Numerical approximation of a one-dimensional elliptic optimal design problem

NUMERICAL APPROXIMATION OF A ONE-DIMENSIONAL ELLIPTIC OPTIMAL DESIGN PROBLEM* J. CASADO-DÍAZ†, C. CASTRO‡, M. LUNA-LAYNEZ†,AND E. ZUAZUA§ Abs ac . We add ess he nume ical app oxima ion by ini e-elemen me hods o an op imal design p oblem o a wo phase ma e ial in one space dimension. This p oblem, in he con inuous se ing, due o high equency oscilla ions, o en does no ha e a classical solu ion, and a elaxed o mula ion is needed o ensu e exis ence. On he con a y, he disc e e e sions ob ained by nume ical app oxima ion ha e a solu ion. In his a icle we p o e he con e gence o he disc e iza ions and ob ain con e gence a es. We also show a as e con e gence when he elaxed e sion o he con inuous p oblem is aken in o accoun when building he dis- c e iza ion s a egy. In pa icula i is wo h emphasizing ha , e en when he o iginal p oblem has a classical solu ion so ha elaxa ion is no necessa y, nume ical algo i hms con e ge as e when implemen ed on he elaxed e sion. Key wo ds. con ol in he coe icien s, composi e op imal design, elaxa ion, nume ical app oxima ion, ini e elemen s AMS subjec classi ica ions. 49M25, 49J20 DOI. 10.1137/10081928X 1. In oduc ion. This pape is de o ed o he ini e-elemen nume ical analysis o a p oblem o op imal mix u e o wo ( he mal o elec ical) ma e ials in o de o mini- mize a gi en unc ional in one space dimension. Le Ωbe a bounded open se o RN,N≥1(al hough ou analysis is limi ed o he case N¼1, he p oblem makes sense in any space dimension), and conside he ollow- ing op imiza ion p oblem: Find ω0∈Usuch ha Jðω0Þ¼min ω∈U JðωÞ: ð1:1Þ He e ω, he con ol, is a measu able subse o Ω,JðωÞ, he cos unc ional, is o he o m JðωÞ¼Zω F1ðx; u; ∇uÞdxþZΩ ω F2ðx; u; ∇uÞdx;ð1:2Þ whe e F1;F2∶Ω×R×RN→Ra e gi en unc ions, and u, he s a e, is he solu ion o *Recei ed by he edi o s Decembe 23, 2010; accep ed o publica ion (in e ised o m) May 26, 2011; published elec onically Sep embe 15, 2011. h p://www.siam.o g/jou nals/mms/9-3/81928.h ml †Dp o. de Ecuaciones Di e enciales y Análisis Numé ico, Facul ad de Ma emá icas, Uni e sidad de Se illa, C. Ta ía s/n, 41012 Se illa, Spain ([email p o ec ed], [email p o ec ed]). The wo k o he i s and hi d au ho s was pa ially suppo ed by p ojec MTM2008-00306 o he MICINN (Spain) and he esea ch g oup FQM-309 o he CICE (Andalusia). ‡Dp o. de Ma emá icas e In o má ica, ETSI caminos, canales y pue os, Uni e sidad Poli écnica de Mad id, Ciudad Uni e si a ia, 28040 Mad id, Spain ([email p o ec ed]). The wo k o he second au ho was pa ially suppo ed by g an MTM2008-03541 o he MICINN (Spain). §Basque Cen e o Applied Ma hema ics, Bizkaia Technology Pa k, Building 500. E-48160 De io, Basque Coun y, Spain ([email p o ec ed]); IKERBASQUE, Basque Founda ion o Science, E-48011 Bilbao, Basque Coun y, Spain. The wo k o he las au ho was pa ially suppo ed by ERC ad anced g an FP7-246775 NUMERIWAVES, g an PI2010-04 o he Basque Go e nmen , ESF Resea ch Ne wo king P og amme OPTPDE, and g an MTM2008-03541 o he MICINN (Spain). 1181 MULTISCALE MODEL.SIMUL. Vol. 9, No. 3, pp. 1181–1216 © 2011 Socie y o Indus ial and Applied Ma hema ics Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. −di ððαχωþβð1−χωÞÞ∇uÞ¼ in Ω; u¼0on∂Ω ð1:3Þ o some gi en sou ce e m ∶Ω→R. The posi i e cons an s α,β ep esen he wo ma e ials, de e mining he coe icien s o he co esponding di usion ma ices. Some es ic ions can and mus be imposed o he con ol ωdepending on he p oblem. Fo example, an in e es ing case is when he ma e ial αis mo e e icien han he ma e ial βbu i is also mo e expensi e. Then, i is usual o conside a es ic ion o he o m jωj≤κ, limi ing he use o he ma e ial α. We include his es ic ion in he admissible se o con ols U, U¼ ω⊂Ω∶ωmeasu able;jωj≤κg:ð1:4Þ The exis ence o an op imal se ω ul illing hese cons ain s, o which he unc ion usolu ion o (1.3) minimizes J, does no hold in gene al (see [15], [16]). In hese cases, i is na u al o look o minimizing sequences, i.e., sequences ωlg∞ l¼1⊂Usuch ha lim l→∞ JðωlÞ¼in ω∈U JðωÞ since hey p o ide nea op imal designs. A usual p ocedu e o ind such sequences is o in oduce a elaxed e sion o he p oblem o which a minimize exis s. Then, a sui able app oxima ion o he minimize s p o ides minimizing sequences o he o iginal p oblem. Fo a sequen ially con inuous unc ional J, in he weak opology o he Sobole space H1ðΩÞ(see [1], [13], [20]), his elaxa ion can be ob ained by eplacing in (1.3) he unc ion χωwi h a measu able unc ion θ aking i s alues in he closed in e al ½0;1and he unc ion ðαχωþβð1−χωÞÞ wi h a ma ix unc ion Ain he se KðθÞo ma ices cons uc ed by homogeniza ion (see, e.g., [17], [19], [21]) mixing he ma e ials α and βwi h espec i e p opo ions θand 1−θ. Rema k ha he se KðθÞis known in he case desc ibed abo e, co esponding o he mix u e o wo iso opic ma e ials (see [14], [22]), bu no in o he in e es ing cases such as he mix u e o mo e han wo ma e ials, aniso opic ma e ials, e c. Hence o h we deno e by ^ U he se o elaxed con ols ðθ;AÞ. No e ha unc ionals o he o m (1.2) a e no sequen ially con inuous in he weak opology o H1ðΩÞ, in gene al. In hose cases, o ob ain he elaxed e sion (see [6]) we mus eplace he se o con ols χωand coe icien s ðαχωþβð1−χωÞÞ wi h he pai s ðθ;AÞ∈^ Uas abo e, and he unc ional Jwi h ano he one o he o m ^ Jðθ;AÞ¼ZΩ Hðx; u; ∇u; A∇u; θÞdx;ð1:5Þ whe e uis solu ion o he homogenized p oblem 8 < : −di A∇u¼ in Ω; u¼0on∂Ω: ð1:6Þ An explici exp ession o he unc ion His only known in some pa icula cases (see he e e ences [2], [6], [7], [8], [11], [12], [18], [23]). I sa is ies Hðx; u; ∇u; A∇u; θÞ¼F1ðx; u; ∇uÞχωþF2ðx; u; ∇uÞχΩ ω;i θ¼χω; 1182 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. and A¼ðαχωþβð1−χωÞÞIand, so, he elaxed unc ional is in ac an ex ension o he o iginal one o he la ge se o elaxed con ols. The elaxed con ol p oblem eads Find ðθ0;A 0Þ∈^ Usuch ha ^ Jðθ0;A 0Þ¼ min ðθ;AÞ∈^ U ^ Jðθ;AÞ: ð1:7Þ In p ac ical applica ions, in o de o sol e nume ically he abo e con ol p oblem (1.1), i is necessa y o in oduce a disc e iza ion o bo h he con ol se and he unc- ional. In he p esen con ex , we ha e a leas wo app oaches o his nume ical ap- p oxima ion issue. One based on he disc e iza ion o he o iginal p oblem and one elying on he disc e iza ion o he elaxed e sion. Recen ly, in [8] and [9], bo h disc e- iza ion p ocedu es ha e been shown o con e ge. In hese a icles, some pa ially elaxed e sions ha e also been s udied in which he class o con ols unde conside a ion is enla ged bu no o he ex en o exhaus ing he class o he elaxed e sion o he p oblem; we e e o [12] o a ela ed esul . We also e e o [26] o he nume ical s udy o he elaxed o mula ion o a pa icula case o p oblem (1.1). In his pape we compa e and ge con e gence a es o he sequences o disc e e minimize s ob ained wi h bo h app oxima ion me hods. These issues a e add essed in he simples one-dimensional se ing, whe e he pa ial di e en ial equa ion (1.3) is educed o an o dina y di e en ial equa ion, he se KðθÞis well known o be educed o he ha monic mean o αand βwi h espec i e p opo ions θand 1−θ, and he unc- ion His explici ly known. No e ha in his case we can w i e ^ Jðθ;AÞ¼ ^ JðθÞin (1.5), since Ais comple ely de e mined by θ, and ^ Uis jus he se o measu able unc ions θ∶Ω→½0;1wi h in eg al less o equal han κ. To make ou esul s p ecise, we i s conside he disc e iza ion o he se o con ols bu no o he s a e equa ion (1.3). In he con ex o ini e-elemen app oxima ion me h- ods, we can conside a decomposi ion o Ωin elemen s wi h maximum size and subse s ωcons i u ed by unions o a subse o such elemen s. I we deno e by U he se o such subse s, he disc e e p oblem eads Find ω 0∈U such ha Jðω 0Þ¼min ω∈U JðωÞ: ð1:8Þ The disc e e space o con ols ob ained in his way U is compac in he s ong opology o L1ðΩÞ, and he co esponding s a e unc ions a e compac in H1ðΩÞ. The e o e, he disc e ized p oblem has a solu ion wi hou he need o a elaxed e sion. In his way we ob ain a sequence o disc e e minimize s ω 0g ha a e likely o cons i u e a minimizing sequence o Jin U,as →0. We show ha his is he case, and we gi e con e gence a es o Jðω 0Þ−in ω∈U JðωÞas →0:ð1:9Þ On he o he hand, ins ead o disc e izing he o iginal con ol p oblem, we can dis- c e ize he elaxed e sion. A e in oducing a decomposi ion o Ωin elemen s, wi h maximal size , we can conside he se ^ U o unc ions θ∈^ Uwhich a e cons an on each elemen . The disc e e elaxed p oblem eads Find ^ θ 0∈^ U such ha ^ Jð^ θ 0Þ¼min θ∈^ U ^ JðθÞ. ð1:10Þ APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1183 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. As abo e, we show ha ^ Jð^ θ 0Þ−in ω∈^ U∇ JðuÞ→0as →0;ð1:11Þ and we gi e con e gence a es. Once a disc e e elaxed minimize is known ^ θ 0we can cons uc a sequence ωk; g∞ k¼1⊂Usuch ha lim k→∞ Jðωk; Þ¼ ^ Jð^ θ 0Þ: This p o ides a minimizing sequence o he o iginal p oblem. As we show, he sequence ωk; g∞ k¼1can be cons uc ed explici ly om ^ θ 0wi h almos no compu a ional cos . Ou esul s show ha i is be e o disc e ize he elaxed p oblem, in he sense ha we ge a as e con e gence a e, as →0, o (1.11) han he one ob ained o (1.9). This is ue e en in he case whe e he o iginal p oblem has a solu ion, and so he e- laxa ion is unnecessa y om a heo e ical poin o iew. Despi e his, he elaxed e sion o he o iginal minimiza ion p oblem can always be o mula ed, and ou esul s show ha i is indeed be e o app oxima e he op imal design p oblem nume ically in hese cases as well. F om a compu a ional poin o iew, besides disc e izing he se o con ols, we mus also disc e ize he s a e equa ion (1.3) o (1.6). This equi es a second decomposi ion o Ωcons i u ed by elemen s o maximum size h. A na u al assump ion is o conside his new decomposi ion as a e inemen o he one used o he con ol se , o ice e sa. In he con ex o he o iginal un elaxed con ol p oblem, deno ing by uh he P1- ini e-elemen app oxima ion o he solu ion o (1.3), and de ining Jhas JhðωÞ¼Zω F1ðx; uh;∇uhÞdxþZΩ ω F2ðx; uh;∇uhÞdx; he ull disc e e con ol p oblem eads Find ω ;h 0∈U such ha Jhðω ;h 0Þ¼min ω∈U JhðωÞ: ð1:12Þ Analogously, we can de ine a ull disc e iza ion o he elaxed p oblem by conside ing ^ JhðθÞ¼ZΩ Hðx; uh;∇uh;A∇uh;θÞdx;ð1:13Þ whe e uhis he P1- ini e-elemen app oxima ion o (1.6). The ully disc e e elaxed p oblem in his case is Find ^ θ ;h 0∈^ U such ha ^ Jhð^ θ ;h 0Þ¼min θ∈^ U ^ JhðθÞ: ð1:14Þ We ocus on he con e gence a es o he sequences ω ;h 0g ;h and ^ θ ;h 0g ;h ob ained wi h he wo app oaches abo e, espec i ely. Mo e p ecisely, we compa e he sequences 1184 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Jðω ;h 0Þ−in ω∈U JðωÞand ^ Jð^ θ ;h 0Þ−in ω∈U JðωÞ; as ; h →0. The ollowing esul s a e p o en: •Disc e izing he elaxed o mula ion, we show ha , sol ing he s a e equa ion wi h he P1- ini e-elemen me hod in a mesh o size hand aking he con ol θ o be piecewise cons an on elemen s o a coa se mesh o size ffiffiffi h p, he e o is o o de h. This cons i u es a big id o mul iscale s a egy, implemen ed on he elaxed e sion, in he sense ha he disc e iza ion o he PDE and ha o he con ol a e pe o med on wo di e en g ids. The PDE is disc e ized in he ine g id o size h, while he con ol is disc e ized in he coa se one o size ffiffiffi h p. •Disc e izing he o iginal un elaxed p oblem, sol ing he s a e equa ion wi h a P1- ini e-elemen me hod in a mesh o size h, and aking he con ol χωpiece- wise cons an in he elemen s o such mesh, we show ha he e o is o o de h1−εwi h εa bi a ily small i he unc ions Fiin (1.2) do no depend on he a iable uand ε¼1∕2o he wise. A big id s a egy consis ing in disc e izing he PDE in he coa se g id (ins ead o he ine one) can p oduce lack o con e gence o bo h he un elaxed and elaxed p o- blems. In pa icula , he minimize s o he disc e e p oblem will possibly gi e a non- minimizing sequence o he con inuous con ol p oblem, as ; h →0. We also gi e an explici example in which he unc ional is independen o u, show- ing ou es ima es a e nea ly sha p. To be mo e p ecise, ou example shows he op imali y o he es ima es in he case in which he elaxed e sion o he p oblem is disc e ized, while an o de ho con e gence is ob ained when he o iginal p oblem is disc e ized, hus showing ha ou es ima es a e nea ly op imal. The e o e he app oach based on he disc e iza ion o he elaxed o mula ion p o- ides a be e app oxima ion and a as e con e gence a e wi h a lowe compu a ional cos . The compu a ional cos and he complexi y o his app oach is lowe since he con ols a e disc e ized in a mesh o o de ffiffiffi h pins ead o h. Fu he mo e, he minimize s o he co esponding disc e e op imiza ion p oblems a e easie o ind nume ically. Indeed, hanks o he con exi y o he elaxed con ol se , g adien -like algo i hms can be implemen ed. This is in con as o he un elaxed p oblem, whe e he con ol se is no con ex and we canno compu e a ia ions. Ins ead, much less e icien me hods such as Mon e Ca lo o gene ic algo i hms should be used. On he con a y, he ad an ages o disc e izing he o iginal p oblem di ec ly a e ha , on one hand, one does no need o know he elaxed o mula ion and, on he o he hand, i p o ides a physical con ol (i.e., a cha ac e is ic unc ion) ins ead o a elaxed one. Howe e , his la e d awback can be o e come by app oxima ing he elaxed con- ols by physical ones, wi h almos no compu a ional cos . This pape p o ides a comple e analysis o he a e o con e gence o he ini e- elemen app oxima ion o he op imal design p oblem unde conside a ion. Whe he his classical enginee ing p ac ice leads o con e gen algo i hms is unknown in many o he op imal design p oblems, excep in some o he pa icula examples as i occu s when dealing wi h he op imal shape design o he domain o Di ichle Laplacian in wo space dimensions (see [10]). No e, howe e , ha , in he la e , he e is no esul abou he con e gence a e. Al hough he p esen a icle is de o ed o he s udy o he 1−dop imal design p oblem, some ema ks abou he N-dimensional case a e gi en in he las sec ion o he pape . As abo e hese ema ks a e de o ed o he case o di usion coe icien s ha APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1185 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. a e uni o mly ellip ic and bounded; he case whe e we conside ma ix di usions such ha hei smalle and/o la ge eigen alues can app oxima e o ze o o in ini y, espec- i ely, is mo e in ol ed. Indeed, e en he de ini ion o solu ion o he s a e equa ion is no clea in his case, whe e in pa icula La en ie ’s phenomenon can occu ; i.e., smoo h unc ions canno be dense in he space o unc ions wi h bounded ene gy (see, e.g., [25] and he e e ences he ein). In his sense, we ema k ha in o de o p o e he con e gence o he ini e-elemen me hod, i is necessa y o ha e he densi y o he Lipschi z unc ions in he space whe e we a e looking o he solu ion o he s a e equa- ion. A ecip oca e o his esul has been ob ained in [5] o a calculus o a ia ions p oblem wi hou es ic ions. As we ha e al eady ema ked, con ol p oblem (1.1) does no ha e a solu ion in gene al. To ha e a well-posed p oblem, such as we do in he p esen pape , an app oach consis s o ob aining a elaxa ion o (1.1) by using homogeniza ion echniques. Howe e , he e exis o he app oaches, o ins ance, he il e ing echnique. Loosely speaking, he idea o he il e ing echnique consis s o eplacing he se o con ols in (1.1) wi h a smoo he class, de ined by mean o a con olu ion ope a o . Mo e p ecisely, in (1.3) he cha ac e is ic unc ions χω, wi h ω∈U, a e eplaced by he smoo h unc ions ρRθ, wi h θ∈L∞ðΩ;½0;1Þ sa is ying he olume es ic ion, whe e ρRis he ypical molli ie unc ion ρRðxÞ¼ρðx∕RÞ∕RNwi h ρa ixed C∞nonnega i e unc ion wi h suppo in he ball o cen e 0 and adius 1, and in eg al equals 1. Thus, we ob ain a new p oblem ( il e ed p oblem) wi h a compac se o con ols in C∞ð¯ ΩÞ, which gua an ees he exis ence o a solu ion when he cos unc ional Jis sequen ially lowe con inuous in he weak opology o H1ðΩÞ. The il e ed p oblems a e hen smoo h app oxima ions o (1.1) when R>0is small, a leas o mally. Gi en R ixed, he ini e-elemen app ox- ima ion o he il e ed p oblem has been s udied in [4], in he amewo k o a con ol p oblem in he coe icien s in elas ici y—namely he compliance p oblem. In [4], he con e gence o he ini e-elemen app oxima ion, as he mesh size ends o ze o, is p o ed bu wi hou explici a es. Some de ini ions and no a ions: •Fo a numbe ∈R, we deno e by ½  he in ege pa o . •Fo a (Lebesgue) measu able subse Eo ð0;1Þ, wi h posi i e measu e, and a unc ion win L1ð0;1Þ, we deno e he mean alue o win Eby ⨍E wdx¼1 jEjZE wdx: •The se o unc ions o bounded a ia ion in ð0;1Þis deno ed by BVð0;1Þ.I ψis in BVð0;1Þand Iis a subin e al o ½0;1, hen VIðψÞ ep esen s he o al a - ia ion o ψin I. •Th oughou he pape , αand βa e wo posi i e cons an s. •Fo p∈½0;1, we deno e by MðpÞ∈R he ha monic mean o αand βwi h p opo ions pand 1−p, espec i ely, gi en by MðpÞ¼p αþ1−p β−1 ¼αβ ð1−pÞαþpβ: No e ha Mð1Þ¼α,Mð0Þ¼β, and α≤MðpÞ≤β∀p∈½0;1:ð1:15Þ 1186 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Fo e e y θ∈L∞ð0;1; ½0;1Þ we de ine Mθ∈L∞ðΩÞby MθðxÞ¼MðθðxÞÞ o a:e:x∈ð0;1Þ: •Fo a ma ix A∈RN×N, we deno e by EigðAÞ he se o i s eigen alues. •Le Φbe a unc ion de ined in he in e al ð0;δÞ o some δ>0. The equali y Φ¼oðhÞ(Landau symbol) means lim h→0 ΦðhÞ h¼0: •We deno e by Ca gene ic posi i e cons an ha can change om line o line. 2. Disc e iza ion and e o es ima es. 2.1. The main esul s. In his sec ion we s a e he main esul s o he pape . They a e e e ed o he nume ical analysis o a con ol p oblem o he 1−dellip ic s a e equa ion in Ω¼ð0;1Þbelow, he con ol being he space-dependen coe icien (−d dx ðαχωþβð1−χωÞÞdu dx¼ inð0;1Þ; uð0Þ¼uð1Þ¼0; ð2:1Þ whe e αand βa e wo ixed posi i e cons an s and a gi en unc ion in (a leas ) L1ð0;1Þ. De ining, o a ixed cons an κ>0, he se o admissible con ols as (1.4), ou aim is o choose ω∈Usuch ha he unique solu ion uω∈H1 0ð0;1Þo p oblem (2.1) minimizes he unc ional J∶UR !de ined as he 1−d e sion o (1.2); i.e., JðωÞ¼Zω F1x; uω;duω dx dxþZð0;1Þ ω F2x; uω;duω dx dx∀ω∈U:ð2:2Þ He e F1;F2∶ð0;1Þ×R×R→Rsa is y Fi∈W1;∞ðð0;1Þ×ð−R; RÞ×ð−R; RÞÞ ∀i∈ 1;2g∀R>0:ð2:3Þ As we said in he in oduc ion, αand β ep esen wo ma e ials ha we wan o mix in o de o minimize J. The cons an κis he maximum quan i y o ma e ial α ha can be used in he mix u e. No e ha aking κ≥1would be equi alen o no imposing any es ic ion in he se o admissible se s ω. Rema k 1. In (2.1), we conside homogeneous Di ichle condi ions o ix ideas, bu ou esul s also hold o nonhomogeneous Di ichle condi ions o o he bounda y condi ions, such as Fou ie o Neumann ones. We can also conside he unc ions Fi sa is ying weake assump ions han (2.3), bu hen he e o es ima es we ind o he nume ical app oxima ions de ined below a e wo se. I is well known ha he o iginal minimiza ion p oblem (1.1) does no ha e a solu- ion in gene al (see [15], [16]). The e o e, i is necessa y o in oduce a elaxa ion. How- e e , as we ha e men ioned in he in oduc ion, o nume ical pu poses i is o en con enien o wo k in he elaxed e sion o he p oblem e en when he o iginal o mu- la ion has a minimize . The elaxed e sion hus plays a key ole in he nume ical analysis we de elop in his a icle. The ollowing esul p o ides a cha ac e iza ion o he elaxa ion. APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1187 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. THEOREM 2.1. A elaxa ion o p oblem (1.1) is gi en by Find θ0∈^ Usuch ha ^ Jðθ0Þ¼min θ∈^ U ^ JðθÞ; ð2:4Þ whe e ^ U¼θ∈L∞ð0;1; ½0;1Þ∶Z1 0 θdx≤κ;ð2:5Þ and ^ J∶^ U→Ris de ined by ^ JðθÞ¼Z1 0θF1x; uθ;Mθ α duθ dx þð1−θÞF2x; uθ;Mθ β duθ dx dxð2:6Þ o e e y θ∈^ Uwi h u¼uθ he solu ion o (−d dx Mθdu dx¼ in ð0;1Þ; uð0Þ¼uð1Þ¼0: ð2:7Þ Rema k 2. Theo em 2.1 also holds ue o e e y ∈H−1ð0;1Þand mo e gene al nonlinea i ies F1,F2. Indeed, i is enough o assume ha F1,F2a e wo Ca a héodo y unc ions (measu able wi h espec o xand con inuous wi h espec o ðs; ξÞ) such ha o e e y R>0, he unc ions φ1;R,φ2;R de ined as φi;RðxÞ¼ sup jsjþjξj≤RjFiðx; s; ξÞj o a:e:x∈ð0;1Þ∀i∈ 1;2g belong o L1ð0;1Þ. Rema k 3. Fo e e y ω⊂ð0;1Þmeasu able, we ha e JðωÞ¼ ^ JðχωÞ: The e o e, ^ Jis in ac an ex ension o he unc ional χω↦JðωÞde ined on he space L∞ð0;1; 0;1gÞ o he elaxed con ol se L∞ð0;1; ½0;1Þ. Rema k 4. Theo em 2.1 is a gene aliza ion o P oposi ion 4.1 and Theo em 4.3 in [6], whe e he mul idimensional case is also conside ed. In he p esen pape , we a e in e es ed mainly in he nume ical analysis o p oblem (1.1). Fo his pu pose, hanks o Theo em 2.1, wo choices a e possible: o disc e ize di ec ly p oblem (1.1) o o disc e ize he elaxed p oblem (2.4). Ou goal is o compa e hese wo possibili ies. To his aim, gi en >0, we ake a pa i ion P ¼ ykgm k¼0o ½0;1, wi h m ∈N, such ha ¼max 1≤k≤m ðyk−yk−1Þ:ð2:8Þ Then, we de ine ^ U and U as he subse s o ^ Ugi en by ^ U ¼θ∈^ U∶θ¼X m k¼1 kχðyk−1;ykÞa:e:inð0;1Þwi h k∈½0;1;1≤k≤m ;ð2:9Þ 1188 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. U ¼ ω⊂ð0;1Þ∶χω∈^ U g:ð2:10Þ Associa ed o hese subse s we can conside he wo disc e iza ions o he con ol p o- blem gi en by (1.10) and (1.8). No e ha p oblem (1.8) is a disc e iza ion o he o iginal minimiza ion p oblem (1.1), while (1.10) is a disc e iza ion o he elaxed p oblem (2.4). The ollowing heo ems p o ide es ima es on he di e ence be ween hese p oblems and (2.4). Some e sions o Theo em 2.2 can also be ob ained in he N-dimensional case; see sec ion 8. THEOREM 2.2. Assuming ∈L1ð0;1Þ, p oblem (1.10) has a solu ion o e e y >0, and we ha e 0≤min θ∈^ U ^ JðθÞ−min θ∈^ U ^ JðθÞ¼oð Þ:ð2:11Þ Mo eo e , i ∈L∞ð0;1Þand p oblem (2.4) has a solu ion θ0in BVð0;1Þ, hen 0≤min θ∈^ U ^ JðθÞ−min θ∈^ U ^ JðθÞ≤C 2:ð2:12Þ THEOREM 2.3. Assuming ∈L1ð0;1Þ, p oblem (1.8) has a solu ion o e e y >0, and we ha e 0≤min ω∈U JðωÞ−in ω∈U JðωÞ≤C 1 2:ð2:13Þ Mo eo e , i o some in ege l≥1, we ha e ha belongs o he space Wl;1ð0;1Þand F1ðx; s; ξÞ,F2ðx; s; ξÞa e independen o sand belong o Cl;1 locð½0;1×RÞ; hen we ha e 0≤min ω∈U JðωÞ−in ω∈U JðωÞ≤C lþ1 lþ2:ð2:14Þ 2.2. Op imali y. We now gi e an example showing ha he p e ious esul s a e nea ly op imal. Example 1. We conside p oblem (1.1) wi h α<β, ¼1,κ¼2∕3, and Jgi en by JðωÞ¼−αZω duω dx  2 dx−βZð0;1Þ ω duω dx  2 dx:ð2:15Þ Fo e e y n∈N, we de ine Pnas he pa i ion o ½0;1gi en by Pn¼ k10−n∶0≤k≤10ng: We de ine ^ Un¼θ∈^ U∶θ¼X 10n k¼1 kχððk−1Þ10−n;k10−nÞwi h k∈½0;1∀k∈ 1; :::;10ng;ð2:16Þ Un¼ ω∈U∶χω∈^ Ung:ð2:17Þ We will p o e in sec ion 6 he ollowing esul . APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1189 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Rema k 7. In Example 3 we a e disc e izing he s a e equa ion (2.1) o (2.7) using a pa i ion o ½0;1o size hbigge han he size ¼h∕2employed in disc e izing he se o con ols. S a emen (2.38) shows ha in his case he minimum o he disc e ized p oblem does no end o he in imum o (1.1). Thus, his ype o disc e iza ion is no con e gen in gene al. 3. P oo o he elaxa ion esul . This sec ion is de o ed o p o ing Theo em 2.1, which cha ac e izes he elaxa ion o p oblem (1.1). To do i , we use he ollowing lemma. LEMMA 3.1. The unc ional ^ J∶^ U⊂L∞ð0;1Þ→Ris sequen ially con inuous o he -weak opology o L∞ð0;1Þ. P oo . Gi en a sequence θn∈^ Uwhich con e ges weakly-in L∞ð0;1Þ o a unc ion θ∈^ U, we ha e o see ha ^ JðθnÞcon e ges o ^ JðθÞ. Fo a such sequence θn, we obse e ha he co esponding solu ion uθno (2.7) is gi en by uθnðxÞ¼−Zx 0 Fð Þ−cn Mθn d ¼−Zx 0ðFð Þ−cnÞαð1−θnð ÞÞþβθnð Þ αβ d wi h Fa p imi i e o in ð0;1Þand cn¼Z1 0 d Mθnð Þ−1Z1 0 Fð Þ Mθnð Þd : The e o e, i is immedia e o show ha kuθnkW1;∞ð0;1Þ≤C; uθn→uθin C0ð½0;1Þ;M θn dun dx −Mθ duθ dx →0inC0ð½0;1Þ wi h uθ he unique solu ion o (2.7). Then, by (2.3) we ob ain lim n→∞ ^ JðθnÞ ¼limn→∞Z1 0θnF1x; uθn;Mθn α duθn dx þð1−θnÞF2x; uθn;Mθn β duθn dx dx ¼Z1 0θF1x; uθ;Mθ α duθ dx þð1−θÞF2x; uθ;Mθ β duθ dx dx¼^ JðθÞ:▯ P oo o Theo em 2.1. Taking in o accoun ha he space o con ols ^ Ugi en by (2.5) is sequen ially compac in he -weak opology o L∞ð0;1Þ, om Lemma 3.1 we deduce ha p oblem (2.4) has a leas a solu ion. On he o he hand, by Rema k 3 i is clea ha in ω∈U JðωÞ¼ in χω∈^ U ^ JðχωÞ≥min θ∈^ U ^ JðθÞ: The e o e, in o de o check ha p oblem (2.4) is a elaxa ion o (1.1), i is enough o p o e ha o e e y θ∈^ U, he e exis s a sequence ωnin Usuch ha 1196 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. χωn⇀ θin L∞ð0;1Þ;ð3:1Þ JðωnÞ→^ JðθÞ:ð3:2Þ The exis ence o his sequence ωnis well known ( o example, i is a consequence o Lemma 5.1 below), while by he con inui y p ope y o ^ Jp o ed in s ep 1, (3.2) is a consequence o (3.1). So, he p oo o Theo em 2.1 is comple e. ▯ 4. P oo o he con e gence es ima es o he disc e ized elaxed con ol p oblem. In his sec ion we p o e Theo em 2.2 e e ed o he con e gence o he dis- c e iza ion o p oblem (2.4) gi en by (1.10). No e ha we a e disc e izing he con ols bu no he s a e equa ion. We also gi e he p oo o P oposi ion 2.5, which pe mi s us o ob ain a physical con ol om a elaxed one. Along his sec ion, we conside a pa i ion P ¼ ykgm k¼0, wi h m ∈N, sa is ying (2.8). The space ^ U is de ined by (2.9). In o de o show Theo em 2.2, we will use he ope a o Π de ined by he ollowing. DEFINITION 4.1. We de ine he p ojec ion ope a o Π ∶L1ð0;1Þ→^ U by Π ψ¼X m k¼1⨍yk yk−1 ψdsχðyk−1;ykÞ∀ψ∈L1ð0;1Þ:ð4:1Þ The ollowing lemma es ima es he di e ence Π θ−θwhen ends o ze o. LEMMA 4.2. Le θbe in L∞ð0;1; ½0;1Þ. Then, o e e y φ∈W1;1ð0;1Þ, i holds ha Z1 0ðθ−Π θÞφdx¼oð Þ;ð4:2Þ Z1 0Zx 0ðθð Þ−Π θð ÞÞφð Þd  dx¼oð Þ:ð4:3Þ Mo eo e , i θis in BVð0;1Þ, and φis in W1;∞ð0;1Þ, we ha e he ollowing imp o emen o he p e ious es ima es: Z1 0ðθ−Π θÞφdx ≤C    dφ dx   L∞ð0;1Þ 2;ð4:4Þ Z1 0Zx 0ðθð Þ−Π θð ÞÞφð Þd  dx≤CkφkW1;∞ð0;1Þ 2:ð4:5Þ P oo . We ake φ∈W1;1ð0;1Þ; o a gi en x∈½0;1, we conside yjde ined by yj¼sup yk∶yk≤x; 0≤k≤m g: Then, using he inequali y  φð Þ−⨍yk yk−1 φds ≤    dφ d    L1ðyk−1;ykÞ ∀ ∈½yk−1;y k we ha e APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1197 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Zx 0ðθ−Π θÞφd  ¼X j k¼1Zyk yk−1θ−⨍yk yk−1 θdsφd þZx yjθ−⨍yk yk−1 θdsφd ¼X j k¼1Zyk yk−1ðθ−⨍yk yk−1 θdsÞðφ−⨍yk yk−1 φdsÞd þZx yjθ−⨍yk yk−1 θdsφd ≤X j k¼1    dφ dx   L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθ−Π θkL1ðyj;xÞ:ð4:6Þ In eg a ing his inequali y in ð0;1Þ,wege Z1 0Zx 0ðθð Þ−Π θð ÞÞφð Þd  dx ≤X m k¼1    dφ dx   L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞ þkφkL∞ð0;1ÞX m −1 j¼0Zyjþ1 yjkθ−Π θkL1ðyj;xÞdx ≤X m k¼1    dφ dx   L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθ−Π θkL1ð0;1Þ :ð4:7Þ I φbelongs o W1;∞ð0;1Þand θbelongs o BVð0;1Þ, using in (4.7)     dφ dx   L1ðyk−1;ykÞ ≤    dφ dx   L∞ð0;1Þ ; kθ−Π θkL1ð0;1Þ≤Vð0;1ÞðθÞ ;ð4:8Þ we deduce (4.5). Inequali y (4.4) is a consequence o (4.6) wi h x¼1¼yjand (4.8). In o de o show (4.2) and (4.3) we now ake a sequence φnin W1;∞ð0;1Þwhich con e ges o φin W1;1ð0;1Þand a sequence θnin BVð0;1Þ, wi h 0≤θn≤1in ð0;1Þ, which con e ges o θin L1ð0;1Þ. Then, we es ima e he igh -hand side o (4.7) as ollows: X m k¼1    dφ dx   L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθ−Π θkL1ð0;1Þ ≤2    dðφ−φnÞ dx    L1ð0;1Þ þkφkL∞ð0;1Þkθ−θn−Π ðθ−θnÞkL1ð0;1Þ þX m k¼1    dφn dx    L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθn−Π θnkL1ð0;1Þ ≤2    dðφ−φnÞ dx    L1ð0;1Þ þkφkL∞ð0;1Þkθ−θn−Π ðθ−θnÞkL1ð0;1Þ þ    dφn dx    L∞ð0;1Þ Vð0;1ÞðθÞþkφkL∞ð0;1ÞVð0;1ÞðθnÞ 2: 1198 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Di iding his inequali y by and passing o he limi i s when ends o ze o and hen when n ends o in ini y, we deduce (4.3). The p oo o (4.2) can be ob ained easoning in a simila way wi h (4.6). ▯ Fo θ∈L∞ð0;1; ½0;1Þ, he ollowing lemma es ima es he di e ence be ween he solu ion o (2.7) and he solu ion o he analogous p oblem when θis eplaced by Π θ. LEMMA 4.3. Assume ∈L1ð0;1Þ. Fo θ∈L∞ð0;1; ½0;1Þ, we conside θ ¼Π θ. Then, he solu ions uθand uθ o (2.7) o θand θ , espec i ely, sa is y kuθ−uθ kL1ð0;1Þ≤oð Þ;ð4:9Þ     Mθ duθ dx −Mθ duθ dx    L∞ð0;1Þ ≤oð Þ:ð4:10Þ I is in L∞ð0;1Þand θis in BVð0;1Þ, hen in (4.9) and (4.10) we can ake oð Þ¼CVð0;1ÞðθÞ 2: P oo . The unc ions uθand uθ a e gi en by uθðxÞ¼−Zx 0 g Mθ dsþcZx 0 1 Mθ ds o a:e:x∈ð0;1Þ;ð4:11Þ uθ ðxÞ¼−Zx 0 g Mθ dsþc Zx 0 1 Mθ ds o a:e:x∈ð0;1Þð4:12Þ wi h ga p imi i e o and c; c ∈Rde ined by c¼Z1 0 1 Mθ dx−1Z1 0 g Mθ dx; c ¼Z1 0 1 Mθ dx−1Z1 0 g Mθ dx:ð4:13Þ Using hese exp essions and aking in o accoun ha min α;βg≤Mθ;Mθ ≤max α;βg; we easily deduce kuθ−uθ kL1ð0;1Þ≤CZ1 0ðθ−θ ÞgdxþZ1 0ðθ−θ Þdx : þZ1 0Zx 0ðθð Þ−θ ð ÞÞgð Þd  dxþZ1 0Zx 0ðθð Þ−θ ð ÞÞd  dx and     Mθ duθ dx −Mθ duθ dx    L∞ð0;1Þ ≤CZ1 0ðθ−θ ÞgdxþZ1 0ðθ−θ Þdx: Lemma 4.3 is hen a simple consequence o Lemma 4.2. ▯ We a e now in posi ion o p o e he ollowing. P oo o Theo em 2.2. The exis ence o solu ion o p oblem (1.10) is a simple con- sequence o he compac ness o (2.9) in L1ð0;1Þ. On he o he hand, using ha F1and F2a e locally Lipschi z, and ha he unc- ions uθ,uθ de ined as in Lemma 4.2 a e bounded in W1;∞ð0;1Þindependen ly o , we ha e APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1199 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. j^ JðθÞ−^ Jðθ Þj ≤Z1 0 F1x; uθ;Mθ α duθ dx ðθ−θ ÞdxþZ1 0 F2x; uθ;Mθ α duθ dx ðθ−θ Þdx þCZ1 0juθ−uθ jþMθ duθ dx −Mθ duθ dx dx: Thanks o Lemma 4.3, we hen deduce (2.11) and (2.12). ▯ To inish his sec ion, we now gi e he p oo o P oposi ion 2.5. P oo o P oposi ion 2.5. Reasoning as in he p oo o Theo em 2.2, we ha e ha he esul is an immedia e consequence o he ollowing lemma, which is simila o Lemma 4.2. ▯ LEMMA 4.4. Assume θand ωas in he s a emen o P oposi ion 2.5; hen o e e y φ∈W1;∞ð0;1Þ, i holds ha Z1 0ðθ−χωÞφdx ≤    dφ dx   L∞ð0;1Þ 2;ð4:14Þ Z1 0Zx 0ðθð Þ−χωð ÞÞφð Þd  dx≤kφkW1;∞ð0;1Þ 2:ð4:15Þ P oo . Since in each in e al ½yk−1þði−1Þsk;y kþisk, wi h 1≤k≤m , 1≤i≤jk, he unc ions θand χωha e he same in eg al, we can eason as in he p oo o (4.6) o deduce ha o e e y x∈½0;1, we ha e Zx 0ðθ−χωÞφd  ≤    dφ dx   L∞ð0;1Þkθ−χωkL1ð0;1Þ 2þkφkL∞ð0;1Þkθ−χωkL1ðIÞ ;ð4:16Þ whe e Iis an in e al o he o m ½yk−1þði−1Þsk;y kþiskcon aining x. Taking x¼1 we ge (4.14). On he o he hand, since θand χωbelong o L∞ð0;1; ½0;1Þ, inequali y (4.16) implies Zx 0ðθ−χωÞφd  ≤ 2kφkW1;∞ð0;1Þ o e e y x∈½0;1. This inequali y immedia ely p o es (4.15). ▯ 5. P oo o he con e gence es ima es o he disc e ized un elaxed con ol p oblem. Le us now p o e Theo em 2.3. As o Theo em 2.2, we will need some p elimina y lemmas. LEMMA 5.1. We conside θ∈L∞ð0;1Þand l∈N; hen, he e exis s ω⊂ð0;1Þmea- su able such ha Z1 0 jθð Þd ¼Zω jd ∀j∈ 0; :::;lg:ð5:1Þ Mo eo e ωcan be chosen in he ollowing way: 1200 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. I l¼2n, wi h n∈N, ω¼ð0;b 0Þ[[ m i¼1ðai;b iÞ; whe e m≤nand 0≤b0<a1<b1<···<am<bm≤1. I l¼2nþ1, wi h n∈N, ω¼[ m i¼1ðai;b iÞ; whe e m≤nþ1and 0≤a1<b1<···<am<bm≤1. P oo . Le us p o e he esul in he case l¼2nþ1, he o he one being simila . We de ine D⊂L1ð0;1Þas D¼ϕ¼X m i¼1 χðai;biÞwi h m≤nþ1;0≤a1<b1<···<am<bm≤1 and Ψ∶D→Rby ΨðϕÞ¼X 2nþ1 j¼0Z1 0 jðθð Þ−ϕð ÞÞd 2 ∀ϕ∈D: Since Dis compac in L1ð0;1Þand Ψis con inuous, we know ha Ψa ains i s minimum in some unc ion ϕ¼X m i¼1 χðai;biÞ∈D: Then, we de ine he polynomial Pas PðλÞ¼X 2nþ1 j¼0Z1 0 jðθð Þ−ϕð ÞÞd λj. We ix k, wi h 1≤k≤m. Fo ε∈R, wi h jεjsmall (εmus also be posi i e i k¼1, a1¼0), he unc ion ϕε¼χ∪i≠kðai;biÞþχðakþε;bkÞ belongs o D. Taking in o accoun ha ΨðϕεÞ¼X 2nþ1 j¼0Z1 0 jðθð Þ−ϕð ÞÞd þZakþε ak jd 2 ; and ha ϕis a minimum poin o Ψ, he de i a i e o ΨðϕεÞwi h espec o εyields PðakÞ¼0i ak≠0;Pða1Þ≥0i a1¼0: Analogously, we can p o e APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1201 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. PðbkÞ¼0i bk≠1;PðbmÞ≥0i bm¼1: I Phas 2nþ2ze os, hen i is he ze o polynomial and we ob ain he conclusion o he lemma. So, we assume in he ollowing ha Phas a mos 2nþ1ze os. By he abo e p o ed we deduce ha m¼nþ1;a 1¼0;and∕o bnþ1¼1; o m<nþ1: Le us p o e ha in all hese cases Psa is ies PðλÞ≥0in [ m i¼1ðai;b iÞ;PðλÞ≤0inð0;1Þ [ m i¼1ðai;b iÞ.ð5:2Þ (i) Case m¼nþ1,a1¼0,bnþ1¼1. Since we a e supposing ha he numbe o ze os o Pis s ic ly less han 2nþ2and P anishes in he 2npoin s akwi h k¼2; :::;nþ1,bkwi h k¼1; :::;n, we ha e ha Phas 2no 2nþ1ze os in ½0;1. I he numbe o ze os is 2nþ1, hen using ha Pð0Þ;Pð1Þ≥0,we deduce ha he o he ze o o Pis in 0 o 1 and ha Psa is ies (5.2). I he numbe o ze os is 2n, hen we ha e Pð0Þ;Pð1Þ>0and (5.2) is sa is ied. (ii) Case m¼nþ1,a1¼0,bnþ1<1. In his case we ha e ha he 2nþ1ze os o Pa e gi en by he poin s akwi h k¼2; :::;nþ1,bkwi h k¼1; :::;nþ1. Since Pð0Þ≥0, we deduce (5.2). (iii) Case m¼nþ1,a1>0,bnþ1¼1. I is simila o he case (ii). (i ) Case m<nþ1. In his case, we ake a poin c∈ðai;b iÞ o some i∈ 1; :::mg. Then o ε>0, small enough, he unc ion ϕε¼ϕ−χðc−ε;cþεÞ belongs o D. Using ha ΨðϕεÞ¼X 2nþ1 j¼0Z1 0 jðθð Þ−ϕð ÞÞd þZcþε c−ε jd 2 ; and de i ing wi h espec o ε, we deduce ha PðcÞ≥0∀c∈[ m i¼1ðai;b iÞ: Analogously, i c∈ð0;1Þ Sm i¼1½ai;b i, aking ϕε¼ϕþχðc−ε;cþεÞ; we deduce ha 1202 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. PðcÞ≤0∀c∈ð0;1Þ [ m i¼1½ai;b i: Thus, (5.2) is also p o en in his case. To inish, le us p o e ha (5.2) implies he conclusion o he lemma. Fo his pu - pose, we jus w i e X 2nþ1 j¼0Z1 0 jðθð Þ−ϕð ÞÞd 2 ¼Z1 0X 2nþ1 j¼0Z1 0 jðθð Þ−ϕð ÞÞd sjðθðsÞ−ϕðsÞÞds ¼Z1 0 PðsÞðθðsÞ−ϕðsÞÞds.ð5:3Þ I s∈Sm i¼1ðai;b iÞ(i.e., ϕðsÞ¼1), hen by (5.2), PðsÞ≥0and since θðsÞ≤1, we ha e PðsÞθðsÞ≤PðsÞϕðsÞ: I s∈=Sm i¼1ðai;b iÞ(i.e., ϕðsÞ¼0), hen by (5.2), PðsÞ≤0and since θðsÞ≥0, we also ha e PðsÞθðsÞ≤PðsÞϕðsÞ: The e o e he las in eg al in (5.3) is nonposi i e, which p o es X 2nþ1 j¼0Z1 0 jðθð Þ−ϕð ÞÞd 2 ¼0: This p o es Lemma 5.1. ▯ As a consequence, we deduce he ollowing. LEMMA 5.2. Le a,bbe in Rwi h a<band le ykgm k¼0be a pa i ion o ½a; bo size δ¼max 1≤k≤mðyk−yk−1Þ: Le also θbe in L∞ða; b;½0;1Þ. Then o e e y l∈N, he e exis s I⊂ 1; :::;mgsuch ha ~ ω¼[ k∈Iðyk−1;y kÞð5:4Þ sa is ies j~ ωj≤Zb a θdx;ð5:5Þ Zb aðθ−χ~ ωÞφdx ≤Cðb−aÞlþ1kDlþ1φkL1ða;bÞþCδkφkL∞ða;bÞ∀φ∈Wlþ1;1ð0;1Þ; ð5:6Þ whe e Cis a posi i e cons an ha depends on l, bu i is independen o θ,δ,a, and b. APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1203 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. P oo . I is enough o show he case a¼0,b¼1. The gene al one ollows using a ansla ion and a dila a ion which ans o ms ða; bÞin ð0;1Þ. Fo a gi en l∈N, by Lemma 5.1 we know he e exis s ω⊂ð0;1Þsa is ying (5.1) and such ha he numbe o discon inui y poin s o χωin ½0;1is a mos lþ1. We hen de ine I¼ k∈ 1; :::;mg∶ðyk−1;y kÞ⊂ωg and ~ ωby (5.4). By he de ini ion o ~ ω, we ha e ~ ω⊂ω, and hen using (5.1) when j¼0, we ob ain (5.5). Mo eo e , using ha χωhas a mos lþ1discon inui y poin s in ½0;1, we ha e jω ~ ωj≤ðlþ1Þδ:ð5:7Þ We now ix φ∈Wlþ1;1ð0;1Þ. Taking a polynomial po deg ee lsuch ha Z1 0jφ−pjdx≤CkDlþ1φkL1ð0;1Þ wi h Cindependen o φ( ake, o example, he Taylo polynomial o deg ee lo φ∈ Wlþ1;1ð0;1Þ⊂Clð½0;1Þ in some poin o ½0;1), we ge Z1 0ðθ−χ~ ωÞφdx ≤Z1 0ðθ−χωÞðφ−pÞdxþZ1 0ðχω−χ~ ωÞφdx ≤CkDlþ1φkL1ð0;1Þþðlþ1ÞδkφkL∞ð0;1Þ:ð5:8Þ This p o es (5.6) o a¼0,b¼1.▯ LEMMA 5.3. Fo >0small we ake a pa i ion P ¼ ykgm k¼0wi h m ∈Nsuch ha (2.8) is sa is ied. We de ine ^ Uby (2.5) and U by (2.10). (a) Fo e e y θ∈^ U, he e exis s ω∈U such ha Zx 0ðθ−χωÞφds ≤C 1 2kφkW1;1ð0;1Þ∀x∈½0;1;∀φ∈W1;1ð0;1Þ;ð5:9Þ whe e Cis a posi i e cons an independen o θand . (b) Fo e e y θ∈^ Uand e e y l∈N, he e exis s ω∈U such ha Z1 0ðθ−χωÞφds ≤C lþ1 lþ2kφkWlþ1;1ð0;1Þ∀φ∈Wlþ1;1ð0;1Þ;ð5:10Þ whe e Cis a posi i e cons an ha depends on l, bu i is independen o θand . P oo . We ake l∈N,γ∈ð2 ; 1Þ, and a subpa i ion Pγ¼ zigmγ i¼0⊂P o P which sa is ies γ− ≤zi−zi−1≤γ∀i∈ 1; :::;m γ−1g; ≤zmγ−zmγ−1≤γ: This implies in pa icula mγ≤1 γ− þ1≤3 γ:ð5:11Þ 1204 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Using ha o e e y i∈ 1; :::;m γ−1g he poin s ykwi h zi−1≤yk≤zia e a pa i ion o ½zi−1;z iwi h mesh , we can apply Lemma 5.2 in each in e al ½zi−1;z i o cons uc a se ω∈Usuch ha o e e y i∈ 1; :::;m γ−1g, we ha e Zzi zi−1ðθ−χωÞφdx ≤Cðγlþ1kDlþ1φkL1ðzi−1;ziÞþkφkL∞ðzi−1;ziÞ Þð5:12Þ o e e y φ∈Wlþ1;1ð0;1Þ. Fo x∈½0;1, we ake he la ge jsuch ha zj≤x; hen, hanks o (5.12) and (5.11), we ha e Zx 0ðθ−χωÞφds¼Zzj 0ðθ−χωÞφdsþZx zjðθ−χωÞφds ≤Cγlþ1kDlþ1φkL1ð0;1ÞþkφkL∞ð0;1Þ3 γþðx−zjÞ:ð5:13Þ Fo l¼0, he abo e inequali y and x−zj<γp o e Zx 0ðθ−χωÞφds ≤CγkD1φkL1ð0;1ÞþCkφkL∞ð0;1Þ γþγ: Minimizing in γ his quan i y, we deduce (5.9). On he o he hand, o x¼1¼zjinequali y (5.13) gi es Z1 0ðθ−χωÞφds ≤Cγlþ1kDlþ1φkL1ð0;1ÞþCkφkL∞ð0;1Þ γ; which minimizing in γp o es (5.10). ▯ Using Lemma 5.3 and easoning simila ly o Lemma 4.3, we easily deduce he ollowing. LEMMA 5.4. Le θbe in ^ Uand ∈L1ð0;1Þ. Then, o e e y >0, he e exis s ω∈U such ha , de ining uθ,u as he solu ions o (2.7) o θand χω, espec i ely, we ha e he ollowing: (a) kuθ−u kL1ð0;1Þ≤Cð1þk kL1ð0;1ÞÞ 1 2:ð5:14Þ (b) I belongs o Wl;1ð0;1Þ, hen     Mθ duθ dx −Mχω du dx    L∞ð0;1Þ ≤Cð1þk kWl;1ð0;1ÞÞ lþ1 lþ2:ð5:15Þ LEMMA 5.5. Le ∈L1ð0;1Þand θbe in ^ U; hen o e e y >0, he e exis s ω∈U such ha j^ JðθÞ−JðωÞj ≤C 1 2ð1þk kL1ð0;1ÞÞ:ð5:16Þ I o some l∈Nwe ha e ha belongs o Wl;1ð0;1Þ,F1ðx; s; ξÞ,F2ðx; s; ξÞa e independen o sand belong o Cl;1 locð½0;1×RÞ; hen APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1205 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. This pe mi s us, o example, o subs i u e in he de ini ion o he elaxed con ol se ^ U he se KðpÞby he (mo e simple) se o symme ic ma ices whose eigen alues a e com- p essed be ween λðpÞand ΛðpÞ. Rema k 9. De ining E¼ ðξ;η;pÞ∈RN×RN×½0;1∶ðη−λðpÞξÞ·ðη−ΛðpÞξÞ≤0g; he unc ion H ha appea s in (8.5) is a Ca a héodo y unc ion wi h domain Ω×R×E. An explici exp ession o Hin he whole o i s domain is no known in gene al. In he pa icula case whe e F1ðx; s; ξÞ,F2ðx; s; ξÞa e a ine unc ions in he a iable ξ, we ha e Hðx; s; ξ;η;pÞ¼pF1ðx; s; ξÞþð1−pÞF2ðx; s; ξÞ∀ðs; ξ;η;pÞ∈R×Ea:e:x∈Ω; while o nonlinea unc ions Fiin he a iable ξ, an exp ession o His only known in some pa icula cases (which essen ially a e conce ned wi h he nonlinea unc ion jξj2); see [3], [6], [8], [11], and [18]. Howe e , an explici ep esen a ion is always known in he bounda y o i s domain ðx; s; ξ;η;pÞ∶∈Ω×R×R×½0;1∶ðη−λðpÞξÞ·ðη−ΛðpÞξÞ¼0g; whe e Hðx; s; ξ;η;pÞis gi en by 8 > > < > > : F1ðx; s; ξÞi p¼1; F2ðx; s; ξÞi p¼0; pF1x; s; βξ−η pðβ−αÞþð1−pÞF2x; s; η−αξ ð1−pÞðβ−αÞi p≠0;1: ð8:7Þ Obse e ha he las line can be aken as he gene al exp ession o H, aking he alues o p¼0and p¼1by con inui y. Analogously as we did in he one-dimensional case, in o de o nume ically sol e p oblem (8.5), o >0we decompose Ωas Ω¼[ m i¼1 Ki;K idisjoin ;measu able;diamðKiÞ< ; i ∈ 1; :::;m g:ð8:8Þ Then, we disc e ize p oblem (8.5) as min ZΩ Hðx; u; ∇u; M∇u; θÞdx −di M∇u¼ in Ω;u¼0on∂Ω; ðθ;MÞ∈^ U;ðθ;MÞcons an in Ki;1≤i≤m ;RΩθdx≤κ: ð8:9Þ As we said in Rema k 9 in he case whe e he unc ions Fiðx; s; ξÞa e nonlinea in he a iable ξ, one o he main di icul ies o sol e p oblem (8.9) is ha His no known. To sol e his di icul y we can eplace Hwi h ano he unc ion. The ollowing esul is p o ed in [8] in he pa icula case F1ðx; s; ξÞ¼F2ðx; s; ξÞ¼FðξÞ. The gene al case ollows simila ly. 1212 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. THEOREM 8.1. We conside a unc ion ^ H∶Ω×R×E→R∪ þ∞gsuch ha ^ Hð:; s; ξ;η;pÞis measu able in Ω∀ðs; ξ;η;pÞ∈R×E;ð8:10Þ ^ Hðx; :; :; :; :Þis lowe semicon inuous in R×E o a:e: x ∈Ω;ð8:11Þ ^ Hðx; s; ξ;αξ;1Þ¼F1ðx; s; ξÞ;^ Hðx; s; ξ;βξ;0Þ¼F2ðx; s; ξÞ;ð8:12Þ ^ Hðx; s; ξ;η;pÞ≥Hðx; s; ξ;η;pÞ∀ðs; ξ;η;pÞ∈R×E a:e: x ∈Ω:ð8:13Þ Fo e e y >0, we decompose Ωby (8.9). Then, he p oblem min ZΩ ^ Hðx; u; ∇u; M∇u; θÞdx −di M∇u¼ in Ω;u¼0on∂Ω; ðθ;MÞ∈^ U;ðθ;MÞcons an in Ki;1≤i≤m ;RΩθdx≤κ ð8:14Þ has a solu ion (no unique in gene al) ðθ ;M Þ. Taking u as he solu ion o −di M ∇u ¼ inΩ;u ¼0on ∂Ω; we ha e ∃lim →0ZΩ ^ Hðx; u ;∇u ;M ∇u ;θ Þdx¼I wi h I he minimum alue o p oblem de ined by (8.5). The sequence ðθ ;M ;u Þis bounded in L∞ðΩÞ×L∞ðΩ;RN×NÞ×H1 0ðΩÞ. E e y unc ion ðθ;M;uÞ∈L∞ðΩÞ× L∞ðΩ;RN×NÞ×H1 0ðΩÞsuch ha he e exis s a subsequence o , s ill deno ed by , sa is ying θ ⇀ θin L∞ðΩÞ;M ⇀ MinL ∞ðΩ;RN×NÞ;u ⇀uinH 1 0ðΩÞ is such ha he unc ion ðθ;σ;uÞwi h σ¼M∇uis a solu ion o (8.6). Rema k 10. A i s choice o unc ion ^ His o ake ^ Hðx; s; ξ;η;pÞ¼8 < : F1ðx; s; ξÞi p¼1;η¼αξ; F2ðx; s; ξÞi p¼0;η¼βξ; þ∞o he wise: In his case, aking in o accoun ha ^ Hðx; u; ∇u; M∇u; θÞ<þ∞a.e. in Ωimplies ha θ is a cha ac e is ic unc ion we ge ha p oblem (8.14) can be w i en as min Zω F1ðx; u; ∇uÞdxþZΩ ω F2ðx; u; ∇uÞdx −di ðαχωþβχΩ ωÞ∇u¼ in Ω;u¼0on∂Ω; ∃I⊂ 1; :::;m gsuch ha ω¼S i∈I Ki;jωj≤κ: The e o e, wi h his choice o unc ion ^ H, Theo em 8.1 gi es he con e gence o he nu- me ical me hod consis ing in disc e izing di ec ly he o iginal (un elaxed) p oblem (8.1). APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1213 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Thanks o (8.7), ano he possibili y o ^ His o ake ^ H¼Hin ∂DðHÞ, and ^ H¼þ∞, o he wise. Fo his choice o unc ion ^ H, aking in o accoun ha o p≠0;1a ma ix M∈KðpÞsa is ies ðMξ−λðpÞξÞ·ðMξ−ΛðpÞξÞ¼ξ o some ξ≠0 ⇔Mis a lamina ion o αI; βIwi h p opo ions pand 1 −p ⇔EigðMÞ¼ðλðpÞ;ΛðpÞ; :::;ΛðpÞÞ: We can w i e p oblem (8.14) as min ZΩpF1x; u; β∇u−M∇u pðβ−αÞþð1−pÞF2x; u; M∇u−α∇u ð1−pÞðβ−αÞdx 8 < : −di M∇u¼ in Ω;u¼0on∂Ω; θ∈L∞ðΩ;½0;1Þ;Msymme ic;EigðMÞ¼ðλðθÞ;ΛðθÞ; :::;ΛðθÞÞa:e:in Ω; θ;Mcons an s in Ki;i¼1; :::;m ;RΩθdx≤κ: In his case, p oblem (8.14) consis s in disc e izing a pa ial elaxa ion o p oblem (8.1) consis ing in conside ing no only he o iginal con ols bu also he ones ob ained by a simple lamina ion. Clea ly, when His known, ano he possibili y is o ake di ec ly ^ H¼H. In his case we a e disc e izing he elaxed con ol p oblem (8.9). Rema k 11. Al hough Theo em 8.1 gi es he con e gence o he disc e ized p oblem (8.14), i does no p o ide any e o es ima e. In pa icula , i does no show which choice o he unc ions ^ Hmen ioned in Rema k 10 is be e . As we saw in he p oo o he es ima es o he one-dimensional p oblem, in o de o ob ain an es ima e o he con e gence a e o he nume ical me hod, one idea is o con- s uc om a elaxed con ol ðθ;MÞano he con ol ðθ ;M Þin he se o disc e ized con ols such ha he solu ions o he s a e equa ions ela i e o ðθ;MÞand ðθ ;M Þa e close. In he case whe e ^ H¼H(which can only be used i His known), one idea is o ake ðθ ;M Þas he mean alue o ðθ;MÞin each elemen o he iangula- ion. Deno ing by uand u he solu ions o −di M∇u¼ in Ω; u¼0on∂Ω;−di M ∇u ¼ in Ω; u¼0on∂Ω wi h in H−1ðΩÞand aking in o accoun ha −di M ∇ðu−u Þ¼−di ðM −MÞ∇uin Ω; we deduce ha ZΩj∇ðu−u Þj2dx≤CZΩjðM −MÞ∇uj2dx; which pe mi s o es ima e he di e ence o u−u depending on he smoo hness p ope - ies o Mand uand hen o es ima e he e o o he disc e ized me hod. When His no known and he e o e we need o disc e ize di ec ly he o iginal p oblem o o conside some pa ial elaxa ion, he choice o ðθ ;M Þis no clea . 1214 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Rema k 12. In Theo em 8.1, we ha e disc e ized he se o con ols, bu he s a e equa ion is di ec ly sol ed. I will be in e es ing o s udy he con e gence when we also disc e ize his equa ion, and in pa icula o s udy wha he ela ion is ha we mus use be ween he iangula ion chosen o he con ols and he one chosen o he esolu ion o he s a e equa ion. A esul in his sense can be ound in [8], showing ha in some cases he me hod con e ges using he same iangula ion o disc e ize he con ols and he s a e equa ion. Acknowledgmen . The au ho s a e g a e ul o he Basque Cen e o Applied Ma hema ics o i s hospi ali y and suppo in se e al isi s. REFERENCES [1] G. ALLAIRE,Shape Op imiza ion by he Homogeniza ion Me hod, Appl. Ma h. Sci. 146, Sp inge -Ve lag, New Yo k, 2002. [2] G. ALLAIRE AND S. GUTIÉRREZ,Op imal design in small ampli ude homogeniza ion, ESAIM Ma h. Model. Nume . Anal., 41 (2007), pp. 543–574. [3] J. C. BELLIDO AND P. PEDREGAL,Explici quasicon exi ica ion o some cos unc ionals depending on de i a i es o he s a e in op imal designing, Disc e e Con in. Dyn. Sys ., 8 (2002), pp. 967–982. [4] B. BOURDIN,Fil e s in opology op imiza ion, In e na . J. Nume . Me hods Eng g., 50 (2001), pp. 2143–2158. [5] C. CARSTENSEN AND C. ORTHNER,Analysis o a class o penal y me hods o compu ing singula minimize s, Compu . Me hods Appl. Ma h., 10 (2010), pp. 137–163. [6] J. CASADO-DÍAZ,J.COUCE-CALVO,AND J. D. MARTÍN-GÓMEZ,Relaxa ion o a con ol p oblem in he coe i- cien s wi h a unc ional o quad a ic g ow h in he g adien , SIAM J. Con ol Op im., 47 (2008), pp. 1428–1459. [7] J. CASADO-DÍAZ,J.COUCE-CALVO,AND J. D. MARTÍN-GÓMEZ,Relaxa ion o op imal design p oblems wi h nonlinea unc ionals in he g adien , o appea . [8] J. CASADO-DÍAZ,J.COUCE-CALVO,M.LUNA-LAYNEZ,AND J. D. MARTÍN-GÓMEZ,Op imal design p oblems o a non-linea cos in he g adien : Nume ical esul s, Appl. Anal., 87 (2008), pp. 1461–1487. [9] J. CASADO-DÍAZ,J.COUCE-CALVO,M.LUNA-LAYNEZ,AND J. D. MARTÍN-GÓMEZ,Disc e iza ion o coe icien con ol p oblems wi h a nonlinea cos in he g adien , in In eg al Me hods in Science and Enginee ing, Compu a ional Me hods 2, C. Cons anda and M. E. Pé ez, eds., Bi khäuse , Bos on, 2010, pp. 55–63. [10] D. CHENAIS AND E. ZUAZUA,Fini e elemen app oxima ion o 2D ellip ic op imal design, J. Ma h. Pu es Appl. (9), 85 (2006), pp. 225–249. [11] Y. GRABOVSKY,Op imal design o wo-phase conduc ing composi es wi h weakly discon inuous objec i e unc ionals, Ad . in Appl. Ma h., 27 (2001), pp. 683–704. [12] R. LIPTON AND A. P. VELO,Op imal design o g adien ields wi h applica ions o elec os a ics,in Nonlinea Pa ial Di e en ial Equa ions and Thei Applica ions, Collège de F ance Semina XIV, D. Cio anescu and J. L. Lions, eds., S ud. Ma h. Appl. 31, No h-Holland, Ams e dam, 2002, pp. 509–532. [13] K. A. LURIE,Applied Op imal Con ol Theo y o Dis ibu ed Sys ems, Plenum P ess, New Yo k, 1993. [14] K. A. LURIE AND A. V. CHERKAEV,Exac es ima es o he conduc i i y o a bina y mix u e o iso opic ma e ials, P oc. Roy. Soc. Edinbu gh Sec . A, 104 (1986), pp. 21–38. [15] F. MURAT,Un con e-example pou le p oblème du con ôle dans les coe icien s, C. R. Ma h. Acad. Sci. Pa is, 273 (1971), pp. 708–711. [16] F. MURAT,Théo èmes de non exis ence pou des p oblèmes de con ôle dans les coe icien s, C. R. Ma h. Acad. Sci. Pa is, 274 (1972), pp. 395–398. [17] F. MURAT AND L. TARTAR,H-con e gence, in Topics in he Ma hema ical Modelling o Composi e Ma e ials, P og . Nonlinea Di e en ial Equa ions Appl. 31, L. Che kae and R. V. Kohn, eds., Bi khäuse , Bos on, 1998, pp. 21–43. [18] P. PEDREGAL,Op imal design in wo-dimensional conduc i i y o a gene al cos depending on he ield, A ch. Ra ion. Mech. Anal., 182 (2006), pp. 367–385. APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1215 Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. [19] S. SPAGNOLO,Sulla con e genza di soluzioni di equazioni pa aboliche ed elli iche, Ann. Sc. No m. Supe . Pisa Cl. Sci. (5), 22 (1968), pp. 571–597. [20] L. TARTAR,P oblèmes de con ôle des coe icien s dan des équa ions aux dé i ées pa ielles, in Con ol Theo y, Nume ical Me hods and Compu e Sys ems Modelling, Lec u es No es in Econ. and Ma h. Sys ems 10, Sp inge -Ve lag, Be lin, 1975. English ansla ion: F. MURAT AND L. TARTAR,On he con ol o coe icien s in pa ial di e en ial equa ions, in Topics in he Ma hema ical Modelling o Composi e Ma e ials, P og . Nonlinea Di e en ial Equa ions Appl. 31, L. Che kae and R. V. Kohn, eds., Bi khäuse , Bos on, 1998, pp. 1–8. [21] L. TARTAR,Cou s Pecco , Collège de F ance, 1977, pa ly w i en in [17]. [22] L. TARTAR,Es ima ions ines de coe icien s homogénéisés, in Ennio de Gio gi Colloquium (Pa is 1983), Res. No es Ma h. 125, P. K ee, ed., Pi man, London, 1985, pp. 168–187. [23] L. TARTAR,Rema ks on op imal design p oblems, in Calculus o Va ia ions, Homogeniza ion and Con inuum Mechanics, Se . Ad . Ma h. Appl. Sci. 18, G. Bu azzo, G. Bouchi e, and P. Suque , eds., Wo ld Scien i ic, Hackensack, NJ, 1994, pp. 279–296. [24] L. TARTAR,An in oduc ion o he homogeniza ion me hod in op imal design, in Op imal Shape Design, 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. [25] V. V. ZHIKOV,Weigh ed Sobole spaces, Sb. Ma h., 189 (1998), pp. 1139–1170. [26] J. GOODMAN,R.V.KOHN,AND L. REYNA,Nume ical s udy o a elaxed a ia ional p oblem om op imal design, Compu . Me hods Appl. Mech. Eng g., 57 (1986), pp. 107–127. 1216 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed. Rep oduced wi h pe mission o he copy igh owne . Fu he ep oduc ion p ohibi ed wi hou pe mission.