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 Socie y o Indus ial and Applied Ma hema ics Vol. 43, No. 1, pp. 216–239 Abs ac . The pu pose o his pape is o a ain some op imali y condi ions o he iden ifica ion o a diffusion ma ix (ma e ial) unde se e al es ic ions. Assuming ha he se o such diffusion ma ices is closed o he H-con e gence, we gi e a me hod o ob ain admissible di ec ions which applies o a no -necessa ily con ex con ol se . Ou esul s pe mi ob aining he diffusion ma ix om he s a e unc ions. Key wo ds. con ol in he coefficien s, admissible di ec ions, noncon ex con ol se AMS subjec classifica ion. 49K20 DOI. 10.1137/S0363012902411714 1. In oduc ion. The p oblem we conside in he p esen pape is ela ed o he choice o an op imal ma e ial unde se e al condi ions, o he iden ifica ion o a ma e ial om a fini e numbe o obse a ions. In a ma hema ical se ing, we ha e he model p oblem min A∈M(Ω) J(y1,... ,y k),(1.1) whe e yi=yi(A), 1 ≤i≤k, a e he solu ions o he equa ions −di A∇yi= iin D(Ω), yi∈H1 0(Ω),1≤i≤k. (1.2) He e Ω is a bounded open se o RN,Jis a smoo h objec i e unc ional in H1 0(Ω)k, 1,... , ka e kfixed elemen s o H−1(Ω), and M(Ω) is a gi en se o measu able unc ions wi h alues in he space o symme ic ma ices o o de N. The elemen s o M(Ω) a e uni o mly ellip ic and bounded. Clea ly, o he gene aliza ions can be conside ed: Jdepending on A, o he bounda y condi ions, e c. A physical example is he iden ifica ion o a ma e ial. Fo his pu pose, we apply a fini e numbe ko ex e nal condi ions (in ou case hey a e ep esen ed by i) and in each case we ealize a measu e o he co esponding s a e. Fo example, we gi e he alue zio he s a e in a subse ω⊂Ω. Then he p oblem can be o mula ed as min k  i=1 ω|yi−zi|2dx, whe e yia e he solu ions o (1.2). Assuming Jis sequen ial lowe semicon inuous o he weak opology o H1 0(Ω)k and M(Ω) is closed o he H-con e gence o he G-con e gence, because we a e ∗Recei ed by he edi o s July 22, 2002; accep ed o publica ion (in e ised o m) No embe 24, 2003; published elec onically June 15, 2004. This wo k has been pa ially suppo ed by he p ojec s BFM 2002-00672 o he D.G.I. o he “Minis e io de Ciencia y Tecnolog´ıa” o Spain and FQM309 o he “Jun a de Andaluc´ıa.” h p://www.siam.o g/jou nals/sicon/43-1/41171.h ml †Dep o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Fac. de Ma em´a icas, Uni e sidad de Se illa, C. Ta fia s/n, 41012 Se illa, Spain ([email p o ec ed], [email p o ec ed], [email p o ec ed]). 216 Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 217 wo king wi h symme ic ma ices and hus he wo concep s a e equi alen (see, e.g., [22], [19], [18], [4], [25]), i is well known ha (1.1) has, a leas , a solu ion (see, e.g., [25], [17]). I M(Ω) does no sa is y his las condi ion and Jis sequen ial con inuous o he weak opology o H1 0(Ω)k, we can ob ain a elaxed p oblem eplacing M(Ω) by i s H-closu e. Thus, i is na u al o assume M(Ω) is H-closed. The calculus o he H-closu e o a se is a e y difficul p oblem and he e a e a lo o wo ks in his field (see, e.g., [23], [14], [13], [8], [25], [3], [16] and he e e ences in hem). In his pape we a e in e es ed in ob aining some necessa y condi ions which mus sa is y he op imal solu ion o (1.1). Fo k= 1, he p oblem has been s udied in [13], [20], [8], [25], [3]. Fo k>1, he e a e ew esul s o ou knowledge (see [25], [6], [7], [3]). The pape is o ganized as ollows: In sec ion 3 we gi e a defini ion o admissible di ec ion (see Defini ion 3.1). Then we p o e ha i Ais a solu ion o (1.1), y1,... ,y k he co esponding s a e unc ions, p1,... ,p k he solu ions o (3.2) ( he adjoin s a es), and C he ma ix defined by (3.3), we ha e Ω H:Cdx≤0(1.3) o e e y admissible di ec ion H. Rela ed esul s can be ound, o example, in [8], [13], [25], [3]. Howe e , in hese pape s, he admissible di ec ions a e o he o m H=B−A, wi h Bin M(Ω), which needs some con exi y assump ions. When k≤N−1 (in pa icula k= 1) and M(Ω) is ob ained by homogeniza ion, mixing a fini e numbe o ma ices wi h fixed p opo ions, a esul o Ta a (see [25]) shows ha al hough M(Ω) is no con ex, o e e y ξ1,... ,ξ k∈RN, he se {(Bξ1,... ,Bξ k)/B∈M(Ω)}⊂L∞(Ω)k (1.4) is con ex, and hus he di ec ions H=B−Acan s ill be conside ed. Howe e , his is no ue o k≥N(o , in p inciple, o o he choices o M(Ω) e en i k≤N−1). This is he eason we ha e gi en a mo e gene al defini ion o admissible di ec ion. In sec ion 4, assuming M(Ω) local (see Defini ion 4.1) and closed o he H- con e gence, we gi e an o iginal me hod o find admissible di ec ions ollowing ou defini ion. As a consequence, we ob ain he main esul o he pape , Theo em 4.5, whe e we p o e ha o e e y A, B ∈M(Ω), l∈{1,... ,N},W⊂RNlinea subspace o dimension l, and e e y bounded measu able se To W, wi h l-dimensional posi i e measu e, he ma ix Hdefined by H(x)ei=(B(x)−A(x)) ei+1 |T|T∇W zˆwi(x, z)d(z) (1.5) is an admissible di ec ion in A. He e e1,... ,e Nis he s anda d basis o RN,∇W z deno es he g adien wi h espec o W, and ˆwiis he solu ion o he pa ial diffe en ial p oblem gi en by (4.4). This di ec ion has he difficul y ha ˆwi(and hen H) canno be explici ly ob ained. Howe e , we hink ha i can be in e es ing, o example, o apply a descen me hod in o de o sol e nume ically p oblem (1.1), whe e we can ob ain ˆwinume ically. Rela ed o his poin , an in e es ing ques ion, one ha we wan o s udy in he u u e, is he op imal choice o Wand T o ob ain he s eepes descen di ec ion. A c i e ion o de e mine his di ec ion (see Rema k 4.9) can be o calcula e he maximum o H:Con he se o ma ices Hob ained by (1.5). Al hough, as we ha e said abo e, i is no possible in gene al o ob ain ˆwiexplic- i ly, we show in Theo em 4.12 ha his can be ca ied ou o a pa icula choice o Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 218 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ T(which p obably is no op imal). This pe mi s us o ob ain a amily o admissible di ec ion, depending on he subspace Wchosen. Essen ially, hey a e o he o m B−Aplus a e m which has a g ow h o o de wo in B−A o e e y B∈M(Ω). When he dimension o Wis equal o 1, he co esponding admissible di ec ion comes jus om a lamina ion. In his case he exp ession o His known and i can be ound, o example, in [25] (i can also be ob ained om he esul s in [10]), bu o ou knowledge i s u ili y o ob ain op imali y condi ions o p oblem (1.1) has no been exploi ed. Mos o he consequences we ob ain in he p esen pape using Theo em 4.12 use only, in ac , l= 1. Howe e , we show ha in some cases (see Rema k 4.17) i is be e o use a subspace o dimension g ea e han one. Using Theo em 4.12 we p o e in Co olla y 4.18 ha o e e y B∈M(Ω), he condi ion C:(A−B)≥0(1.6) (which is he condi ion we find i he admissible di ec ions a e o he o m B−A) is s ill ue on he se whe e Chas a nonposi i e eigen alue o whe e Ke (A−B)={0}.In pa icula (see Co olla y 4.20) he condi ion (1.6) holds a.e. in Ω o e e y B∈M(Ω) when k≤N−1. When M(Ω) comes om he mix u e o a fini e numbe o ma e ials wi h fixed p opo ions, his esul can also be ob ained om he con exi y o he se defined by (1.4), bu we no e ha ou se M(Ω) is mo e gene al. In sec ion 5 we s udy he case whe e M(Ω) is in a iable by o a ions, which is a na u al assump ion in he applica ions. Then we show ha condi ion (1.3) implies ha Cand Aa e mu ually diagonalizable a.e. in Ω. Mo eo e , assuming u he hypo heses (in pa icula i M(Ω) is H-closed and N≥3), we p o e in P oposi ion 5.4 ha he eigen alues o Aand Ca e mu ually o de ed. As applica ion o he esul s s a ed abo e, i is possible o ob ain, in some si u- a ions, he ma ix A om Cand hen o educe he se o op imali y condi ions o a nonlinea pa ial diffe en ial sys em wi h a iables yi,pi,1≤i≤k. The main p oblem o ca ying ou his poin is ha in gene al, he H-closu e o a gi en se is unknown. In sec ion 6, we apply ou esul s o wo examples: The fi s one is he mix u e o wo homogeneous iso opic ma e ials, which has also been s udied in [3] (see also [25] o k= 1). In his case M(Ω) is con ex. In second p oblem we conside a polyc ys al in dimension 2, whe e M(Ω) is no con ex. 2. No a ion. Fo a linea subspace W⊂RN, we define L(W, W ) as he space o he linea applica ions om Win o Wand by Ls(W, W ) he subspace o he symme ic applica ions. When W=RNwe w i e MN=L(RN,RN), Ms N=Ls(RN,RN). The o hogonal p ojec ion o RNin o Wis deno ed by PW. Fo a ma ix A∈M N, we define AW∈L(W, W )byAW=PWA|W. The o hogonal subspace o Wis deno ed by W⊥. Fo u:W→R, we deno e ∇Wu:W→W he g adien o uwi h espec o W, i.e., ∇Wuis defined by ∇Wuξ=Dξu∀ξ∈W, whe e Dξuis he de i a i e o uin he di ec ion ξ. We deno e by {e1,... ,e N} he s anda d basis o RN. The g oup o he o hogonal ma ices in RNo de e minan 1 is deno ed by ON. The scala p oduc o wo ma ices A, B ∈M Nis w i en A:B. The enso ial p oduc o wo ec o s ξ,η ∈RNis deno ed as ξ⊗η. Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 219 Fo a bounded open se Ω ⊂RN, we deno e by M(Ω) a fixed subse o he space L∞(Ω,Ms N) such ha he e exis α, β > 0 which sa is y B(x)ξξ ≥α|ξ|2,|B(x)ξ|≤β|ξ|∀B∈M(Ω) a.e. x∈Ω.(2.1) Fo a ma ix A∈M(Ω), KA(M(Ω)) and ¯ KA(M(Ω)) a e he cones o admissible di ec ions o M(Ω) in A; see Defini ion 3.1. Fo T⊂RNand ∈(0,N], we deno e by |T| he -dimensional Hausdo ff measu e o T. The in eg al o a unc ion u:T→R, wi h espec o he -dimensional Hausdo ff measu e, is w i en T u(z)dz. When =N, we simpli y he no a ion by w i ing |T|and T u(z)dz. We use he subindex  o mean pe iodici y. Fo example, o a cube Y⊂RN, H1 (Y) is he space o unc ions o H1 loc(RN) which a e Y-pe iodic. 3. Op imali y condi ions. In his sec ion we in oduce he defini ion o admis- sible di ec ion. Using i , we ob ain he fi s op imali y esul o he con ol p oblem (1.1). De ini ion 3.1. Fo A∈M(Ω), le us define he cone o admissible di ec ions ¯ KA(M(Ω)) as he closu e in he weak-∗ opology o L∞(Ω,Ms N)o he se KA(M(Ω)), whe e KA(M(Ω)) is he se o H∈L∞(Ω,Ms N)such ha he e exis a cons an c>0 and Aε∈M(Ω),ε>0, such ha ⎧ ⎨ ⎩ Aε−AL∞(Ω,Ms N)≤cε, lim ε→0 Aε−A ε=Ha.e. in Ω. (3.1) Theo em 3.2. We conside J:H1 0(Ω)k→R,F ´eche de i able, 1,... , k∈ H−1(Ω)k.Le A∈M(Ω) be a solu ion o (1.1) and y1,... ,y k he solu ions o (1.2). We define he adjoin s a es p1,... ,p kas he solu ions o −di (A∇pi)=∂iJ(y1,... ,y k)in D(Ω), pi∈H1 0(Ω),1≤i≤k, (3.2) and he ma ix C∈L1(Ω,Ms N)by C=1 2 k  i=1 (∇yi⊗∇pi+∇pi⊗∇yi).(3.3) Then we ha e Ω H:Cdx≤0∀H∈¯ KA(M(Ω)).(3.4) Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 220 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ P oo . Le us fi s p o e he esul o H∈KA(M(Ω)). Fo ε>0 small enough, we define y∗ i,ε,1≤i≤k, as he solu ion o −di ((A+εH)∇y∗ i,ε)= iin D(Ω), y∗ i,ε ∈H1 0(Ω). (3.5) Then i is easy o check ha o 1 ≤i≤k,weha e lim ε→0 y∗ i,ε −yi ε=˙yiin H1 0(Ω),(3.6) wi h ˙yi he solu ions o −di (A∇˙yi+H∇yi)=0 inD(Ω), ˙yi∈H1 0(Ω). (3.7) Now, o ε>0, we conside Aε∈M(Ω) in he condi ions o (3.1). Then o 1≤i≤k, we define yi,ε as he solu ions o −di (Aε∇yi,ε)= iin D(Ω), yi,ε ∈H1 0(Ω). (3.8) Taking y∗ i,ε −yi,ε as es unc ion in he diffe ence o (3.5) and (3.8), and di iding by ε,wege 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 he ellip ici y o A+εH ( o εsmall enough) and (3.1), we deduce om (3.9) he exis ence o c>0 such ha 1 εy∗ i,ε −yi,ε2 H1 0(Ω) ≤cy∗ i,ε −yi,εH1 0(Ω)µεL2(Ω), wi h µε=Aε−(A+εH) ε∇y∗ i,ε. F om (3.1), (3.6), and he Lebesgue-domina ed con e gence heo em, we deduce ha µεcon e ges s ongly o ze o 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 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 221 On he o he hand, since Ais a solu ion o (1.1) and Aε∈M(Ω), we ha e J(yε)−J(y) ε≥0∀ε>0, wi h yε=(y1,ε,... ,y k,ε) and y=(y1,... ,y k). F om (3.10) and he F ´eche de i - abili y o J,wege k  i=1∂iJ(y),˙yi= lim ε→0 J(yε)−J(y) ε≥0.(3.11) Bu aking ˙yias he es unc ion in (3.2) and pias he es unc ion in (3.7), we ha e k  i=1∂iJ(y),˙yi= k  i=1 Ω A∇pi∇˙yidx =− k  i=1 Ω H∇yi∇pidx. This p o es Ω H:Cdx= k  i=1 Ω H∇yi∇pidx ≤0∀H∈KA(M(Ω)).(3.12) Now le Hbe in ¯ KA(M(Ω)). Fo δ>0 we define Gδ=M∈L∞(Ω,Ms N)/Ω C:(M−H)dx <δ . Since Gδis a neighbo hood o Hin he weak-∗ opology o L∞(Ω,Ms N), he e exis s Hδin Gδ∩KA(M(Ω)) and hen, om (3.12), we ge Ω C:Hdx=Ω C:Hδdx +Ω C:(H−Hδ)dx<δ o e e y δ>0. This p o es (3.4). Rema k 3.3. The abo e heo em is s ill ue i he elemen s o M(Ω) a e no necessa ily symme ic by changing A o A in he defini ion (3.2) o he unc ions pi and aking C= k  i=1 ∇pi⊗∇yi. Rema k 3.4. I M(Ω) is con ex, he condi ion (3.4) implies Ω A:Cdx= max Ω B:Cdx/B∈M(Ω).(3.13) Rema k 3.5. Theo em 3.2 s ill holds i we ake KA(M(Ω)) as he cone o ma ices H∈L∞(Ω,Ms N(Ω)) such ha o e e y sequence Φ1 ε,... ,Φk ε,which espec i ely con e ges in L2(Ω)N o Φ1,... ,Φk, he e exis s Aε∈M(Ω) such ha Aε−A εΦi ε→HΦiin L2(Ω)N,i=1,... ,k. Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 222 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ The ad an age o his defini ion is he ollowing: I M(Ω) is he se we ob ain by mixing ma e ials wi h p opo ions fixed, hen o e e y ξ1,... ,ξ N−1∈RN, he se {(Bξ1,... ,Bξ N−1)/B ∈M(Ω)}⊂L∞(Ω)N−1 is con ex (see [24], [25]). So, wi h his defini ion o KA(M(Ω)), he ma ices o he o m B−Abelong o KA(M(Ω)) i k≤N−1. Thus, (3.13) s ill holds in his case. La e we will deduce his esul (see Co olla y 4.20) o mo e gene al choices o M(Ω), using simply Defini ion 3.1 o admissible di ec ions. 4. Calculus o admissible di ec ions. In he ollowing, le us calcula e ex- plici ely some admissible di ec ions by imposing addi ional hypo heses abou M(Ω). De ini ion 4.1. We say ha M(Ω) is local i he e exis s a mul i alua ed appli- ca ion F:x∈Ω→F(x)⊂M s Nsuch ha M(Ω) = {B∈L∞(Ω,Ms N)/B(x)∈F(x)a.e. x∈Ω}, whe e Fis measu able in he sense ha {x∈Ω/F(x)∩G=∅} is measu able ∀G⊂M s Nopen. As i is p o ed in [21], he local p ope y is sa isfied in se e al ypical examples o M(Ω). A fi s consequence o assuming M(Ω) is local ollows. P oposi ion 4.2. We assume M(Ω) is local. We conside A∈M(Ω),H1,... , Hm∈KA(M(Ω)),ω1,... ,ω m⊂Ωmeasu able such ha |ωi∩ωj|=0i i=j. Then he ma ix H=m i=1 Hiχωibelongs o KA(M(Ω)). P oo . By Defini ion 3.1, o e e y i∈{1,... ,m} he e exis s Ai ε∈M(Ω) and c>0 (which can be aken independen o i) such ha Ai ε−AL∞(Ω,Ms N)≤cε, lim ε→0 Ai ε−A ε=Hia.e. in Ω. Taking hen Aε= m  i=1 Ai εχωi+AχΩ ∪m i=1ωi, which belongs o M(Ω) because M(Ω) is local, we ha e Aε−AL∞(Ω,Ms N)≤cε, lim ε→0 Aε−A ε=Ha.e. in Ω, and hen Hbelongs o KA(M(Ω)). Rema k 4.3. I is no difficul o show ha he abo e esul emains ue i we eplace KA(M(Ω)) by ¯ KA(M(Ω)). Using P oposi ion 4.2, we ge he ollowing. P oposi ion 4.4. In he assump ions o Theo em 3.2,i M(Ω) is local, we ha e H:C≤0a.e. in Ω∀H∈¯ KA(M(Ω)).(4.1) P oo . By P oposi ion 4.2, o e e y H∈KA(M(Ω)) and e e y ω⊂Ω measu able, he ma ix Hχωbelongs o KA(M(Ω)). So, using (3.4), we ge ω H:Cdx≤0.(4.2) Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 223 Since ¯ KA(M(Ω)) is he closu e o KA(M(Ω)) in he weak-∗ opology o L∞(Ω,Ms N), we deduce ha (4.2) holds, in ac , o e e y H∈¯ KA(M(Ω)) and e e y ω⊂Ω measu able, which implies (4.1). Le us now see how assuming M(Ω) is local pe mi s us o ob ain admissible di ec ions. Theo em 4.5. We suppose ha M(Ω) is local and closed o he H-con e gence. We conside a linea subspace W⊂RNo dimension and a measu able bounded subse T⊂Wsuch ha |T|is posi i e. Then, o e e y A, B ∈M(Ω), he ma ix 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) o 1≤i≤Nand a.e. x∈Ωbelongs o KA(M(Ω)).In(4.3), he unc ion 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 zˆ d (z) =T (A(x)−B(x))ei∇W zˆ d (z), ∀ˆ ∈H1 loc(W),∇W zˆ ∈L2(W, W )a.e. x∈Ω. (4.4) P oo . We conside A,B,W, and Tas in he s a emen o he heo em. Fo an o hono mal basis {e 1,... ,e  }o W, we deno e Y=  i=1 λie i/−1 2<λ i<1 2,1≤i≤⊂W. Fo ε>0 small enough, we deno e 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 o he H-con e gence, he ma ix Aεob ained by ak- ing, o εfixed, he H-limi when δ ends o ze o o he ma ices x→˜ Aε(x, 1 δPW(x)) belongs o M(Ω). Since he ma ices ˜ Aεa e a enso ial p oduc o unc ions which only depend on xand unc ions which only depend on y, i is well known (see, e.g., [5], [2]) ha Aεis gi en by Aε(x)ei=Y ˜ Aε(x, y)(∇W ywi,ε +ei)d(y),(4.5) whe e wi,ε is he unique solu ion o ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ wi,ε ∈L2(Ω,H1 (Y)/R), Y ˜ Aε(x, y)(∇W ywi,ε(x, y)+ei)∇W y (y)d(y)=0 ∀ ∈H1 (Y)/Ra.e. x∈Ω. (4.6) Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 224 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ Le us s udy he asymp o ic beha io o Aε. Fi s , we ema k ha o e e y ∈ H1 (Y)/Rand a.e. x∈Ω, we ha e Y ˜ Aε(x, y)ei∇W y (y)d(y)=Y A(x)ei∇W y (y)d(y) +Tε (B(x)−A(x))ei∇W y (y)d(y), bu o a.e. x∈Ω, he fi s e m on he igh -hand side anishes. So, wi,ε sa isfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Y ˜ Aε(x, y)∇W ywi,ε(x, y)∇W y (y)d(y) =Tε (A(x)−B(x))ei∇W y (y)d(y) ∀ ∈H1 (Y)/Ra.e. x∈Ω. (4.7) Fo 1 ≤i≤N, we ake wi,ε as he es unc ion in (4.7). Then we ge 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) o a.e. x∈Ω. Using hen (2.1) and he Cauchy–Schwa z inequali y, we deduce he e exis s c>0 such ha Y|∇W ywi,ε(x, y)|2d(y)≤c|Tε|=cε(4.8) o 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). F om (4.8), we deduce ha ∇W zˆwi,ε χε−1 Yis bounded in L∞(Ω,L 2(W, W )). So he e exis s a subsequence o ε, which we s ill deno e by ε, which con e ges weak-∗in L∞(Ω,L 2(W, W )). Since he cu l o he limi is ze o, i is he g adien o a unc ion ˆwi∈L∞(Ω,H1 loc(W)). Once we p o e ha ˆwiis he solu ion o (4.4), we conclude ha he whole o he sequence con e ges. We conside ˆ ∈D(W) and ε>0 small enough, such ha ε1 supp(ˆ )⊂Y, hen we define ε∈H1 (Y)by ε(y)=ε1 ˆ (ε−1 y) a.e. y∈Y. Taking εas he es unc ion in (4.7), using he change o a iables z=ε−1 y, and in eg a ing wi h espec o xin a measu able se ω,wege ωε−1 lY (A(x)χW T(z)+B(x)χT(z))∇W zˆwi,ε(x, z)∇W zˆ (z)d(z)dx =ωT (A(x)−B(x))ei∇W zˆ (z)d(z)dx. Passing o he limi in his equali y and aking in o accoun he a bi a iness o ˆ and ω, and he densi y o D(RN)/Rin he ac o space o unc ions wi h g adien in L2(RN) o e R(see, e.g., [9]), we show ha ˆwiis he solu ion o (4.4) o 1 ≤i≤N. Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 231 Now, i m−m−= 0, hen clea ly k+≥m−m−. In ano he case, we conside E= Span({φ− j/1≤j≤k}∪{φ− j,φ + j/1≤j≤k}⊥), which has dimension m−+N−m.I φ∈E⊥, hen, as abo e, s a emen (i) gi es ˜ Cjφφ ≥0, 1 ≤j≤k, and hen ˜ Cφφ ≥0. Mo eo e , ˜ Cφφ =0iff ˜ Cjφφ =0,1≤j≤k, which, by s a emen (i), implies ha φis o hogonal o φ− j,φ+ j,1≤j≤k; i.e., φis in Eand so φ= 0. Thus, aking i=m−+N−min (4.19) and using he compac ness o he uni a y ball in RN,we ge λm−+N−m+1 ≥min φ∈E⊥ |φ|=1 ˜ Cφφ > 0, and hus k+≥m−m−. The o he inequali ies in (4.18) ollow analogously. As a consequence we ge he ollowing. Co olla y 4.20. In he assump ions o Theo em 3.2,i M(Ω) is local and closed o he H-con e gence and k≤N−1, hen condi ion (4.14) holds. P oo . We apply Theo em 4.19 o ξi=∇yi(x), ηi=∇pi(x), 1 ≤i≤k, a.e. x∈Ω. In his case ˜ C=C(x). Since, clea ly, he numbe m+which appea s in his esul is less han o equal o k≤N−1, we deduce ha he numbe o posi i e eigen alues o Cis less han o equal o N−1, and hen he e exis s a leas a nonposi i e eigen alue o Ca.e. in Ω. Co olla y 4.18 gi es hen (4.14). 5. In a iabili y by o a ions. In he applica ions, i is a na u al hypo hesis o assume ha M(Ω) is in a iable by o a ions. We show in his sec ion ha his assump ion implies ha he eigen ec o s o Aand Cag ee. De ini ion 5.1. We say ha M(Ω) is in a iable by o a ions i o e e y B∈ M(Ω) and e e y Q∈L∞(Ω,MN), wi h Q∈O Na.e. in Ω, he ma ix QBQ belongs o M(Ω). We ha e he ollowing esul . P oposi ion 5.2. In he assump ions o Theo em 3.2,i M(Ω) is in a iable by o a ions, hen Aand Ca e mu ually diagonalizable a.e. in Ω. P oo . Le us fi s p o e ha gi en a skew-symme ic ma ix Rand a measu able se ω⊂Ω, he unc ion (RA +AR )χωbelongs o KA(M(Ω)). To his pu pose we define G:MN→M N×Rby G(M)=(MM ,de (M)). Since Ke (G(I)) coincides wi h he space o skew-symme ic ma ices, i is known (see, e.g., [1]) ha o ε∈R wi h |ε|small enough, he e exis s Pε∈M Nsuch ha G(Pε)=G(I) o , equi alen ly, Pε∈O N, and (Pε−I)/ε con e ges o R. Defining hen Aε=PεAP εχω+AχΩ ω and using ha M(Ω) is in a iable by o a ions, we deduce ha Aεbelongs o M(Ω) and (Aε−A)/ε con e ges o (RA +AR )χωin L∞(Ω,Ms N). Thus (RA +AR )χω belongs o KA(M(Ω)). Using now ha he se o skew-symme ic ma ices is a ec o ial space, condi ion (3.4), and he a bi a iness o ω, we deduce 2(RA):C=(RA +AR ):C= 0 a.e. in Ω. Fo i, j ∈{1,... ,N},i=j, we ake in he abo e equa ion Ras he ma ix Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 232 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ defined by Rlk =⎧ ⎪ ⎨ ⎪ ⎩ 1i l=i, k =j, −1i l=j, k =i, 0 in ano he case. Then we ge (AC)ij −(CA)ij = 0 a.e. in Ω, i.e., Aand Ccommu e, and hen hey a e mu ually diagonalizable. Rema k 5.3. F om P oposi ion 5.2, assuming ha he ma ix Cis known and ha hei eigen alues a e all diffe en , we mus look o he op imal solu ion Aon he se o ma ices o M(Ω) which ha e he same eigen ec o s as Ca.e. So i can be in e es ing o w i e condi ion (4.13) assuming ha Bis also mu ually diagonalizable wi h C.I we es ic ou sel es o he spaces Wwhich a e gene a ed by eigen ec o s o C,we ge he ollowing esul : In he condi ions o P oposi ion 5.2, i ci,ai,i∈{1,... ,N}, a e, espec i ely, he eigen alues o Cand A, hen o e e y B∈M(Ω) mu ually diagonalizable wi h Aand C, wi h eigen alues b1,... ,b N,weha e 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 no e ha by Rema k 4.16, o A,B,Cas abo e, he condi ion (5.1) implies in pa icula (4.15). Assuming s onge hypo heses, we can imp o e P oposi ion 5.2. P oposi ion 5.4 below is ela ed o a heo em due o Lewis [11], which applies o he op imiza ion o a unc ion h:Ms N→Rcon ex and in a iable by o a ions (see also [12], whe e he e is a e iew o esul s co esponding o op imiza ion p oblems on symme ic ma ices). P oposi ion 5.4. In he assump ions o Theo em 3.2, we assume M(Ω) in a i- able by o a ions and a leas one o he ollowing hypo heses: (i) M(Ω) is con ex. (ii) M(Ω) is H-closed and N≥3. (iii) M(Ω) is H-closed, N=2and k=1. Then he e exis s Q∈L∞(Ω,MN), wi h Q∈O Na.e. in Ω, such ha QAQ = diag(a1,... ,a N), (5.2) QCQ = diag(c1,... ,c N), and a1≤···≤aN,c1≤···≤cN. P oo . F om P oposi ion 5.2 he e exis s Q∈L∞(Ω,MN), wi h Q∈O Na.e. in Ω, such ha (5.2) holds. Clea ly, we can also assume c1≤···≤cNa.e. in Ω. We conside i, j ∈{1,... ,N},i=j, and we ake L∈O N, defined by Lei=ej,Le j=−ei,Le l=el∀l=i, j. Since M(Ω) is in a iable by o a ions, he ma ix B=(LQ) diag(a1,... ,a N)LQ belongs o M(Ω). Le us now see ha i one o he hypo heses (i), (ii) o (iii) hold, hen C:(A−B)≥0 a.e. in Ω.(5.3) Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 233 Fo his pu pose, we define ˜ M(Ω) as he con ex hull o {SAS /S∈L∞(Ω,MN),S∈O Na.e. in Ω}, i (i) holds and as he H-closu e o his se in cases (ii) and (iii). Then ˜ M(Ω) is con ained in M(Ω) and Abelongs o ˜ M(Ω), so Ais also a solu ion o (1.1) wi h M(Ω) eplaced by ˜ M(Ω). In case (i), ˜ M(Ω) is con ex and local, so om (4.1) and B−Ain KA(M(Ω)), we deduce (5.3). In cases (ii) and (iii), ˜ M(Ω) is local and H-closed. Mo eo e , i (ii) holds, hen Ke (B−A)= 0 a.e. in Ω, while i we ha e (iii) hen, om Theo em 4.19, Chas a leas a nonposi i e eigen alue. So, in bo h si ua ions, we deduce (5.3) om (4.17). F om (5.3) we ge 0≤C:(A−B)=ci(ai−aj)+cj(aj−ai)=(ci−cj)(ai−aj) a.e. in Ω. This finishes he p oo o P oposi ion 5.4. Rema k 5.5. As we ha e seen in he p oo o P oposi ion 5.4, he o de ela ion be ween he eigen alues o Cand Ais a consequence o (5.3) wi h Bdefined as abo e. So i N= 2 and M(Ω) is H-closed, by Co olla y 4.18, he hesis o P oposi ion 5.4 s ill holds on he se whe e Chas a leas a nonnega i e eigen alue. Whe e he wo eigen alues a e posi i e, assuming c1<c 2, he condi ion (4.15) implies a1<a 2o a1>a 2and c2a2≤c1a1.(5.4) Rela ed o his inequali y, we also ema k ha i in he condi ions o P oposi ion 5.4 (ii) he e a e wo eigen alues o C,ci,cjsuch ha ci≤cj≤0, hen besides ai≤aj,weha e|cj|aj≤|ci|ai. 6. Applica ions. In his sec ion le us show how he condi ion (3.4) and he consequences we ha e ob ained om i can be used, in some cases, o ob ain A∇yi, A∇pi,1≤i≤k, as explici unc ions o ∇yi,∇piand hen, om (1.2) and (3.2), o educe he op imali y condi ions gi en in Theo em 3.2 o a nonlinea sys em in ∇yi, ∇pi. The main difficul y in ca ying ou his idea is ha ob aining he H-closu e o a subse o L∞(Ω,MN) is a e y difficul p oblem, which has only been sol ed in some pa icula cases (see [23], [14], [13], [8], [16], [25]). To simpli y he exposi ion, we ha e chosen wo simple p oblems whe e he H-closu e is well known. The fi s consis s o he mix u e o wo homogeneous iso opic ma e ials in dimension wo ( he p oblem can also be s udied analogously o highe dimensions). This p oblem has also been s udied in [3] and [7]. In his case he se M(Ω) is con ex. In he second p oblem we conside a noncon ex si ua ion co esponding o a polyc ys al in dimension wo. Fi s p oblem. We s a by ecalling he ollowing esul which has been p o ed in [23] and [14]. Theo em 6.1. We assume N=2.Fo 0<α≤βand θ∈L∞(Ω) wi h 0≤θ≤1 a.e. in Ω, he se Mθ(Ω) o he H-limi s o he sequences αI χωn+β(1−χωn)I, such ha ωn⊂Ωa e measu able se s and sa is y χωncon e ges weakly-∗in L∞(Ω) o θ, is cha ac e ized as ollows: Mθ⊂L∞(Ω,Ms 2)is he se o ma ices such ha hei eigen alues λ1,λ2sa is y Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 234 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ he ollowing inequali ies 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) whe e λ−,λ+a e gi en by λ+=αθ +β(1 −θ),λ −=θ α+1−θ β−1 .(6.2) Le us see how he esul s ob ained in he p e ious sec ions pe mi us o ob ain A om C, and hen om ∇yi,∇pi,1≤i≤k, o some choices o M(Ω) ela ed o Mθ(Ω). In he ollowing, we define ˇ Ω={x∈Ω/(c1(x),c 2(x)) =(0,0)},(6.3) whe e c1,c2a e he eigen alues o he ma ix C. P oposi ion 6.2. In he assump ions o Theo em 3.2,i M(Ω) is he se o ma ices defined in Theo em 6.1 o a fixed unc ion θand c1,c2,c1≤c2, a e he eigen alues o C, hen he e exis s an associa ed basis {µ1,µ 2}o eigen ec o s o C such ha a.e. in Ω, we ha e Aµi=aiµi,i=1,2,(6.4) whe e a.e. in ˇ Ω, he unc ions a1,a2a e gi en by he ollowing: I 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. I 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 ano he case a1=λ−,a2=λ+, whe e λ−,λ+a e defined by (6.2). P oo . In his case, he se M(Ω) is local, con ex, and in a iable by o a ions. So we can apply P oposi ion 5.4 and (4.14), which imply ha a.e. in Ω, Asa isfies (6.4) o a basis o eigen ec o s o C, and a1c1+a2c2is he maximum o λ1c1+λ2c2, wi h λ1,λ2in he se defined by (6.1). Sol ing his maximum p oblem we ge he exp essions a1and a2gi en in P oposi ion 6.2. The abo e esul assumes ha θis known, bu clea ly his is no a ealis ic si ua ion. Nex , we conside wo examples whe e θalso a ies. In he fi s one we impose he condi ion 1 |Ω|Ω θdx=s,(6.5) Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 235 wi h s∈(0,1). This means we know he p opo ion o he ma e ials defined by αand βbu no i s local dis ibu ion. This usually holds when one ma e ial is be e han he o he bu i is also mo e expensi e. In he second si ua ion we conside he case whe e we do no ha e any es ic ion on θ. P oposi ion 6.3. In he assump ions o Theo em 3.2, i o s∈(0,1) gi en, M(Ω) is he se o ma ices defined in Theo em 6.1, wi h θ∈L∞(Ω),0≤θ≤1a.e. in Ωand such ha (6.5) holds, hen he ma ix Asa isfies he hesis o P oposi ion 6.2, whe e he co esponding unc ion θis such ha defining F∈L∞(ˇ Ω) by F(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α(β2−α2)(√−c1−√−c2)2 (2α+(β−α)θ)2i c2<0and c2 c1≥α α+(β−α)θ, −β(β2−α2)(√c1+√c2)2 (α+β+(β−α)θ)2i c1>0and c2 c1≤β α+(β−α)θ, −(β−α)αβ (α+(β−α)θ)2c1+c2in ano he case, he e exis s ∈Rwhich sa isfies ⎧ ⎨ ⎩ F(x)≤ a.e. in {θ=0}∩ˇ Ω, F(x)= a.e. in {0<θ<1}∩ˇ Ω, F(x)≥ a.e. in {θ=1}∩ˇ Ω. (6.6) P oo . We ema k ha i θis he co esponding unc ion associa ed wi h Ain he defini ion o he elemen s o M(Ω) and we define Mθ(Ω) as in he s a emen o Theo em 6.1, hen Ais also he solu ion o (1.1) wi h M(Ω) eplaced by Mθ(Ω). So P oposi ion 6.2 applies. Le us now a y θ. Fo e e y θ∗∈L∞(Ω) such ha 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 he unc ion gi en by P oposi ion 6.2 applied o θ∗a.e. in ˇ Ω and Aθ∗=Aa.e. in Ω ˇ Ω. De i ing Aθ∗wi h espec o θ∗, we ob ain an admissible di ec ion. Then, using condi ion (3.4), we ge Ω Fϑdx≤0(6.7) o e e y ϑ∈L∞(Ω) such ha 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. I is easy o check ha his implies he exis ence o ∈R, which sa isfies he s a emen o he p oposi ion. Rema k 6.4. The exp ession o Fis s ic ly dec easing wi h espec o θ. Then, om (6.6), i is possible o ob ain θas a unc ion o c1,c2and . Rema k 6.5. In P oposi ion 6.3, i ≥0, hen θ= 0 a.e. in he se {c1>0}c2 c1≤β α∩ˇ Ω. Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 236 J. CASADO-D´ IAZ, J. COUCE-CALVO, AND J. D. MART´ IN-G´ OMEZ Analogously, i ≤0, hen θ= 1 a.e. in he se {c2<0}c2 c1≥α β∩ˇ Ω. We finish wi h he ollowing esul . P oposi ion 6.6. In he assump ions o Theo em 3.2, le M(Ω) be he se o ma ices defined in Theo em 6.1, whe e θis any unc ion in L∞(Ω) such ha 0≤θ≤1a.e. in Ω, and deno e by c1,c2,c1≤c2, he eigen alues o C. Then he e exis s an associa ed basis {µ1,µ 2}o eigen ec o s o Csuch ha a.e. in Ωwe ha e (6.4). Mo eo e , a.e. in ˇ Ω, he unc ions a1,a2a e gi en by he ollowing: I 0≤c1≤c2, hen a1=a2=β. I c1≤c2≤0, hen a1=a2=α. I c1<0<c 2, hen ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 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. P oo . We p oceed simila ly o P oposi ion 6.3, bu now, in he condi ion (6.7), he unc ion ϑdoes no necessa ily sa is y Ω ϑdx=0. This implies ha he unc ion Fgi en in P oposi ion 6.3 sa isfies (6.6) wi h =0, which easily gi es he esul . Second p oblem. Gi en a diagonal ma ix Λ = diag(α, β) wi h 0 <α<β, le us now conside he op imiza ion p oblem (1.1) when M(Ω) is he H-closu e o he ma ices o he o m R(x)ΛR(x) , whe e Ris measu able, and R(x) belongs o O2 o a.e. x∈Ω (obse e ha o assume Λ diagonal is no a es ic ion). This se M(Ω) is known (see, e.g., [25], [16]) and ag ees wi h he se o unc ions B∈L∞(Ω,Ms 2) such ha o a.e. x∈Ω, he eigen alues b1(x) and b2(x)o B(x) sa is y α≤b1(x), b2(x)≤β,b1(x)b2(x)=αβ. Fo his choice o M(Ω), we ha e he ollowing esul . P oposi ion 6.7. In he assump ions o Theo em 3.2,i c1and c2, wi h c1≤c2, a e he eigen alues o C, hen he e exis s an associa ed basis {µ1,µ 2}o eigen ec o s o Csuch ha a.e. in Ω, we ha e Aµi=aiµi,i=1,2,(6.8) whe e a.e. in he se ˇ Ωdefined by (6.3), he unc ions a1,a2a e gi en by c2<0and c2 c1 >α β⇒⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a1=αβc2 c1 , a2=αβc1 c2 . Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php NONCONVEX CONTROL PROBLEMS IN THE COEFFICIENTS 237 I c1>0and c2 c1≤β α, hen he e exis h ee possibili es: a1=α, a2=β, o a1=β, a2=α, o ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a1=√αβc2 c1 , a2=√αβc1 c2 . In ano he case a1=α,a2=β. P oo . Since he se M(Ω) is in a iable by o a ions, we can apply P oposi ion 5.2 o deduce ha o a.e. x∈Ω, he e exis s a basis {µ1(x),µ 2(x)}o R2such ha C(x)µi(x)=ci(x)µi(x),A(x)µi(x)=ai(x)µi(x),i=1,2, wi h c1(x)≤c2(x), a1(x),a 2(x)∈R. F om he defini ion o M(Ω), we also ha e ha o a.e. x∈Ω, he e exis s ∗(x)∈[1,β α] such ha a1(x)=α ∗(x),a 2(x)= β ∗(x). Mo eo e , om (5.1), we deduce ha ∗sa isfies ( ∗(x)− )αc1(x)−βc2(x) ∗(x)  (6.9) +( ∗(x)− )2min αc1(x) ,βc2(x) ( ∗(x))2 ,1 ∗(x)+ αc1(x)+βc2(x) ∗(x) ≥0 o e e y ∈[1,β α] and a.e. x∈Ω. In he se whe e ∗(x)=1,weha e ∗(x)− <0 o e e y ∈(1,β α]. So, di iding by 1 − and aking con e ging o 1 on he igh , we deduce αc1−βc2≤0 a.e. in {x∈Ω: ∗(x)=1}.(6.10) Analogously, we deduce βc1−αc2≥0 a.e. in x∈Ω: ∗(x)=β α,(6.11) αc1−βc2 ( ∗(x))2= 0 a.e. in x∈Ω: 1< ∗(x)<β α,(6.12) whe e he s a emen (6.12) implies c1c2>0, ∗=βc2 αc1 a.e. in x∈ˇ Ω: 1< ∗(x)<β α.(6.13) Analyzing he diffe en cases which appea depending on he sign o c1o c2, we easily conclude om (6.10), (6.11), and (6.13) he p oo o P oposi ion 6.7. Rema k 6.8. We ha e deduced (6.10), (6.11), and (6.13) om inequali y (6.9). One could conjec u e ha his inequali y gi es, in ac , mo e in o ma ion. Howe e , a simple calculus shows ha he s a emen s (6.10), (6.11), and (6.13) also imply (6.9). Rema k 6.9. P oposi ion 6.7 does no gi e he exp essions o a1and a2in he se whe e c1>0 and c2 c1≤β α; i gi es h ee possibili ies. The possibili y a1=α,a2=β Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 238 J. 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Ta a ,An In oduc ion o he homogeniza ion me hod in op imal design, in Op imal Shape Design (CIM/CIME, Summe School, T ˆoia, 1–6 June 1998), Lec u e No es in Ma h. 1740, A. Cellina and A. O nelas, eds., Sp inge -Ve lag, Be lin, 2000, pp. 47–156. Downloaded 06/10/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php