that if θis the corresponding function associated with Ain the definition of the elements of M(Ω) and we define Mθ(Ω) as in the statement of Theorem 6.1, then Ais also the solution of (1.1) with M(Ω) replaced by Mθ(Ω). So Proposition 6.2 applies. Let us now vary θ. For every θ∗∈L∞(Ω) such that 0 ≤θ∗≤1 a.e. in Ω, 1 |Ω|Ω θ∗dx =s, and θ∗=θa.e. in Ω \ˇ Ω, we define Aθ∗∈M(Ω) as the function given by Proposition 6.2 applied to θ∗a.e. in ˇ Ω and Aθ∗=Aa.e. in Ω \ˇ Ω. Deriving Aθ∗with respect to θ∗, we obtain an admissible direction. Then, using condition (3.4), we get Ω Fϑdx≤0(6.7) for every ϑ∈L∞(Ω) such that ϑ= 0 a.e. in Ω \ˇ Ω, ϑ≥0 a.e. in {θ=0}∩ˇ Ω, ϑ≤0 a.e. in {θ=1}∩ˇ Ω, and Ω ϑdx=0. It is easy to check that this implies the existence of r∈R, which satisfies the statement of the proposition. Remark 6.4. The expression of Fis strictly decreasing with respect to θ. Then, from (6.6), it is possible to obtain θas a function of c1,c2and r. Remark 6.5. In Proposition 6.3, if r≥0, then θ= 0 a.e. in the set {c1>0}c2 c1≤β α∩ˇ Ω. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 236 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ Analogously, if r≤0, then θ= 1 a.e. in the set {c2<0}c2 c1≥α β∩ˇ Ω. We finish with the following result. Proposition 6.6. In the assumptions of Theorem 3.2, let M(Ω) be the set of matrices defined in Theorem 6.1, where θis any function in L∞(Ω) such that 0≤θ≤1a.e. in Ω, and denote by c1,c2,c1≤c2, the eigenvalues of C. Then there exists an associated basis {µ1,µ 2}of eigenvectors of Csuch that a.e. in Ωwe have (6.4). Moreover, a.e. in ˇ Ω, the functions a1,a2are given by the following: If 0≤c1≤c2, then a1=a2=β. If c1≤c2≤0, then a1=a2=α. If c1<0<c 2, then ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α β≥−c1 c2⇒a1=a2=β, β α>−c1 c2 >α β⇒a1=−αβc2 c1 ,a 2=α+β−−αβc1 c2 , −c1 c2≥β α⇒a1=a2=α. Proof. We proceed similarly to Proposition 6.3, but now, in the condition (6.7), the function ϑdoes not necessarily satisfy Ω ϑdx=0. This implies that the function Fgiven in Proposition 6.3 satisfies (6.6) with r=0, which easily gives the result. Second problem. Given a diagonal matrix Λ = diag(α, β) with 0 <α<β, let us now consider the optimization problem (1.1) when M(Ω) is the H-closure of the matrices of the form R(x)ΛR(x)t, where Ris measurable, and R(x) belongs to O2for a.e. x∈Ω (observe that to assume Λ diagonal is not a restriction). This set M(Ω) is known (see, e.g., [25], [16]) and agrees with the set of functions B∈L∞(Ω,Ms 2) such that for a.e. x∈Ω, the eigenvalues b1(x) and b2(x)ofB(x) satisfy α≤b1(x), b2(x)≤β,b1(x)b2(x)=αβ. For this choice of M(Ω), we have the following result. Proposition 6.7. In the assumptions of Theorem 3.2,ifc1and c2, with c1≤c2, are the eigenvalues of C, then there exists an associated basis {µ1,µ 2}of eigenvectors of Csuch that a.e. in Ω, we have Aµi=aiµi,i=1,2,(6.8) where a.e. in the set ˇ Ωdefined by (6.3), the functions a1,a2are given by c2<0and c2 c1 >α β⇒⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a1=αβc2 c1 , a2=αβc1 c2 . Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 237 If c1>0and c2 c1≤β α, then there exist three possibilites: a1=α, a2=β, or a1=β, a2=α, or ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a1=√αβc2 c1 , a2=√αβc1 c2 . In another case a1=α,a2=β. Proof. Since the set M(Ω) is invariable by rotations, we can apply Proposition 5.2 to deduce that for a.e. x∈Ω, there exists a basis {µ1(x),µ 2(x)}of R2such that C(x)µi(x)=ci(x)µi(x),A(x)µi(x)=ai(x)µi(x),i=1,2, with c1(x)≤c2(x), a1(x),a 2(x)∈R. From the definition of M(Ω), we also have that for a.e. x∈Ω, there exists t∗(x)∈[1,β α] such that a1(x)=αt∗(x),a 2(x)= β t∗(x). Moreover, from (5.1), we deduce that t∗satisfies (t∗(x)−t)αc1(x)−βc2(x) t∗(x)t (6.9) +(t∗(x)−t)2min αc1(x) t,βc2(x) (t∗(x))2t,1 t∗(x)+tαc1(x)+βc2(x) t∗(x)t≥0 for every t∈[1,β α] and a.e. x∈Ω. In the set where t∗(x)=1,wehavet∗(x)−t<0 for every t∈(1,β α]. So, dividing by 1 −tand taking tconverging to 1 on the right, we deduce αc1−βc2≤0 a.e. in {x∈Ω: t∗(x)=1}.(6.10) Analogously, we deduce βc1−αc2≥0 a.e. in x∈Ω: t∗(x)=β α,(6.11) αc1−βc2 (t∗(x))2= 0 a.e. in x∈Ω: 1<t ∗(x)<β α,(6.12) where the statement (6.12) implies c1c2>0,t ∗=βc2 αc1 a.e. in x∈ˇ Ω: 1<t ∗(x)<β α.(6.13) Analyzing the different cases which appear depending on the sign of c1or c2, we easily conclude from (6.10), (6.11), and (6.13) the proof of Proposition 6.7. Remark 6.8. We have deduced (6.10), (6.11), and (6.13) from inequality (6.9). One could conjecture that this inequality gives, in fact, more information. However, a simple calculus shows that the statements (6.10), (6.11), and (6.13) also imply (6.9). Remark 6.9. Proposition 6.7 does not give the expressions of a1and a2in the set where c1>0 and c2 c1≤β α; it gives three possibilities. The possibility a1=α,a2=β Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 238 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ seems to be the most natural in order to stick continuously with the values of a1and a2in the other zones. We also note that if M(Ω) was convex then, using that B−A is an admissible direction for every B∈M(Ω), we should obtain in place of (6.9) that αt∗+β t∗= max αt +β t:t∈1,β α, which implies that t∗= 1 (and then a1=α,a2=β) a.e. on the set where c1>0 and c2 c1≤β αas well as the expressions of a1and a2given in Proposition 6.7 in the other cases. However, since M(Ω) is not convex, this reasoning is not good and thus the only conclusion we obtain is that stated in Proposition 6.7. REFERENCES [1] V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin,Optimal Control, Consultants Bureau, New York, 1987. [2] G. Allaire,Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), pp. 1482–1518. [3] G. Allaire,Shape Optimization by the Homogenization Method, Appl. Math. Sci. 146, Springer-Verlag, New York, 2002. [4] H. Attouch,Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, London, 1984. [5] A. Bensoussan, J. L. Lions, and G. Papanicolau,Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North–Holland, Amsterdam, 1978. [6] J. Casado D´ ıaz, J. Couce Calvo, and J. D. Mart´ ın G´ omez,Sobre el control de la matriz de difusi´on, in Actas encuentro de matem´aticos andaluces (Sevilla, 13–17 November 2000), E. Briales-Morales, A. Carriazo-Rubio, T. Chac´on-Rebollo, P. Real-Jurado, and A. Romero- Jim´enez, eds., Publicaciones de la Universidad de Sevilla, 2001, pp. 237–244. [7] J. Casado D´ ıaz, J. Couce Calvo, and J. D. Mart´ ın G´ omez,Sobre la identificaci´on de la matriz de difusi´on mediante varios experimentos, in Actas XVII CEDYA, VII CMA (Salamanca, 24–28 September 2001), L. Ferragut and A. Santos, eds., 2001, CD-ROM. [8] A. V. Cherkaev,Variational Methods for Structural Optimization, Appl. Math. Sci. 140, Springer-Verlag, New York, 2000. [9] J. Deny and J. L. Lions,Les espaces de Beppo Levi, Ann. Inst. Fourier (Grenoble), 5 (1953– 1954), pp. 305–370. [10] G. Francfort and G. W. Milton,Sets of conductivity and elasticity tensors stable under lamination, Comm. Pure Appl. Math., 47 (1994), pp. 257–279. [11] A. S. Lewis,Convex analysis on the Hermitian matrices, SIAM J. Optim., 6 (1996), pp. 164–177. [12] A. S. Lewis and M. L. Overton,Eigenvalue optimization, in Acta Numerica, Acta Numer. 5, Cambridge University Press, Cambridge, UK, 1996, pp. 149–190. [13] K. A. Lurie,Applied Optimal Control Theory of Distributed Systems, Plenum Press, New York, 1993. [14] K. A. Lurie and A. V. Cherkaev,Exact estimates of the conductivity of a binary mixture of isotropic materials, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), pp. 21–38. [15] G. W. Milton,A link between sets of tensors stable under lamination and quasiconvexity, Comm. Pure Appl. Math., 47 (1994), pp. 959–1003. [16] G. W. Milton,The Theory of Composites, Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, UK, 2002. [17] F. Murat,Th´eor`emes de non-existence pour des probl`emes de contrˆole dans le coefficients,C. R. Acad. Sci. Paris S´er. A-B, 274 (1972), pp. 395–398. [18] F. Murat and L. Tartar,On the control of coefficients in partial differential equations,in Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, A. Cherkaev and R. Kohn, eds., Birkh¨auser Boston, Boston, 1997, pp. 1–8. [19] F. Murat and L. Tartar,H-convergence, in Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, A. Cherkaev and R. Kohn, eds., Birkh¨auser Boston, Boston, 1997, pp. 21–43. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 239 [20] F. Murat and L. Tartar,Calculus of variations and homogenization, in Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, A. Cherkaev and R. Kohn, eds., Birkh¨auser Boston, Boston, 1997, pp. 139–173. [21] U. Raitums,On the local representation of G-closure, Arch. Ration. Mech. Anal., 158 (2001), pp. 213–234. [22] S. Spagnolo,Sulla convergenza di soluzioni di equazione paraboliche ed elliptiche, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), pp. 571–597. [23] L. Tartar,Estimations fines de coefficients homog´en´eis´es, in Ennio de Giorgi colloquium (Paris, 1983), Res. Notes Math. 125, P. Kree, ed., Pitman, London, 1985, pp. 168–187. [24] L. Tartar,Remarks on optimal design problems, in Calculus of Variations, Homogenization and Continuum Mechanics, Ser. Adv. Math. Appl. Sci. 18, G. Bouchitt´e, G. Buttazzo, and P. Suquet, eds., World Scientific, Singapore, 1994, pp. 279–296. [25] L. Tartar,An Introduction to the homogenization method in optimal design, in Optimal Shape Design (CIM/CIME, Summer School, Trˆoia, 1–6 June 1998), Lecture Notes in Math. 1740, A. Cellina and A. Ornelas, eds., Springer-Verlag, Berlin, 2000, pp. 47–156. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php