Exact Controllability and Asymptotic Analysis for Shallow Shells

Chin. Ann. Math.28B(1), 2007, 93- -122Chinese Annals ofDOI: 10. 1007/s11401-005-0183-zMathematics, Series BCTheEditorial Oficeof CAM andcorlal oice.of T AN andSpringer- Verlag Berlin Heidelberg 2007Exact Controllability and AsymptoticAnalysis for Shallow ShellsS. KAIZU*N. SABU*A bstract The authors consider the exact controllability of the vibrations of a thin shal-low shell, of thickness 2e with controls imposed on the lateral surface and at the top andbottom of the shell. Apart from proving the existence of exact controls, it is shown thatthe solutions of the three dimensional exact controllability problems converge, as the thick-ness of the shell goes to zero, to the solution of an exact controllability problem in twodimensions.Keywords Exact controllability, Asymptotic analysis, Shallow shells2000 MR Subject Classification 74K251 IntroductionThe problem of exact controllability has been studied extensively by J. L. Lions [11, 12]and the asymptitic behaviour of thin plates and shells has been studied by P. G. Ciarlet, VLods, B. Miara and others (cf. [1- 6]). I. Figueiredo and E. Zuazua [7] have studied the exactcontrollability and asymptotic behavior for thin plates and in this paper, we study the exactcontrollability problem for thin shallow shells and the limiting behaviour of the solutions.We begin with a brief description of the problem and describe the results obtained.Let 0e =重(2*), n°=wx (-e,e) withw C R2, and the mapping φ :0 - + R3 be givenpy重(x°)= (x1,x2, e4(x1,x2)) + xgaE(x1,x2)for all xe = (x1,x2,x4)∈0，where中is an injective mapping of class C8 and af is a unitnormal vector to the middle surface 重(丙) of the shell. Let 70 be the boundary of w and letT=(rox(-e,e)) andis = φ°(wX {<}).The exact controllability problem may be formulated as follows: given initial data {气,中}in a suitable energy space, does there exists a time T> 0 and controls ie = (i号) on Tδ and0= (0) on生such that the unique solution ψe of the problem (2.7) reaches equilibrium attime T, that is ψ(T)= ψe(T)= 0.In this article we first show, using Hilbert Uniqueness Method (HUM), that this problem isexactly controllable by assuming the validity of the regularity result described in Lemma 3.1.We then make appropriate scalings on the data and the unknowns and transfer the problemto a domain几= w x (-1,1) which is independent of E and study the asymptotic behaviourManuscript received May 17, 2005.* Department of Mathematics, Faculty of Education, Ibaraki University, 2-1-1, Bunkyo, Mito, Japan.E-mail: sabu@math.itb.ac.in中国煤化工YHCNM HG.)4s. Kaizu and N. Sabuof the scaled controlled solutions. The key to the asymptotic analysis lies in establishing theweak convergence (4.32) of the scaled solutions φ(e) of the homogeneous problem (3.9) andthe strong convergence (4.55) of the scaled solutions 0(e) of the forward Cauchy problem (3.1).We then show that the limit (4) of the scaled controlled solutions (ψ(e)) of (3.18) is of theKirchhff-Love form; that is, ψ3 is independent of x3,ψa = ψa - x30a43,ψa is independent of x3.Moreover ψ3 is the solution (in the transposition sense) of a two dimensional problem with con-trols on the boundary and interior of the shell and the functions ψa can be uniquely determinedin terms of a known function.2 The Three-Dimensional ProblemThroughout this paper, Latin indices vary over the set {1, 2, 3} and Greek indices over theset {1, 2} for the components of vectors and tensors. The summation over repeated indices willbe used.Let w C R2 be a bounded domain with a Lipschitz continuous boundary 70 and let w lielocally on one side of 70. For each E > 0, we define the sets02° =wx(-e,)，TE=wx{e}， Tr=7ox(-ε,e).Let x°= (21,x2,x2) be a generic point onMe andletθa=δ=x andδξ=品We assume that for each E, we are given a functionφ° :w→R of class C3. We then definethemaphe:w→Rbyh*(x1,x2)= (x1, x2,4*(x1,x2)) for all (x1,x2)∈w.(2.1)At each point of the middle surface Se = h"(w), we define the normal vectora°= (|8ψ°|2 +|2ψ9|2+1)-*(-81ψφ°,-02φ°,1).For each E > 0, we define the mapping φ :0°- + R3 by重(x°)= (x1,x2,4* (x1,x2))+ xga*(x1,x2) for allxl∈0°.(2.2)It can be shown that there exists an∈o > 0 such that the mapping φ :9e→重“(S°) isaC1diffeomorphism for all 0 <∈≤∈o. The set∩° =重(25) is the reference configuration of theshell. We denote by e the standard basis in R3.For 0< ε≤Eo, we define the setsf4=重(T生)，f=亚(FS)and we define vectors 9{ and g'.e by the relationsg{=δφ° and gi°.9g{=8娃which form the covariant and contravariant basis respectively at重(x°). The covariant andcontravariant metric tensors are given respectively by9%=g;g5 and ge=g's.gs*.中国煤化工MYHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells)5The Christoffel symbols are defined byr =g"9.8g9.Note however that when the set ne is of the special form 0° =wx (-e, e) and the mapping重is of the form (2.2), the following relations holdr38=T =0.The volume element is given by V g"dxe whereg° = det(g;).It can be shown that for E suficiently small, there exist constants 91 and 92 such that0<91≤g°≤92.(2.3)Let Aikl,e denote the elastic tensors. We assume that the material of the shell is homoge-neous and isotropic. Then the elasticity tensor is given byAikl.e = X8vskl + (ik8ji + 8i8ik),(2.4)where λ and μ are the Lame constants of the material.It satisfies the following coercive and symmetry relations. There exists a constant c > 0such that for all symmetric tensors (ij)Aijkletxrtj≥c 2 (tig)2,(2.5)i,j=1Aik,e = Ahi,e = Aikl.e.(2.6)Then the system of equations which govern the vibrations of the medium ne is:ρ°ξ - δ暗(ψ)=0inQ°=i° x (0,T),ψ=0{on2g=fx (0,T),(2.7)的(9)的=谐on=f生x (0,T),ψ<(0)=心，ψ←(0)=幄 ini*,where DE is the unit normal vector along the boundary ofn*, ρ∈is the density of mass and的(2)= 4il.e(e),的()= (孵+ 85的)。(2.8)The controls are ie on the lateral surface Tg through Dirichlet action and心on the upperand lower faces 14 through Neumann action.We define the spacesH。(029)={0*∈H(02):的。=0},(2.9)V(2*)= [。(2)]",(2.10) .x(D2)= {9=∈L'(0,T;2(2):g∈L'(0,T;H。(29)])}.(2.11)中国煤化工MYHCNM HG.)6s. Kaizu and N. SabuWe introduce the functionqi = (G) byq*()=°-站=φ(x°)-φ°(x8) Va° ∈0°,where话is a fixed point in the middle surface of the shell, and the constants R(5) and Te are2√βC(eR(站)=|x°一(心，T°=VP max {R(路),R(2) J√mwhere C(R) is the constant of continuity of the trace map tr : V(2)→[L2(e*129)]3. Wedenote by号the j-th component of the tangential gradient on 8e .Throughout this paper, we denote by Ci,i = 1,2,3,... various constants which are inde-pendent of e.3 Preliminary ResultsIn this section, we will first recall some existence, regularity and energy estimate results forthe forward Cauchy problem associated with (2.7) and then we will deduce some identities forthe 3D problem.Let θe be the solution of the following forward Cauchy problem; that is,ρ°θ≤- η号(@*)=f°in Q,θ=0on 26,(3.1)的(@)跨=0on生,0*(0)=θg，0∈(0)=θq inie.Ife=0then we use心,ξ and φ in place of的,时and 0e respectively.Let E0 (t) denote the energy of the solutionθe at timet∈[0, T]; that is,E”()=16 pf Ser()PdDe + (()),(3.2)wherea*(0*,$9)= | 0(9)8(9)dne.(3.3)When t= 0, we haveE"*(0)=分[ p乙iiPdSe +za(s,的).(3.4)Remark 3.1 For u∈C2([0,T],[V(2)') and v∈V(N2) we denote by | ivdax the dualityproduct between i∈[V(2)' and V(82).Lemma 3.1 (a) Assume that的g∈V(2), 09∈[2(2)3 and fe∈L(0,T:(2(2)).Then there exists a umique solution θ of (3.1) with0e∈C"([0,T], V(2)(0)()(9()nw2(0,T](2).(3.5)中国煤化工YHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells .)7(b) If θ∈H2(29)∩v(09),θs∈V(D2) and fe∈L+(0,T;V(82)) then0e∈C"([0,T], H号+°(29)∩V(2))∩C"(0,T], V(52))∩W2(0,T],(2(D2)3)) (3.6)for someδ> 0.(c) The following energy estimate holds:(3.7)D”()sa{*(0)+本二[。lim}Moreover, if fa,e∈X(S2) and f3.e∈L'(0,T;L2(2)), thenB4 ()SC{e"(0)+之I和眼+=[。leuni]}(3.8)a=1Proof The proof of (a) is classical and for (b) we refer the reader to the work of Grisward[8] and Nicaise [14]. The proof of (c) is similar to the proof of [7, Lemma 2.1].3.1 Identities related to the 3D shell problemLetting fe= 0 in (3.1), we see that φ satisfiesρ°φS-的的(φ)=0imQ,φ=0on 2,(3.9)号(中)j=0on生,φ(0)=帖，φ*(0)=φ insit.Then we have the following identity.Lemma 3.2 Letθe be the solution of (3.1) with的∈H2(R9)∩V(2e),的∈V(22) andf°∈L(0,T;V(29)) andφ be the solution of (3.9) with φ∈H2(22)∩V(029), φs∈V(12).ThenI (.9)g的(),5att +(q° .D9)[°sφs -的(和)8jof]df$dt的(9)idS"dt一(3.10)Proof The proof follows by multiplying the first equation of (3.1) by .8;φ and the firstequation of (3.9) by g.0jθ and integrating by parts.Notice that1。=f" fp°gφsdsLedt- |的(和 )gediedt.(3.11)中国煤化工YHCNM HG.)8s. Kaizu and N. SabuHence the equation (3.10) can be written asl (".i )的(市* )8g0;dgdt +(q° i°)\p°0qφ - oζ(σ )ago明]d(dtJs。[/.的(游8)时。+[/6. (:56).1g。+2[fρ9oφ:d8e]°pogadredt. (3.12)Corollary 3.1 Let φ be the solution of (3.9) with initial data in H2(D2)∩V(2e)x V(22).Then the following identity holds.(。(。们的的的能始水十会(.0[p Z()2 -的(面)间利宝山=[/。(G.,g站+)die]。+ Eol (0)dt.(3.13)Proof The proof follows by taking fe =0 andθe =φ in (3.12) and noting that E" (t) =3.2 The exact controllability problem for 3D shellWe will now prove the exact controllability result for 3D shell using the Hilbert UniquenessMethod. In order to do that we will first establish some a priori estimates for the energy Eψ* (t)of problem (3.9). We will also introduce the transposition formulation and define the HUMoperator and show that it is an isomorphism between V(S22) x L2(0e) and its dual.Theorem 3.1 (Direct Inequality) Let 0 < ε≤1 and T > 0 be fixed. Assume thatδ∈H2(2e)∩V(2e) andφ∈V(2e). Then the solution φ of (3.9) with initial data {中心, φξ }satisfiese (".的)的(和)Bζafdt+ (°.D的)[p* 2 (9)2-号(而)&邴ddt≤C3E0° (0). (3.14)Proof The proof follows from the above corollary.Theorem 3.2 (Inverse Inequality) Let0<ε≤ 1 andT > T°. Then for every solution φ°of (3.9) with initial data {%,φi}∈H2(n2e)∩V(09)x V(52), we have[T- T]Eo°(O)≤C{ [。(G。C门)的(edt(3.15)Proof Proceeding as in [7, Theorem 3.2], it can be shown thatp“中(街8的的+的)d2e;max{(i),C.2, R(站)}(3.16)The result then follows from (3.13) and the above estimate.中国煤化工YHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells .)9From the above two theorems it follows that for a fixed E and T with0<∈≤1 andT > Te,the mapping{%,鸭}∈H2(29)nV(2)x V(52)→I{o↑f}I，where川{心,所}1={。(° ，的)路(和* )f8dt .+ I (g.D的)[p* 2(的)2 -哈(*)eqr生at}(3.17)is a norm in H2(D2)∩V(29)x [V(02 )]", is equivalent to the usual norm in V(829)x [2(2)]3.3.3 Transposition formulationFor a given {ξ,φ1}∈V(Qe) x [L2(ne ), we first solve the homogeneous problem (3.9) forφ° with initial data {,φ }. Then we introduce the backward Cauchy problem: find ψe suchthatρ°9 - 8号(4*)=0in Q,的=%的on 5,(3.18)的(09所=粥s市一行的(中9)] on 生，ψ(T)=0，ψ(T)=0in介*.The transposition formulation of (3.18) can be obtained as follows.We multiply the first equation of (3.18) by 0, the solution of (3.1), and integrate by partson Qe and we obtain the following identity+ (.0)p°φ90q - (φ° )明ddt.(3.19)Definition 3.1 The function ψe is a solution of the problem (3.18) in the sense of transpo-sitionifψe∈L∞(0,T;(L2(12)), the traces {ie (0), ive(0)} makes sense in [L2(52*)]3x V(ne)'and ψ∈satisfies(0-)(1(*(0)}({,69}).- IA的严edx*=(q° .D门)的(0 )8;dTSdt+.(Q° .D9)[*°2go -号()的ddt (3.20)for any f°∈L(0,T;L2(he)3) and for any {,0}∈V(0e)x (L2(2))3 with({4(),-4<(0)}, {的, 0})e = <{p" (4(0),-0°9(0)}, {0os, 0i1}),(3.21)where <, ) denote the dual product between V(2e) x [L2(ie )]3 and its dual.中国煤化工YHCNM HG.100s. Kaizu and N. SabuTheorem 3.3 Let 0 < E≤1 and T> Te be fiaxed. Then there exists a unique solutionψe∈L∞(0, T;(L2(N2))&) of (3.18) in the sense of transposition.Proof The proof follows by duality arguments.3.4 The HUM operatorLet0<∈≤1 be fixed and {帖,的}∈V(29)x [z2(2e)3. First we solve the problem (3.9)with initial data {, φ$} and then we solve the problem (3.18) in the transposition sense, i.e,we solve (3.19). Let ψ∈be the solution of (3.19) and let ψ脂= ψ(0) and ψ = ψ=(0). Then wedefine(({元,程})= {r,一8}, .that is,(《({品,的},{的, 6))) = ({站,一路}, {05,61})efor any {0s,時}∈V(2°)x [2(23)]3.Theorem 3.4 Let0< ε≤1 and T> Te be fixed. Then the operator Ae is a continuousisomorphism betueen V(2*)x [L2(De )]3 and its dual. Moreover if {Po，0} = (^*)-({qwi,-S})where {qq,→48}∈V(0e)'x [2(529)]3, we have{j" 112(0) +(中心, %)}}声≤T-Tl{4q,- 4l(y(x1(092x)3.(3.22)Proof The proof is similar to the proof of [7, Theorem 3.4].Theorem 3.5 (Controllability Result) Let0<ε≤ 1 be fixed. IfT > Te, then the elasticitysystem (2.7) is exactly controllable. More precisely, if {vg, }∈[L2(ie )]3 x[V(2e )]' then thereexist controls of the formeieon(3.23)的=明叮[*一号号(中9)] on 重,where φ° is the solution of (3.9) with initial data{邮,φ}= (-4({v5,-45})(3.24)such that the solution ψe of (2.7) satisfies ψ(T)= ψe(T)=0.Proof We first solve the problem (3.9) with initial data {酯,φ{} = (^9)-({v$q, -路})and we obtain the function φ°. Then we define the controls ie and心as in (3.23). In view of(3.24), the solution ψe of (3. 18) satisfiesψ<(0)=站，ve(0)=w,or, equivalently, the solution of (2.7) satisfies ψ(T)= ψ<(T)=0.中国煤化工MYHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells .101Note that because of the relation (3.10), the equation (3.20) is equivalent to[/. ((a)a,n,。+[ ()en'。+ /:3prosq;dsredt(3.25)4 The Scaled P roblemTo study the asymptotic behaviour of the solutions ψe as E→0, we first transform theproblem (2.7) toS°=wx(-e,e) and then to the domainM=wx (-1, 1) which is independentof E.Since the mappings亚:Se -→n are assumed to be Cl-diffeomorphisms, the correspondenceo(x° ,t)g"e = o(x",t)e'induces a bijection between V(2*) and V(Se ) whereV(2)= {ve∈(H(2))3:09 = 0on FH}.Then we have (cf. [2)hμ=X咋- Fie(x°)%,臨(6)= el(*g9)o∈V(2) x [L2(2)}3 ofthe problem (3.9) satisfyE(0) = llevp11(e),e√pφ12(e),√3()l2(2)]) + a()(),o()}≤C1. (4.31)Let {φ(E)}e>0 be the scaled (weak) solutions of (3.9) with initial data {oo(e),φ1()}. Then thereexrists a subsequence {o(e)}e>o (still indexed by∈for notational convenience) satisfying thefollowing.(i) There eristsφ∈L∞(0,T;V(S2))∩W1,∞(0,T;[L2(2)]3) such that, as∈→0,φ(e)→φweakly* in L°(0,T; V(2))，φ3(e)→P3weakly* in L(0,T; L2(2)),∈φa(e)→0.weakly* in L∞(0,T; L2(2)),ealls(中()) →ealls()weakly* in L°(0,T; L2(2)), .(4.32)eal|3(φ(e))→0e()→x+2ella(中) weakly* in L∞(0,T;2(2)).(ii) The limit function φ= {φa,φ3} is a Kirchhoff- Love displacement, that is, φ3 is inde-pendent of x3, .φa=φa - x3Oap3，φa is independent of x3. .(4.33)中国煤化工YHCNM HG.106s. Kaizu and N. SabuMoreover, φa = (S()3))a where for a given φ3∈H3(w), (S(P3)) = (中a, φ3) is uniquely deter-mined by ./.[、4),e0on0 + 4sea()]agnadwJwlλ+ 2μ=] [(+1(0rx403)800 + 4u(0x)apgnadw Vno ∈H(w)(4.34)and φ3∈C'([0, T]; H3(w))∩C'(0, T]; L2(w)) is the unique solution of the 2D shell problem2pφ3 - Aapma3(中3)- (n%g(()3))8a34)=0 inwx (0,T),φ3=8φ3=0on8wX (0,T),8v(4.35)中3(0)=寸|. φo3dx3， φ3(0)=， |iφ13dx3 in w，where_4Xμmas(3)=-(3x+ 2元AS30aβ +-0aSs},(4.36)n&()=A1Leo()oas+4hueag()(4.37)+ 2pand {中o3, φ13} is the weak: limit of {o3(e), φ13(e)}e>o in H。(Q2)x L2(S2).Proof Letting fi(e)= 0 in (4.26), we see that φ(e) satisfies| p[e2$a()ug*f(e) + e中a(e)v3g°3() + e$s()a9^3(e) + 3(e)v3g33(e)]√g(e)dx+ I (le)(e()e)()()>()dx=0，Vu∈ V(2).(4.38)Taking v= φ(e)(x, t) in the above equation, we have1da I pe(a()a(e)g() + 2e()()() + (a()}*()Vg(.)y*xx+ 1d(4.39)+o年I H(l()()()(=()()V g(e)dx = 0.Using the positive definiteness of (g"(e)) and integrating from0 tot,0 0 such that for 0< E≤Eo,{IpvulI.a}≤Cis{ 2 l((ulola} Vo∈ V().(4.42)，jUsing the assumption (4.31) and the above inequality we havepla(ell. + pl(a(all. + l(ll≤p(4(laIal(e)l., + l()lY + E l(()l)≤pCis( J(Cea() + (o()2) + (le(e(e(=(e)e)()l(x‘x≤C16.(4.43)Hencel(lal()lo≤C16，l(3l)o ≤C16，l()≤C16，l(()Cln< C16. (4.44)From this, it can be shown, by using the same arguments as in [1], that for each fixedt∈[0, T], the weak * convergence (4.32) holds, φ is of the form (4.33) and| (l()()()(v)v9(e)dx+- | map()})8agn3dw- | ()ap4n3sdw| n%a()8znadw(4.45)for allv= (7a - x3OaI3, n3)∈VkL(2).Since (∈φa(e), φ3(e))→(0,中3) weak * in L∞(0, T; L2(2)), it follows that for fixedv= (v;) =(Na - x3Han3,n3)∈Vkc(S2),_∈φa(e)va √g(e)dx→0weak * in L°(0,T),| p3(e)v3√g()dx→φ3gv3dx weak* in L∞(0,T).This implies that[",.eial()as√vgedxdt =-coa(eras√vqe)dxdt→0，Vζ∈ D(0,T) (4.4)andφ3()v35√g(e)dxdt→- I_ φ33ζ d.xdt| φzv3ζdxdt，Vζ∈ D(0,T),(4.47)i.e,|. e中a()a√g(e)dx→0 and |_ 中3()v3√g(e)dx→ |_ φzv3zdx in D'(0,T). (4.48)中国煤化工YHCNM HG.108s. Kaizu and N. SabuHence passing to the limit in (4.38) by taking v= (na - x3OaN3, n3)∈VkL(02), we getp | Q373dwma(3)agnsdw- |唱()asondw+ | n2p()8gnadw=0 (4.49)for all v= (na - x30an3,n3)∈VkL(2). This is equivalent to .2p| 0373dw- | ma(3)ngf7rdw -| n%g(q)8asonsdw=0， Vη3∈H3(w)， (4.50)| n&e(φ)8gnadw=0， Vna∈H}(w). (4.51)Since φ is of the form (4.33), the equation (4.51) can be written as°4λμ. Lxreoo(。)δap + 4uea(Pa )]3gmadw4λμ下((an403)ap + 4(ag403)j8gnodw，V1a ∈H(w).(4.52)The left-hand side of the above equation is elliptic over H}(w) and for a given φ3∈H2(w),the right hand side defines a linear functional over H (w) and hence by Lax-Milgram Theorem,φa can be uniquely determined in terms of φ3.Theorem 4.2 Assume that the initial data {0o(e), θ1(e)}∈V(2)x [L2(2)]3 and the appliedbody forces {f*(e)} of the variational problem (4.26) satisfy the following. .(i) f(e)→f°∈X(S2), f3(e)→f3∈L(0,T; L(0)(ii) The sequence {0o()} verifies0o()→0ostrongly in V(02),eal|3(0o))→0 .strongly in L2(92),(4.53)el|}(o())→x +2-Falla(o) strongly in L2(2).(ii) The sequence {0r(c)} satisfies∈θ1a(∈)→0strongly in L2(Q),(4.54)013(e)→θ13 strongly in L2(2), θ13∈ L2(w). .Then the solutions {0(<)}e>o of (4.26) satisfy the fllowing .(i) There exists a functionθ∈L∞(0,T;V(2))∩H'(0,T;[L2(S2)]3) such thatθ(e)→θstrongly in L2(0, T; V(2)),03(e)→03strongly in L2(0, T; L2(2)),e0a(e)→0strongly in L2(0, T; L(2)),ealls(0(e))→eal|s(0)(4.55) .ea|3(0(e))→0 .strongly in L2(0,T; L2(2)),e(3())→x+hralla(0) strongly in LD(0:,1,(L(2)).中国煤化工MHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells .109(i) The limit function 0 = {0a,03} is a Kirchhoff- Love displacement, that is 03 is indepen-dent of x3,θa= θa- x38a03，Ba is independent of x3.(4.56)Moreover, Oa = (Sfa (3))a, where for a given fa∈X(82) and 03∈H(w), (Sfa (03)) = (@a,03)is uniquely determined by{「_ 4Xμ=] lx+2p((a3$p03)8ap + 4u(a3403)|Ognadw+/.(_. fis)oadw Vna∈H()(4.57)and 03∈C°([0,T], H(w))∩C'([0, T];L2(w)) is the unique solution of the 2D shell problem2p0s - agmaq(03) - ((3)a34)= | fadxrs +8。./”。x3f^dx3 inwx (0,T),803.03==0 on 8wx (0,T),(4.58)03(0)=)/. Oadx3， 6<(0)=)/.013dx3 in w,where {0o3, 013} is the weak limnit of {0o3(e),013(e)}e>o in H(9) x L2(2).ProofUsing the boundedness of fQ(e)∈X(0) and f"(E)∈L'(0,T,L2(2)), it can beshown by proceeding the same way as in Theorem 4.1 that the convergences (4.55) holds weak*in L∞(0, T, L2(Q)) (hence weakly in L2(0, T, L2(2))), (0;) is of the form (4.56) and (0;) satisfies2 | 037n3dx- | map(03)dasn3dw- | n&g(0)8aj1n3dw+ | ngg(B)8gMadw(4.59)for all v= (na - x38aN3, n3)∈VkL(Q2). This is equivalent to (4.57)-(4.58).To show the strong convergence of (e0a(e), 03(e), ei:()(e)) in (L2(0, T; L(2))2 x L2(0,T;(L2(2))9)，it is enough to show that they converge in norm as we already know that theyconverge weakly.For σij,Tj∈(L2(0,T; L2(2))9, we define(oij,To)=(4.60)Using (4.26), we have。" | p(e(e)2xdr +.p(<()2dxdr .+ | | Al((()()e)()(x))d~= . | pe(0a()8()gx$(e)√g(e)dxdr +o(e()g3()V g(e)dxdr中国煤化工MHCNM HG.110s. Kaizu and N. SabuAl(le(e(e()(e)(e)()v)()dx。(ASM(e√q(e)- 4(l((e((e)e(()xdr>=f。" J"e021(eg(e(a(e)drdr+peθ1a(<)013(e)g^()√g(e)dxdr+2peθna(<)013(e)g^(e)V 9() d.xdr-/"1。o-2f (r(ae(addd +2[" [。f().(e()v aedrdtpea(e)<(e)g"3(e)√g(e)dxdr(4.61)Letting E→0 and using (4.18)-(4.20), (4.53), (4.54) and (4.55) the right hand side of the aboveequation goes to(4.62)wherea(0o, 0o)=- | ma3(0o)8ap0o3dw -a(Oo)daspOo3dw +n品a(Oo)8s0oadw. (4.63)中国煤化工YHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells111On the other hand,=]。[J0101s)Pdx +(0.0)+2/ frdaraldra=f [/。p03)Adr + a0,.0) -2。f()0.(O](r +2 I f(10:()dx]dr-2%。f" Jjoodud+2=[ 1。(a3)2rdr+ [" a(0.0)d7-2[ [.1f()。(0)dxdr +2(4.64)-2f”f" Jjoodud+2,From (4.62) and (4.64) it follows that I|(e0a(e), 03(e), e()())|))→I|(0, 03, eiil.That 0;(e)→0; in V(S2) can be proved using the strong convergence of eilj(e)(0(e)) to eilj;in (L2(2))9.Lemma 4.2 For 73,53∈L2(0,T; H(w)) defineb(,m)= f" I (()ndwd.(4.65)Then b(., :) is symmetric.Proof We first claim that forη= (n;), ζ=(6;)∈L2(0,T;(H}(w))2 x L2(0, T; H2(w)) thebilinear form B(, ) defined byB(i,n)= | n()[),840n + Ognoldrat(4.66)is symmetric. We haveB(i;,ni)=n%g(S)[8as4n3 + agna]dwdt_ 2λμ_ 2入μ+]。.(x21(0()8a + 2p&()jinadwdt[_ 2Xμ=。L((2(ep() + Opra)iasa + 21(ea() + asS3). aosMdudt_ 2Xμ+。/ Lx+2(epo() + Op0p)a3 + 2u(ea() + agS3){snadwdt_2λμ= ]。.xt ，[(O,So + 8p0(1Caa4. + 8analdudt中国煤化工YHCNM HG.112s. Kaizu and N. Sabu+J"[22u/[ea3() + as34S31](0-1073 + agna]dwdt(4.67)which is symmetric in (5;) and (7z).Let (Sa), (na)∈L2(0,T; H'(w)) be such that (S(3)) = (a,5S3),(S(n3)) = (7a,n73). Thenb(s, n3)=S。" J."n(3))as4nsdwdtn&a(S(3))8gna dwdtsJ。" J."," .n9()(0a84O73 + a)gna]dwdtB(Sζ3, Sn3) = B(Sn3, Sζ3) = b(n3, (3). .(4.68)Hence b(., ) is symmetric.Lemma 4.3 Let 03 be the solution of (4.58) with fa∈L'(0,T;H:(w))， f3∈L4(0,T;H2(w)) and {0o3,013} ∈H3(w)∩H3(w) and let φ3 be the solution of (4.35) with {中o3, φ13}∈H8(w)∩H2(w). Then the following identity holds-I(qcv)map(03)dapOsdndtJawx(0,T)=| | 2p(03(aOaP3) + q3(3Do3))dw" +2 |2p03φsdwdt0'Jwx(0,T)-2map(03)8aBdadwdt +2n品(3))a3403dwq<(m&(3))8a34)Ozdw - -/ n2a(Spu (03)8asp)q<&)-<.J,fIoaodx3,●3)f2s:(0.0.<.-<.^. f91da30.), (4.75)where (, .> denotes the dual product between L' (0, T; L2(w)) and its dual.Also,(4.76)Adding the above two equations, we getlin[]。" J["f(a)()()()(*()); + ef(,(()*(e;}>(<)>x2}f"qaoaovpdx3,q3>. (4.77)中国煤化工YHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells115It can be shown as in[1] that for fixed t∈[0,T],((le(+(())()(a)()ax.a→q^-map()ap0sdxdt -/ n2p()8a3403dxdtJwx(0,T)-m(()30.dxdt.(4.78)The result follows from (4.71)- (4.73), (4.77) and (4.78).Lemma 4.5 LetT> T(e) and ψ(e) be the sequence of solutions of the problem (4.28) withinitial data {4o(e), ψ1()} satisfyingI{4o(e), ()}l(](](x](x)≤C.(4.79)Assume also that the sequence of functions {0o(e),0r(e)} and {fa(e), f3(e)} satisfy the assump-tions of Theorem 4.1. Then there exists ψa∈[X(S2)]' and ψ3∈L∞(0,T; L2(2)) such that<{p>/13, -p4>o3}, {0o3, 013})=(f)+[2p03φ3 + 3mas(03)8apo3]dwdtJawx(0,T)-[(<(n"g(3))a34)Ozdwdt +→o((3)a4qcocoadwdtwx(0,T)where-{v13,4o3} is the weak limit in[V(Q)]' x L2(S2) of a subsequence of {4n3(e), 4>o3()}e>0,-ψa is the weak limit of the subsequence of {4a(e)} in [X(2)', ψ3 is the weak: * limit of thesubsequence of {43()} in L∞(0,T; L(2))，-03 is the solution of (4.58),-中3 is the solution of (4.35) with initial dataoisdaz}and {φo3,φ13} is the weak limit of {φo3(e), φ13(e)} in V(2) x L2(2) and{oo(),中r(e)} = (A9)-({ψ(e), -o0o()).Proof Note that the solution ψ(e) of (4.28) with initial data {4o(e), ψ1(e)} satisfies (4.29).We want to compute the limit as∈→0 of (4.29).中国煤化工MHCNM HG.116s. Kaizu and N. SabuNote that because of (4.79), there exists a subsequence of {4o(e), ψhn()}e>o (still denotedby ε for notaional convenience) that weakly converges in [L2(2)]3 x [V(2)]'. Also since {0o(e),0()}e>o converges strongly in V(92) X [L2(2)}, we havep (a()0os()g^(e) + ea()()0o3(e)g"3(e)√g(e)dx+ρ |. (cv)3(e)(0)0oa(e)g3a (e) + ()()03()33()V) g(e)dx-ρ (2(0()()0a()g8(e) + (al()()g*())v ()dx-ρ I (3()(0)a(e)g*(e) + y)()0)(1()g}()√a(e)dx→_ρ(ψ⊥3θo3 - ψo3θ13)dx asε→0.(4.81)From the assumption (4.79) and estimate (3.22), it follows that the initial data {oo(e),φ1(e)}= A-1(e)({q41(e), -yo(e)} satisfyl(√p1u(e),e√pφ12(e), √(3)()(2()3 + a(e)(中o(e), o())≤C.(4.82)Hence {中o(e), φ1 (e)}e>o satisfy the assumption (4.31). Let {中o, φ1} be the weak limit of thesubsequence of {φo(e), φ1(e)}e>o in V(S2) x [L2(2)]3.Choosing {0o(e), 0r(e)} = {0,0} in (4.29) and using (4.82) it follows that(<3(),f()>\≤C | l(()y()uat iff9(e)=0, β=1,2,(4.83)()()f9())≤C。l(eax(x)tif(e)=0, f*(e)=0 forβ≠a.(4.84)Hence there exists ψa∈[X(2)',a= 1,2 and ψ3∈L∞(0, T; [2(92)) such that(),f()→2f9(4.85)The identity (4.80) fllowvs from (4.81), (4.85), (4.69) and (4.70).Lemma 4.6 The limit displacement ψ = (ψi) of the controlled displacement ψ(e) is aKirchhoff- Love displacement, that is,ψ)3 is independent of x3,ψa= ψa- x3OaV3，ψa is independent ofx3 and is a function of φ3.(4.86)Proof (i) To show that ψ3 is independent of x3, it is enough to show that(<43,-83f)=0，Vf∈ D(Ix (0,T)).(4.87)In (4.28), we consider sequences {0o(e), 01(e)}e>o, {f(e)}e>0 such thatθo(e)→0 inV(2)， 0r(e)→0 in [(2)]",(f()(f(),f*())→(0,0, -03f) in [()]2 x L'(0,T; L2(2))中国煤化工MHCNM HG.Exact Controllability and Asymptotic Analysis for Shallow Shells .117It can be verified that the above sequences satisfy all the hypothesis of Theorem 4.2. Notethat when fa =0, we haveSfa = S. At the limit we obtain (cf. (4.80))0= (43,-03f) +/ [2p03φ3 + 3mas(03)8as$s]dwdt-(cx)a(03)as303adwdt-/ (()ay0sdudtJawx(0,T)/wx(0,T)q(8(n"a(3))8as4)0sdwdt +/_ (03))a3pqc&+ (43, -8af*).(4.97)(ii) To prove that ψa = Z(φ3), let us consider in (4.80) 0o3= 013=0,f3=0and fa≠0,and fa∈D(wx (0,T)). Then we have0=(pa- xa43,f*>+/ [2po0303 + 3ma(3)apo$3ldwdt-. (x)ma(0).gdordwdt-. (()aogawuatJawx(0,T)q<(n%a(3))8as4)0sdwdt +/ n%a(Sfa (3))apqc&c=+[2ρ03φ3 + 3ma3(03 )8apo3]dwdt -(qcUζ, )mas(03)JaposdwdtJawx(0,r)中国煤化工MYHCNM HG.120s. Kaizu and N. Sabuwx(0,T)Jwx(0,T)n%a(S(qc8cφ3))8a340zdwdt(4.103)for any93∈L'(0,T;L2(w)) and any {0o3,013}∈H3(w)x L2(w) with 03 the solution of2p03 +8μ(λ + 2p)?203 - m8(S(03))8a34=93 inwx (0,T),3(+ 2u)(4.104)03= 8,03= 0on 8wx (0,T),03(0)= 0o3，03(0)= 0n3inw.Theorem 4.3 There exists a unique solution to the problem (4.101) in the transpositionsense.Proof Multiplying the first equation of (4.101) by 03, a solution of (4. 104), and integrateby parts, it can be shown using duality arguments the existence of a unique solution y3∈L∞(0, T; L2(w)).Theorem 4.4 Let T> T(e) and ψ(e) be the scaled (weak) solutions of problem (2.7) withcontrols (3.23). Suppose that the scaled initial data {4o(e), ψr(e)} verifies|{4r(), - 4()}()x[2()]3≤C.(4.105)Then there exists a subsequence of ψ$(e) (still indexed by e) and functions {4a, 43} in [X(8)]2xL∞(0,T; L2(82) such that, for any {f1,f2, fs}∈[X(2)2 x L'(0,T;L2(2))(4:(<),f)→(V%,f”〉as∈→0.Moreover, the limit function ψ = (4:;) satisfies(i) ψ= (ψ;) is a Kirchhoff- Love displacement, that is, ψ43 is independent of x3 and ψa =ψa- x38a43, where ψa is independent of x3.(i) ψ3 is the solution (in the transposition sense) of the following 2D problem2pψ3 +8μ(入+ 24)Sψ3 - ngg(S(43))8a343(\ + 2μ)= 2p*3 + 84(1t, 2s2d3 -(3))Da3ψ - (5(()3)(AX+ 2μ)+ n2a(S(qo and φ3 is the unique solution of the homogeneous 2D problem2pφ3 +84(2+321)0303- (1)08=00 inwx (0,T),3(入+ 2μ)(4.107)中3=0，8φ3=0on 8wx (0,T),φ3(0)= φo3，中3(0)= φ13in w,where {中o3, φ13} is the weak: limit in the space [V(Q2)]x L2(Q2) of the sequence {oo(e),φ1(e)} =A-1(e)({4r(e), -vo(e)}).(ii) ψa= Z(中3).Proof To prove the theorem, it is enough to prove that 43 = Y3, where Y3 satisfies (4. 103).Note that (4.80) will coincide with (4.103) if we are able to prove that (4.80) is valid for fa = 0,any pair {0oa,013}∈H2(w)x L2(w) and any 93∈L'(0, T; L2(w)). Choosingf(e)= (0,0,93)，θr()= (0,0, 013),0o()= (-x38r0o3, -x3020o3, 0o3 + 2v(e))such thatEv(E)→0in H'(S2),03v(e)→、f。x3△03 in L2(2)λ+ 2μ(note that this is possible because HH (S2) is dense in L2(2)), it follows thatf(e)→(0,0, g3) strongly in [X(2)]2 x L'(0,T; L(2)),0o(e)→(-x381 0o3, -x3020o3, 0o3)strongly in V(92),(4.108)al|())→0，e313(0()) →x+:x3△0o3 strongly in L2(92),∈01a(e)→0，θ13(e) →θ13strongly in L2(S2).Hence we conclude that (4.80) is valid for ay pair {0o3,013} ∈H(w)x L2(w) and anyg3∈L'(0,T;L2(w)).Acknowledgment This work was completed when the second author was visiting theUniversity of Ibaraki under the JSPS fellowship whose support is greatfully acknowledged.References[1] Busse, S., Ciarlet, P. G. and Miara, B., Justifcation d'un modele lineaire bi-dimensional de coques“faibl-ment courbees" en coordonnees curillignes, Model. Math. Anal. Num., 31(3), 1997, 409- -434.[2] Ciarlet, P. 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