Boundary Controllability of Thermoelastic Plates via the Free Boundary Conditions

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1 Univerity of Nebraka - Lincoln igitalcommon@univerity of Nebraka - Lincoln Faculty Publication, epartment of Mathematic Mathematic, epartment of January Boundary Controllability of hermoelatic Plate via the Free Boundary Condition George Avalo Univerity of Nebraka-Lincoln, gavalo@mathunledu Irena Laiecka Univerity of Virginia Follow thi and additional work at: Part of the Mathematic Common Avalo, George and Laiecka, Irena, "Boundary Controllability of hermoelatic Plate via the Free Boundary Condition" Faculty Publication, epartment of Mathematic hi Article i brought to you for free and open acce by the Mathematic, epartment of at igitalcommon@univerity of Nebraka - Lincoln It ha been accepted for incluion in Faculty Publication, epartment of Mathematic by an authorized adminitrator of igitalcommon@univerity of Nebraka - Lincoln

2 SIAM J CONROL OPIM Vol 38, No, pp c Society for Indutrial and Applied Mathematic BOUNARY CONROLLABILIY OF HERMOELASIC PLAES VIA HE FREE BOUNARY CONIIONS GEORGE AVALOS AN IRENA LASIECKA Abtract Controllability propertie of a partial differential equation PE model decribing a thermoelatic plate are tudied he PE i compoed of a Kirchoff plate equation coupled to a heat equation on a bounded domain, with the coupling taking place on the interior and boundary of the domain he coupling in thi PE i parameterized by α> Boundary control i exerted through the two free boundary condition of the plate equation and through the Robin boundary condition of the temperature hee control have the phyical interpretation of inerted force and moment and precribed temperature, repectively, all of which act on the edge of the plate he main reult here i that under uch boundary control, and with initial data in the baic pace of well-poedne, one can imultaneouly control the diplacement of the plate exactly and the temperature approximately Moreover, the thermal control may be taken to be arbitrarily mooth in time and pace, and the thermal control region may be any nonempty ubet of the boundary hi controllability hold for arbitrary value of the coupling parameter α, with the optimal controllability time in line with that een for uncoupled Kirchoff plate Key word partial differential equation, exact-approximate controllability AMS ubject claification 35B37 PII S Introduction Statement of the problem hroughout, Ω will be a bounded open ubet of R with ufficiently mooth boundary Γ = Γ Γ, with both Γ and Γ being open, with Γ being poibly empty, and atifying Γ Γ = Furthermore, Γ will be any open and nonempty ubet of Γ With thi geometry, we hall conider here the following thermoelatic ytem on finite time,: { ωtt ω tt ωα θ= on, Ω; βθ t η θ σθ α ω t = ω = ω = on, Γ ; { ω µb ωαθ = u µ B ω ω τ ω tt α θ = u on, Γ ; θ λθ = { u3 on, Γ, on, Γ\Γ, λ ; ωt ==ω,ω t t==ω,θt==θ on Ω Received by the editor June 5, 998; accepted for publication in revied form April 7, 999; publihed electronically January, epartment of Mathematic and Statitic, exa ech Univerity, Lubbock, X 7949 avalo@mathttuedu he reearch of thi author wa partially upported by NSF grant MS epartment of Mathematic, hornton Hall, Univerity of Virginia, Charlotteville, VA 93 ilv@amunapmavirginiaedu he reearch of thi author wa partially upported by NSF grant MS-9848 and Army Reearch Office grant AAH

3 338 GEORGE AVALOS AN IRENA LASIECKA Here, α, β, η, and σ are poitive contant he poitive contant i proportional to the thickne of the plate and aumed to be mall with < M he boundary operator B i are given by and B ω ν ν ω x y ν ω ν y ω x ω ω ω B ω ν ν x y ν y ν x he contant µ, i the familiar Poion ratio, and ν =ν,ν denote the outward unit normal to the boundary Here and throughout we hall make the following geometric aumption on the uncontrolled portion of the boundary Γ : 3 with hx, y x x,y y, {x,y } R uch that hx, y ν on Γ he PE model, with boundary function u = u = and u 3 =, mathematically decribe an uncontrolled Kirchoff plate ubjected to a thermal damping, with the diplacement of the plate repreented by the function ωt, x, y and the temperature given by the function θt, x, y ee for a derivation of thi model he given control variable u t, x and u t, x are defined on the portion of the boundary, Γ ; the control u 3 t, x i defined on, Γ Making the denotation 4 { HΓ k Ω ϖ H k Ω : j ϖ j } = for j =,, k, Γ we will throughout take the initial data ω,ω,θ tobeinh Γ Ω H Γ Ω L Ω For initial data in thee pace and control u = u = and u 3 =, one can how the well-poedne of with the correponding olution ω, ω t,θ being in C,; H Γ Ω H Γ Ω L Ω ee, eg, and In thi paper, we will tudy controllability propertie of olution of under the influence of boundary control function in the following pace: u,u,u 3 L,;L Γ H Γ C r Σ,, where r> and Σ, =, Γ 5 For arbitrary u,u,u 3 of uch moothne, the correponding olution ω, ω t,θ will be in the large pace C,; A ee the definition of A in 49 In particular, we intend to addre, on the finite time interval,, the quetion of exact-approximate controllability thi term being originally coined in 6 hat i to ay, for given data ω,ω,θ initial and ω,ω,θ terminal in H Γ Ω HΓ Ω L Ω, and arbitrary ɛ>, i there a uitable control triple u,u,u 3 L,;L Γ H Γ C r Σ, uch that the correponding olution ω, ω t,θ of atifie the following teering property at terminal time : ω,ω t = ω,ω and θ θ ɛ? L Ω In thi regard, we pot our main reult here for which we need the number { max µ max hx, y } 6, d x, y, Γ, x,y Ω up x,y Ω where, above, dx, y, Γ denote the ditance between x, y and Γ

4 CONROLLABILIY OF HERMOELASIC PLAES 339 heorem Let aumption 3 and 6 tand hen for >, the following controllability property hold true: For given initial data ω,ω,θ and terminal data ω,ω,θ in the pace H Γ Ω HΓ Ω L Ω, and arbitrary ɛ >, one can find control function u,u,u 3 L,;L Γ H Γ C r Σ, where arbitrary r uch that the correponding olution ω,ωt,θ to atifie at terminal time, ω,ωt = ω,ω, θ θ <ɛ L Ω heorem i almot a corollary from the following controllability reult for the mechanical variable only, which comprie the bulk of our effort here heorem With the coupling parameter α in being arbitrary, and 3, 6 in place, then for >, the following property hold true: For all initial data ω,ω,θ HΓ Ω HΓ Ω L Ω and terminal data ω,ω HΓ Ω HΓ Ω, there exit u,u,u 3 L,;L Γ H Γ H Σ,, where arbitrary, uch that the correponding olution ω, ω t,θ to atifie ω,ω t = ω,ω Remark 3 Note that the point x,y can be elected in uch a way o that max hx, x,y Ω y diam Ω, and o, ultimately, in 6 can be rechoen a = diam Ω Remark 4 Note that in our tatement of controllability, no geometric condition are impoed on the controlled region of the boundary Γ, only on the poibly void boundary portion Γ Literature o date, the only work dealing with the boundary control of thermoelatic plate, in dimenion greater than one, had been that of J Lagnee in indeed, thi preent paper i principally motivated by In thi paper, Lagnee how that if the coupling parameter α i mall enough and the boundary Γ i tar haped, then the boundary controlled ytem i partially exactly controllable with repect to the mechanical variable ω, ω t Alo in, a boundary-controlled ytem of thermoelatic wave i tudied, with a coupling parameter α likewie preent therein, and a reult of partial exact controllability for thi PE i cited again, controllability with repect to the hyperbolic component hi controllability reult i quoted in to be valid for all ize of α; however, in the erratum 3, the author of ha acknowledged a flaw in the controllability proof, the correction of which will neceitate a mallne criterion on α Ultimately, then, the paper produce a controllability reult if the coupling parameter i mall enough, a reult in the tyle of he chief contribution of the preent paper i to remove retriction on the ize of α ee heorem above For a one-dimenional verion of, S Hanen and B Zhang in 8, via a moment problem approach, how the ytem exact null controllability with boundary control in either the plate or the thermal component Other controllability reult for the thermoelatic ytem, which do not aume any mallne condition on the coupling parameter, involve the implementation of ditributed/internal control ubject to clamped or hinged boundary condition hee reult include that in 6, in which interior control i placed in the Kirchoff plate component ubject to clamped boundary condition

5 34 GEORGE AVALOS AN IRENA LASIECKA With uch control, one obtain exact controllability for the plate ω, ω t and approximate controllability for the temperature θ ie, exact-approximate controllability In addition, the work in 9 deal with obtaining a reult of null controllability for a linear ytem of thermoelaticity, in which both the hyperbolic and the parabolic component can be driven to zero by mean of interior control placed in the hyperbolic wave component Another reult of internal control for the thermoelatic PE i in 5, wherein interior control i placed in the heat equation only ie, βθ t η θ σθ α ω t = u o a to obtain exact controllability for both component ω and θ he novelty of thi reult i that thi total exact controllability obtain for all value of the rotational inertia parameter : in the limiting cae =, one i then preented with a reult of exact controllability for a PE modeled by the generator of an analytic emigroup ee 8 hi controllability hold for all value of α Again, the main contribution of thi paper i that we conider boundary control acting via the higher order free mechanical boundary condition, and we do not aume any ize retriction on the coupling parameter α Moreover, we do not impoe any geometric tar-haped condition on the controlled portion of the geometry At thi point, we attempt to compare the degree of difficulty in obtaining controllability reult for thermoelatic plate under mechanical interior control with lowerorder mechanical boundary condition enforced uch a clamped or hinged, veru that involved in the preent tudy, where, again, boundary control i exerted upon the econd and third order free boundary condition hi comparion i appropriate, ince the novelty of our work i touted to be mechanical exact controllability for the PE, whatever α may be; and excluding the paper 5, the only other available controllability reult for thermoelatic ytem, which require no ize contraint on α, concerned thermoelatic ytem under ditributed interior mechanical control and with lower mechanical boundary condition in place An underlying trategy in control theoretic tudie of thermoelatic plate ha been to exploit, if poible, previouly known controllability reult for uncoupled Kirchoff plate o thi end, one attempt to treat the thermoelatic ytem a a ort of perturbation of the Kirchoff plate It i well known that if the underlying controllability map can be decompoed into the um of a compact map and a urjective controllability map, correponding to a impler ubcomponent of the PE ytem, then the exact controllability of the original problem i equivalent to it approximate controllability hi favorable cenario occur in equation of thermoelaticity with either clamped or hinged boundary condition and interior, ditributed control ee, eg, Indeed, the part of the impler component i played by the claical and much-tudied Kirchoff plate, for which many reult on exact controllability are already available in the literature aking the boundary condition to be clamped or hinged allow for a known tructural decompoition of the thermoelatic ytem into a group aociated with the Kirchoff plate and a compact perturbation Combining thi decompoition with the boundedne of interior control action immediately yield the deired decompoition of the original controllability map into the um of a urjective controllability map correponding to the Kirchoff plate and a compact perturbation hi popular trategy wa ued in 6, where an exact-approximate controllability reult wa etablihed for the thermoelatic ytem with clamped homogeneou boundary condition and internal control he ituation i dratically different in the preent paper, involving the cae of boundary control Here, in thi cae of free mechanical boundary condition, the

6 CONROLLABILIY OF HERMOELASIC PLAES 34 correponding controllability operator cannot be taken to be a compact perturbation of the controllability map for the uncoupled boundary-controlled Kirchoff plate In the firt place, the aociated input tate pace map, defined explicitly in 4, i an inherently unbounded operator with repect to the natural energy pace ee 7 for recent harp regularity reult for correponding olution, which are till, however, below the level of energy Moreover, in the preent cae of free boundary condition, there i a decompoition of the underlying thermoelatic emigroup, but it i into the um of a Kirchoff plate emigroup and an unbounded not compact operator ee 6 hi complication i due to the fact that the Lopatinki condition are not atified for the Kirchoff model under free boundary condition, and to the intrinic nature of the coupling between the mechanical and thermal variable within the free boundary condition hee two complication above, again an artifact of the free cae, explain why there have been o few reult regarding the boundary control of thermoelatic plate and why a decoupling of the thermoelatic PE into a ole Kirchoff plate can only go o far Our goal here i to dipene with thi mallne aumption and, in addition, how that a control can be contructed that provide exact controllability of the mechanical variable and approximate controllability of the thermal component We note that the thermal control u 3 preent in wholly abent in play no part at all in the removal of the ize retriction on α; it i in place only to exploit, in a compactne-uniquene argument, recently obtained approximate controllability propertie of the thermoelatic plate under the action of boundary control in the free mechanical boundary condition ee At thi point in time, the thermoelatic ytem cannot be hown to be approximately controllable with control in the free boundary condition only and no thermal control herefore, the preence of the thermal boundary control here i not an artificiality; it appear to be neceary for approximate controllability We do not know if the future will bring a unique continuation reult for the thermoelatic plate in the abence of the thermal component However, the reult of heorem ay that the thermal control may be taken to be very mooth and with arbitrarily mall upport Γ Again, thi benign ituation i a conequence of our employing thermal control at the compactne-uniquene level only; it play no part whatoever in generating the main obervability etimate etimate 5 of heorem, thi being free of any ize retriction on α he trategy adopted in thi paper conit of the following tep Initially, a uitable tranformation of variable i made and applied to ; ubequently, a multiplier method i invoked with repect to the tranformed equation he mulitipler employed here are the differential multiplier ued in the tudy of exact controllability for the Kirchoff plate model inpired by, together with the nonlocal ΨO multiplier ued in the tudy of thermoelatic plate in 3 and 4 he controllability time in heorem ultimately depend in part upon the radial vector field aociated with the differential Kirchoff multiplier ee Lemma 5 below hi multiplier method allow the attainment of preliminary etimate for the energy of the ytem However, thee etimate are polluted by certain boundary term that are not majorized by the energy o cope with thee, we ue the harp trace etimate etablihed in 5 for Kirchoff plate he ue of thi PE reult introduce lower order term into the energy etimate, which are eventually eliminated with the help of a new unique continuation reult in It i only at the level of invoking thi uniquene reult that the thermal control u 3 on Γ mut be introduced he controllability time in 6 i optimal

7 34 GEORGE AVALOS AN IRENA LASIECKA 3 Operator theoretic formulation and analyi 3 Preliminary definition In obtaining our controllability reult heorem, it will be ueful to conider the PE ytem a an abtract evolution equation in a certain Hilbert pace, to which end we introduce the following definition and notation With HΓ k Ω a defined in 4, we define Å: L Ω Å L Ω to be Å=,with domain { Å = ω H 4 Ω HΓ Ω: ω µb ω= on Γ and 7 ω µ B ω τ = on Γ } Å i then poitive definite and elf-adjoint, and conequently from 7 we have the characterization 4 Å = HΓ Ω, 8 Å = HΓ Ω, 3 4 Å = { } ω H 3 Ω HΓ Ω: ω µb ω= on Γ Note that without lo of generality, we are here taking Γ to be nonempty in order to have the equivalence of the H Ω norm with that induced by the Å In the cae that Γ =, we would imply modify Å by enforcing ω µ Bω τ Γ = ω Γ intead of ω µ Bω τ Γ = in 7 hi modification would not change the problem Moreover, uing Green formula in, we have that for ω, ω mooth enough, 9 Ω ω ωdω =aω, ω Γ Γ ω µ B ω ωdγ τ ω µb ω ω dγ, where a, i defined by a ω, ω ω xx ω xx ω yy ω yy µ ω xx ω yy ω yy ω xx µω xy ω xy dω Ω In particular, thi formula and the econd characterization in 8 give that for all ω, ω Å, Åω, ω = Å ω, Å ω = a ω, ω Å Å L L Ω, Ω ω Å = ω = a ω, ω Å L Ω We define A : L Ω A L Ω to be A =, with irichlet boundary condition, viz, A =H Ω H Ω

8 CONROLLABILIY OF HERMOELASIC PLAES 343 A i alo poitive definite, elf-adjoint, and by 7 3 A = H Ω We denote the operator A R : L Ω A R L Ω by the following econd order elliptic operator: A R = σ { η I, with A R= ϑ H Ω : ϑ } 4 λϑ = A R i elf-adjoint, poitive definite on L Ω, with it fractional power therefore being well defined In particular, we have again by 7 that for, 4 3, A R=H Ω, ϑ, ϑ = A R ϑ, A R ϑ H Ω L Ω = ϑ, ϑ LΩ λ ϑ, ϑ σ ϑ, L Γ η ϑ 5 L Ω We denote the operator A N : L Ω A N L Ω by the following econd order elliptic operator: { A N =, with A N = ϑ H Ω : ϑ Γ = ϑ } 6 = Γ Once again by 7, we have for 4, A N = { ϑ H Ω uch that ϑ Γ = }, will denote the claical Sobolev trace map, which yield for f C Ω f = f Γ ; f = f 8 Γ We define the elliptic operator G,G, and a follow: v = on Ω, 9 G h = v v = v = on Γ, v µb v=h v µ B v τ v = on Ω, = on Γ, G h = v v = v = on Γ, v µb v= v µ B v τ = h on Γ,

9 344 GEORGE AVALOS AN IRENA LASIECKA ση { v = on Ω, I v= on Ω, h = v Rh = v v v Γ = h on Γ; λv = h on Γ, v λv = on Γ\Γ he claic regularity reult of, p 5 then provide that for all q real, L H q Γ,H q Ω, R L H q Γ,H q3 Ω, G L H q Γ,H q5 Ω, G L H q Γ,H q3 Ω enoting the topological dual of H q a H q pivotal with repect to the L -inner product, then with the elliptic operator A R and R a defined above, one can how that for q, the Banach pace adjoint R A R LA R, H q Γ atifie R A R ϑ = ϑ Γ for all ϑ A R Moreover, with the operator Å and G i a defined above, one can readily how with the ue of Green formula 9 that ϖ Å the Banach pace adjoint G i Å,H LÅ i Γ atify for i =,, 3 G i Åϖ = { i i ϖ Γ on Γ, on Γ With A N given by 6, we define the operator P : P L Ω L Ω by 4 P I A N i With the parameter >, we define a pace HΓ, Ω equivalent to HΓ Ω with inner product 5 ω,ω H Γ, Ω ω,ω L Ω ω, ω L Ω ω,ω HΓ Ω and with it dual denoted a H Γ Ω After recalling that, H Γ Ω = A N by 7, two extenion by continuity will then yield that P L HΓ,Ω,H Γ Ω, 6 with P ω,ω H Γ, Ω H Γ, Ω =ω,ω H Γ, Ω Furthermore, the obviou HΓ Ω-ellipticity of P, and Lax Milgram give u that P LHΓ, Ω,H Γ, Ωi boundedly invertible, with 7 P L H Γ, Ω,H Γ,Ω

10 CONROLLABILIY OF HERMOELASIC PLAES 345 Moreover, becaue P i poitive definite and elf-adjoint a an operator P : L Ω P L Ω, the quare root P i conequently well defined with P =HΓ, Ω, by 7 It then follow from 5 and 6 that for ω and ω HΓ Ω,, 8 P ω = ω L L Ω Ω ω L Ω = ω HΓ, Ω, 9 P ω, P ω =ω, ω L H Ω Γ, Ω ii Finally, by Green formula we have for ω, ω Å, ÅG ω, ω H Γ, Ω H Γ, Ω ω = ω, ω L Ω, ω ω, G Å ω L Γ L Γ 3 = ω, ω L Ω = A Nω, ω H Γ, Ω H Γ, Ω after uing 3 We thu obtain after two extenion by continuity to H Γ, Ω that P = I 3 ÅG a element of L HΓ,Ω,H Γ Ω, In obtaining the equality above, we have ued implicitly the fact that for every ϖ H Γ Ω and ϖ Å,, 3 ϖ,ϖ H Γ, Ω H Γ, Ω = ϖ,ϖ Å Å We denote the Hilbert pace H to be 33 H with the inner product ω ω ω, ω θ θ 34 Å H = H Γ,Ω L Ω, Å ω, Å ω L Ω P ω,p ω β θ, θ L Ω L Ω With the above definition, and making the denotation 35 A R σ η ÅG λåg, we then et A : H A H to be I I A P Å α I α β A I η β A R 36 { with A = ω,ω,θ Å Å A R } uch that Åω αåg θ H Γ Ω,

11 346 GEORGE AVALOS AN IRENA LASIECKA We make the following denotation for the pace of controllability: 37 U = L Γ H Γ H Γ, U = L,;L Γ H Γ H, Γ, where We define the control operator B on U by having for every u =u,u,u 3 U, 38 Bu= P ÅG u ÅG u η β A RRu 3 Note that a priori the mapping B only make ene a an element of LU, A, where H A Indeed, for fixed u =u,u,u 3 U one ha, upon uing the expreion for the invere A given in 4 below, and the definition of the elliptic operator G, G, and R in 9 and above, that 39 Bu = A A = A where i a defined in 35 By duality, we have P ÅG u ÅG u η β A RRu 3 G u G u αå Ru3 Ru 3 A, 4 U = L Γ H Γ H Γ, U = L,;L Γ H Γ H,;L Γ, and B L A,U 3 Abtract operator formulation If we take the initial data ω,ω,θ to be in H, and control u U, where U i a defined in 37, then conidering the operator definition above, the coupled ytem can be rewritten a fortiori a the operator theoretic model 4 d ωt ω t t θt = A ωt ω t t But, θt ω ω t θ = ω ω, θ with thi equation having ene in A a pace trictly larger than H Given the operator definition for A and B above, the olution ω, ω t,θ to the OE 4 and o to the PE i given by 4 ω ω t θ = e A ω ω θ e A Bud,

12 CONROLLABILIY OF HERMOELASIC PLAES 347 which by 39 and the convolution theorem i an element of C,; A With thi repreentation of the olution ω, ω t,θ in mind, we define the input terminal tate map L LU, A a 43 L u= e A Bud aken a an unbounded operator from U into H, then L : L U H i cloed and denely defined, with it domain of definition L given to be 44 L ={u U :L u H } It adjoint L : L H U, where U i a given in 4, i likewie cloed and denely defined, with 45 L = φ,φ,ψ H : L φ φ ψ U A we are concerned with obtaining exact controllability of the diplacement ω, ω t only, we accordingly define the projection operator Π : H Å HΓ, Ω by ϖ Π ϖ ϖ 46 = ϖ ϑ Henceforth, the work here will be concerned with determining the urjectivity of the cloed operator ΠL, ΠL U Å HΓ, Ω, with 47 ΠL u =Π e A Bud, and with ΠL =L etermining the urjectivity of the operator ΠL for ome > become our concern here, ince it i equivalent to howing the exact controllability of the mechanical component ω, ω t to heorem hi urjectivity for ΠL i in turn equivalent to the exitence of a certain obervability inequality pertaining to the range of the adjoint L Π the inequality below, where L Π : L Π Å HΓ Ω H, i likewie a cloed denely defined operator a L i, with it domain given by { } L Π 48 = φ,φ Å HΓ Ω:φ,φ, L It i the injectivity condition that we intend to directly verify In order to rewrite thi abtract inequality in PE form ie, a the inequality below, we need the following two propoition, the firt of which i proved in the appendix below Propoition 5 he Hilbert pace adjoint A of A, a defined in 36, i given to be 49 I A = P I with A = I Å α α β A I η β A R Å Å, { φ,φ,ψ } uch that Åφ αåg ψ H Γ Ω, A R

13 348 GEORGE AVALOS AN IRENA LASIECKA above, i the ame denotation made in 35 Remark 6 Uing the emigroup {e A t } t generated by A, then for terminal data φ,φ,ψ H, 5 φt φ t t ψt =e A t φ φ ψ C,; H i the olution to the following backward problem: { φtt φ tt φα ψ= on, Ω, βψ t η ψ σψ α φ t = 5 φ = φ = on, Γ, { φ µb φαψ = φ ψ µ B φ τ φ tt λψ = on, Γ, λ, φ,φ t,ψ=φ,φ,ψ α ψ = on, Γ, Remark 7 For terminal data φ,φ,ψ ina, the two equation of 5 may be written pointwie a 5 P φ tt = Åφ αåg ψ αλåg ψ α ψ in H Γ Ω,, 53 βψ t = η ψ σψα φ t in L Ω, 54 φ,φ t,ψ=φ,φ,ψ Remark 8 Since Γ Γ =, and Γ i mooth, we can aume throughout that A i dene in the graph topology of L Propoition 9 he adjoint L : L H U of L i computed to be 55 L φ φ ψ = φ t, φ t Γ,η ψ Γ Γ for all φ φ ψ L, where φt Γ,φ t Γ,ψ Γ are boundary trace of the olution φ, φ t,ψto the coupled ytem 5 Proof By Remark 8, it i enough to how the characterization in 55 for φ,φ,ψ A With thi in mind, one ha readily the claic repreentation 56 L φ φ ψ = B e A t φ φ ψ for every φ φ ψ A, where again, B L A,U i the adjoint of B We mut how that the righthand ide of thi equality may be written explicitly in PE form a 55 o thi end, for every u,u,u 3 U and φ,φ,ψ A,wehave 57

14 CONROLLABILIY OF HERMOELASIC PLAES 349 L u u, φ φ u 3 ψ A A = e A B u u d, φ φ u 3 ψ A A = e A A A B u u, φ φ d u 3 ψ A A = A B u u,e A A d u 3 H = φ φ ψ G u G u αå Ru3 Ru 3, A e A φ φ ψ d H Noting that φt φ t t ψt e A t φ φ ψ give the olution to the backward problem 5, we then ue thi relation, the definition of the adjoint A in 49, and Propoition 4 of the appendix to obtain 58 = = = = L u u u 3, φ φ ψ A A G u G u αå Ru3 Ru 3 φ t P Åφ αp ψ α β A I φ t η β A Rψ Å G u, Å φ t u,g Åφ t L Ω H d L Γ u,g Åφ t u, φ t u,φ t L H Γ Γ H Γ thereby completing the proof of Propoition 9 Immediately, we have Corollary, Å G u, Å φ t η Ru 3,A R ψ L L Ω Ω η u 3,ψ L Γ H Γ H Γ η u 3,ψ H, Γ H, Γ,

15 35 GEORGE AVALOS AN IRENA LASIECKA Corollary he adjoint operator L Π : L Π Å HΓ Ω, U i given by L Π φ φ t 59 = φ, φ t Γ,ηψ Γ Γ for all φ,φ L Π, where φt Γ,φ t Γ,ψ Γ are boundary trace of the olution φ, φ t,ψ to the following backward ytem: { φtt φ tt φα ψ= on, Ω, βψ t η ψ σψ α φ t = 6 φ = φ = on, Γ, { φ µb φαψ = φ ψ µ B φ τ φ tt λψ = on, Γ, λ, φ,φ t,ψ=φ,φ, α ψ = on, Γ, We conclude thi ection with a regularity reult for the thermal component of the olution φ, φ t,ψ to 5, thi being originally derived in and for the forward problem Auming terminal data φ,φ,ψ A, we have, by uing 5, the equality 6 d φt φ t t ψt H = A φt φ t t ψt, φt φ t t ψt H Integrating thi equation from to, performing computation imilar to thoe performed for the proof of Propoition 9, recalling the characterization 5, and ubequently invoking a denity argument, we have the following propoition Propoition With terminal data φ,φ,ψ H, we have that the component ψ of the olution of 5 i an element of L, ; A R Indeed, we have the following relation valid for all >: 6 φ φ φ φ t =η ψ H ψ H A R ψ L Ω Proof of heorem he neceary inequality A tated above, howing the partial exact controllability of the ytem for ome time > i equivalent to howing the urjectivity of the operator ΠL : L U Å H Γ, Ω, where ΠL i a defined in 47 and with L a defined in 44 Uing the claical functional

16 CONROLLABILIY OF HERMOELASIC PLAES 35 analyi eg, couple Lemma 388i and heorem 65ii of 9, the urjectivity of ΠL for ome time > i tantamount to the exitence of a contant C > uch that following inequality i atified for all φ,φ L Π, where L Π i a defined in 48: L φ φ C φ φ Å H Γ, Ω U Corollary then give that thi abtract inequality above may be rewritten by having for all φ,φ L Π, φ t H Γ φ t η ψ H, Γ C φ, φ L Γ Å H Γ, Ω where φt Γ,φ t Γ,ψ Γ are trace of the olution φ, φ t,ψ to the backward ytem 6 thi being adjoint with repect to So to prove the tatement of partial exact controllability of the thermoelatic ytem heorem, it will hence uffice to etablih the inequality for >large enough With thi end in mind, we make the following denotation for the mechanical energy of the ytem for t : E φ t = Å φt P 3 φ t t, L Ω L Ω where again φ, φ t,ψ olve the backward ytem 6 In addition, we will denote by lotφ, φ t,ψ lower order term any um of term that obey the following etimate for ome contant C : lotφ, φ t,ψ C φ φt L, ;H 3 ɛ Ω L, ;H ɛ Ω ψ 4 ψ L, ;H ɛ Ω L, ;H ɛ Ω By way of etablihing, the bulk of the work will entail the derivation of the following etimate heorem For > large enough, the olution φ, φ t,ψ to 5 with terminal data φ,φ,ψ L atifie the following inequality: E φ t A R ψ E φ L Ω 5 C ψ L Ω φ t L Γ lotφ, φ t,ψ hi theorem will follow from a chain of reult Given the denity of A in L ee Remark 8 and the fact that the olution of 5 ha the repreentation φt φ t t = e A t φ 6 φ, ψt ψ

17 35 GEORGE AVALOS AN IRENA LASIECKA it will be enough to how inequality 5 for olution φ, φ t,ψ to 5 correponding to terminal data in A aking φ,φ,ψ, A we then have that φ, φ t,ψ i an element of C,; H C,; A C,; A and a uch ha the additional regularity ee 3, heorem and alo :,; 7 φ C,; H 4 Ω; φ t C,; H 3 Ω; φ tt C ψ t C,; A R, φ G φ tt αg ψ αλg ψ C,; Å Å hi extra regularity of φ, φ t,ψ, correponding to mooth initial data, will jutify the computation to be done below Proof of heorem A mentioned above, the terminal data φ,φ,ψ will be conidered to be in ; A accordingly the correponding olution φ, φt,ψ of 5 will be a claical one, with the regularity poted in 7 With the end in mind of deriving the etimate, we tart by making the ubtitution, 8 φt =e ξt φt and ψt =e ξt ψt, where parameter ξ R i to be determined Necearily then φ, φ t, ψ olve the coupled backward ytem ξ φ ξ φ t φ tt ξ φ ξ φ t φ tt φα ψ= on, Ω, β ξ ψ ψ t η ψ σ ψ α ξ φ φ t = φ = φ = on, Γ, φ µb φα ψ= φ µ B φ τ ξ φ ξ φ t φ tt α ψ = on, Γ, ψ λ ψ = on, Γ, λ, φ, φt, ψ = e ξ φ, ξe ξ φ e ξ φ,e ξ ψ 9 Since φ,φ,ψ, A the extra regularity in 7 give that φ, φt, ψ ia claical not jut weak olution of 9; accordingly, we can rewrite 9 abtractly a ee Remark 7 and 3 ξ φ ξ φt φ tt ξ φ ξ φt φ tt ÅG ξ φ ξ φt φ tt Å φ αåg ψ αλåg ψ α ψ = inh Γ Ω,, β ξ ψ ψ t η ψ σ ψ α ξ φ φ t = inl Ω, φ, φt, ψ = e ξ φ, ξe ξ φ e ξ φ,e ξ ψ

18 CONROLLABILIY OF HERMOELASIC PLAES 353 Now multiplying the heat equation by α η and adding it to the Kirchoff plate, and ubequently taking the parameter ξ to be ξ α η, we obtain the ingle equation φ tt φ tt Å φ ÅG ξ φ ξ φt φ tt αåg ψ αλåg ψ 3 4 = c ψ c ψt c φ c3 φt c 4 φ, φ, φt, ψ = e ξ φ, ξe ξ φ e ξ φ,e ξ ψ, where the contant c = α3 β η ασ η, c = αβ η, c = α4 4 η, c 3 = α η, and c 4 = α4 4η Note that the particular choice of ξ made here eliminate the higher order term φ t Sytem 3 4 may be rewritten in PE form a the Kirchoff plate equation φ tt φ tt φ=c ψc ψt c φc3 φt c 4 φ on, Ω, φ = φ = on, Γ, φ µb φ= α ψ φ µ B φ φ tt τ = ξ φ ξ φ t α ψ on, Γ, φ, φt, ψ = e ξ φ, ξe ξ φ e ξ φ,e ξ ψ 5 A φ G ξ φ ξ φ t φ tt αg ψ αλg ψ C,; Å uing the lat containment in 7, then φ, φ t i a claical olution of 5 We note at thi point that one can readily derive the trace etimate Lemma 45 of the appendix below for the plate component φ Γ of the olution φ, φ t, ψ of 9 he proof of thi i relegated to the appendix, ince it i entirely analogou to that hown for the forward problem in 3 and 4 hi etimate will be critical in the proof of the following lemma, which give an energy relation for the mechanical variable Lemma a he olution φ, φ t, ψ to 9 atifie the following relation for all and τ,: 6 t =F, τ, E φ t=τ t= where τ i the mechanical energy function defined in 3 and F, i a function E φ defined below in 34 that obey the following etimate for all and τ, and ɛ>: 7 φ t ɛ L Γ F, τ C ɛ ɛ τ lot φ, E φ E φ φ t ψ t A E φ R ψt L Ω b For ɛ>mall enough, the olution φ, φ t, ψ to 9 atifie the following

19 354 GEORGE AVALOS AN IRENA LASIECKA etimate for all and τ,: ɛ τ C E φ ɛ E φ ɛ ɛ t A 8 ɛ E φ R ψt φ t L Ω L Γ lot φ, φ t ψ Above, the contant C ɛ i independent of time Proof We take the duality pairing of the abtract equation 3 with φ t and integrate in time and pace o a to get 9 φtt φ tt ÅG φtt, φ t = H Γ Ω H Γ Ω Å φ, φt Å Å ÅG ξ φ ξ φt αåg ψαλ ÅG ψ, φt Å Å c ψ c ψt c φ c3 φt c 4 φ, φ t L Ω Note that here we are uing implicitly the fact that the terminal data φ,φ,ψ being in A implie that Å φåg ξ φξ φ t φ tt αåg ψαλåg ψ i an element of C,; H Γ Ω Second, denoting A to be the invere of the elliptic operator defined in, we multiply the PE 5 by c A ψ, and ubequently integrate in time and pace o a to get φtt φ tt φ c ψc ψt c φc3 φt c 4 φ ψ L Ω = c,a A Rewriting 9 Uing equality 3 and the characterization in 3, we have upon the taking of adjoint that 9 may be rewritten a t t=τ = c E φ ψ c φ c3 φt c 4 φ, φ t c ψt, φ t t= L Ω L Ω τ ξ φ ξ φ t αλ ψ, φ t α ψ, φ t L Γ L Γ A Rewriting i An integration by part, the ue of the heat equation, and the fact that A R ψ = ψ ψ σ η ψ = A I ψ σ η ψ yield c c φtt,a ψ = c L Ω φt,a ψ t L Ω φt,a ψ τ L Ω

20 = CONROLLABILIY OF HERMOELASIC PLAES 355 c c c φt,a ψ τ L Ω σ φ t, β ξ A ψ η β I ψ φ t, αξ β I φ α β I φ t L Ω L Ω ii An integration by part and employment of Green theorem yield 3 c φ tt,a ψ = c φ t, A ψ = c φ t, A ψ τ c φ t, A ψ t = c φ t, A ψ c c σ φ t, = c φ tt, A L Ω τ c L Ω τ L Ω L Γ τ L Ω A c β ξ φ t, αξ I φ α β β ψ L Ω φ t, A ψ t L Ω φt,a A ψ t L Ω c φt, ψ t ψ η I ψ β I φ t L Ω L Γ L Γ iii hrough the ue of Green theorem 9 and the boundary condition in 5, we obtain 4 c = c c φ, A ψ L Ω a φ, A ψ αc φ, A ψ L Γ ψ, A ψ L Γ where we have ued the fact that φ Γ = φ Γ = implie B φ Γ = ; ee Jointly then, equalitie and 4 give the relation 5

21 356 GEORGE AVALOS AN IRENA LASIECKA = c φt, ψ t c σ φ t, L Ω β ξ c φ t, αξ β I φ α β I φ t c c c c c φ t, σ β ξ A ψ φ t, αξ I φ α β β a φ, A ψ αc φ, A ψ η I ψ β I φ t ψ ψ, A L Γ c ψ c ψt c φ c3 φt c 4 φ, c σ β ξ A τ A ψ η β I ψ L Ω L Ω L Γ L Γ L Γ ψ c φ t, A ψ φt,a L Ω ψ L Ω Summing the relation and 5, we obtain 6 t t=τ τ = c E φ ψ c φ c3 φt c 4 φ, φ t c t= τ ξ φ ξ φ t αλ ψ, φ t α L Γ φ t, A ψ η β I ψ c c c φt, αξ β I φ α β I φ t L Ω A L Ω ψ ψ, φ t L Ω σ A φ t, β ξ ψ η I ψ β φ t, αξ I φ α I φ t β β L Γ α A ψ, ψ L Γ φ, A ψ L Γ L Ω L Γ c φ t, A ψ L Γ c c a L Ω τ φ, A ψ L Ω φt,a ψ τ L Ω note the cancellation of the high order term ψ t, φ t L Ω We now proceed to etimate the right-hand ide of thi relation In o doing, we will be uing implicitly, in B B7 below, the inequality ab ɛa C ɛ b

22 7 CONROLLABILIY OF HERMOELASIC PLAES 357 B We have by trace theory τ ξ φ ξ φ t αλ ψ, φ t C φ t L Γ lot φ, φt, ψ L Γ α ψ, φ t L Γ B A A i a bounded operator, we have c ψ c φ c3 φt c 4 φ, φ t c A ψ ɛ Å φ lot φ, φt, 6 ψ 8 L Ω B3 A LH Ω for > by tandard elliptic theory, and A LL Ω,A, we then have in conjunction with trace theory 9 c c σ φ t, φ t, αξ β ξ A ψ η β I ψ β I φ α β I φ t τ σ A c φ t, β ξ ψ L Γ c a φ, A ψ αc τ ψ, A ψ ɛ 6 Å φ φ, lot φt, ψ L Ω B4 Uing the fact that LH Ω for > have along with trace theory that τ c φ t, η ψ β αξ I φ α β β C φ t lot φ, φt, ψ 3 L Γ L Ω L Ω L Γ, and ψ t Γ = λ ψt Γ,we φ t L Γ B5 By, p 3, heorem 3 and trace theory we deduce that LH Ω,H Γ, and o accordingly we have 3 c φ t, η ψ β ɛ 6 α β φt P φ A t ψ L R Ω L Γ L Ω C ɛ φ t H Γ

23 358 GEORGE AVALOS AN IRENA LASIECKA B6 A A LH Ω,HΩ, by the characterization of elliptic operator given in 7, we then have for all t, φ t t, A ψt C φ t t A ψt L Ω L Ω L Ω ɛ P φt t φ, 6 lot φt, ψ L Ω We thu have c φ t t, A ψ t=τ L Ω t= ɛ P 6 φ t τ P 3 φt L Ω c L Ω φt t,a ψt lot φ, φt, ψ L Ω B7 Finally, we can ue the trace reult Lemma 45 of the appendix and the fact that A LH ɛ Ω,H 3 ɛ Ω again by 7 to have 33 c τ φ, A ψ L Γ C φ ψ ɛ φ C L Γ H ɛ ɛ Ω 6C ψ L Γ H ɛ Ω where the contant C above i the very ame a that in 43 ɛ t ɛ τ lot φ, φt, 3 E φ 3 E φ E φ ψ herefore, if we define F, τ tobe t=τ t= 34 F, τ right-hand ide of 6, etimate 7 33, then we have F, τ C ɛ φ t ɛ t A L Γ E φ R ψt ɛ τ lot φ, φt, E φ E φ ψ 35, L Ω where the contant C ɛ doe not depend on time hi and equality 6 prove a o prove b, we combine 6 and 7 and ubequently take ɛ> mall enough he proof of Lemma i concluded With the radial vector field h defined in 3, one ha the following relation, which i eentially demontrated in the complete proof i carried out in Propoition 46 of the appendix below Propoition 3 With the vector field h a defined in 3, the olution φ, φ t, ψ to 5, correponding to terminal data φ,φ,ψ, A atifie the following equality for arbitrary ɛ,: 36

24 ɛ CONROLLABILIY OF HERMOELASIC PLAES 359 ɛ t = E φ ɛ ɛ ɛ c ψ, h φ t φ t ɛ ɛ ɛ ɛ ɛ φ t α L Ω ψ, c ψ c φ c3 φt c 4 φ, h φ φ ɛ ɛ h φ φ L Ω h ν Γ L Γ ɛ ɛ φ dγ ξ φ ξ φt α ψ,h φ φ L Ω h ν φ t φ t dγ Γ L Γ φ t, h φ L Ω φ t, h φ L Ω φ t, φ L Ω φ t, φ c ψ, h φ t= ɛ L Ω φ ɛ h ν ɛ Γ φ x φ y L Ω t=ɛ t= ɛ t=ɛ φ φ µ x y µ φ dγ x y So a to derive another intermediate energy inequality, we will now etimate the right-hand ide of the relation 36 In the coure of thi work, we will make critical ue of the following trace etimate for uncoupled Kirchoff plate, which wa derived in 5 It i thi regularity reult that allow the controlled portion Γ of the boundary to be free of geometric contraint race theorem ee 5 Let the function ϕt, x atify the following Kirchoff equation on an open, bounded domain Ω R n, with mooth boundary Γ, Γ=Γ Γ, where each Γ i i open and nonempty, with Γ Γ = : ϕ tt ϕ tt ϕ=f on, Ω, 37 ϕ= ϕ = on, Γ, ϕ µb ϕ=g ϕ µ B ϕ τ ϕ tt = g on, Γ here the boundary operator B and B are a given in Let <ɛ < ɛ>be arbitrary hen the following inequality hold true for the olution ϕ: 38 ɛ ɛ ϕ τ L Γ ϕ L Γ ϕ τ L Γ and

25 36 GEORGE AVALOS AN IRENA LASIECKA { C,ɛ, f H 3 g ɛ Ω L Γ ϕ H 3 ϕ ɛ Γ t L Γ } ϕ t L Γ g H, Γ Remark 4 In the original tatement of thi theorem ee heorem in 5, the term f H 3 in the inequality 38 i replaced by ɛ Ω f H q, Ω, where q< However, if one replace the H q, Ω pace with L,;H q Ω, the value of allowed parameter extend to q<3/ ɛ hi i in line with elliptic theory correponding to free boundary condition By the ue of thi trace reult in part, we have the following energy etimate Lemma 5 For all ɛ, and ɛ >arbitrary, the olution φ, φ t, ψ to 5 atifie ɛ t C E φ ɛ E φ ɛ E φ ɛ C φ t ɛ A 39 R ψ lot φ, φ t, ψ, L Γ L Ω where the time independent contant C µ max x,y Ω hx, y where, again, µ i Poion ratio and h atifie 3 Proof We proceed to majorize the right-hand ide of 36 A Handling the term ɛ ɛ c ψ c φ c3 φt c 4 φ, h φ φ L Ω: Firt, by Green theorem and the fact that L H Ω,H Ω and h L H Ω,H Ω, we obtain φ, h φ φ = φ, h φ L Ω φ L Ω φ, φ φ = φ, h φ φ lot φ, φt, ψ, H ɛ Ω H ɛ Ω L Γ φ, φ φ L Γ where in the lat tep we have alo ued Cauchy Schwarz and the trace theory We thu have ɛ c ψ c φ c3 φt c 4 φ, h φ lot φ, φt, ɛ φ ψ 4 L Ω A Likewie uing Sobolev trace theory, the fact that ψ divergence theorem, we have = λ ψ, and the 4

26 CONROLLABILIY OF HERMOELASIC PLAES 36 ɛ c ψ, h φ t φ t φ t ɛ L Ω L Ω ɛ ξ φ ξ φt α ψ ɛ,h φ φ α ψ, φ L Γ L Γ ɛ c ψ, h φtx h φty C φ t lot φ, φ t, ψ ɛ L Ω L Γ ɛ = c div ψh φt dω ɛ Ω ɛ c h ν ψ φ t dγ C φ t lot φ, φ t, ψ ɛ Γ L Γ C φ t ɛ A R ψ lot φ, φ t, ψ L Γ L Ω A3 Uing 3, we have ɛ 4 h ν φ ɛ Γ A4 We now etimate the term φ t, h φ L Ω φ t, h φ t= L Ω φ t, φ ɛ L Ω t=ɛ φ t, φ c ψ, h φ t= ɛ 43 L Ω φ L Ω Firt, a h φt H ɛ Ω for all t,, we have 44 ψt, h φt φt lot φ, φt, ψ L Ω = t=ɛ ψt, h φt φt H ɛ Ω H ɛ Ω Second, we have pointwie in time φ t, h φ φ t, φ L Ω L Ω = φt x, y x x φ xx y y φ xy, x x φ xy y y φ yy dxdy 45 Ω 3 φ t, φ L Ω Now, to handle the firt term on the right-hand ide of 45, we ue the inequality ab δ a δ b with δ µ where, again, Poion ratio µ, 46

27 36 GEORGE AVALOS AN IRENA LASIECKA φt x, y x x φ xx y y φ xy, x x φ xy y y φ yy dxdy Ω = x x φ t φ xx,φ xy dxdy y y φ t φ xy,φ yy dxdy Ω Ω max hx, y { φ t dω φ xx φ yy dω x,y Ω µ Ω µ { P max hx, y µ x,y Ω Ω } φ xydω φ t µ µ µ L Ω } φ xydω Ω Ω Ω φ xx φ yy dω From thi inequality, the definition of a, in, and the characterization in, we obtain 47 φt x, y x x φ xx y y φ xy, x x φ xy y y φ yy dxdy Ω max hx, y { P φ Å t φ µ x,y Ω L Ω L Ω o deal with the econd term on the right-hand ide of 45, we can ue the fact that L H Ω,H Ω for all real, oatohave φ t t, φt = φ t t, φt L Ω H ɛ Ω H ɛ Ω C φ t t φt φ, lot φt, ψ 48 H ɛ Ω H ɛ Ω Combining 45, 47, and 48 with the definition of E φ in 3, we then obtain 49 5 h φt φ t t, L Ω µ max hx, y tlot E φ x,y Ω } φ t t, φt L Ω φ, φt, ψ Coupling 44 and 49 in turn, we arrive at the etimate 43 { µ max x,y Ω A5 Handling the term ɛ ɛ 5 hx, y } E φ ɛ E φ ɛ α ψ, h φ and noting that lot φ, φt, ψ h φ =ν φx ν x x φ xx ν y y φ xy ν x x φ xy ν φy ν y y φ yy,

28 CONROLLABILIY OF HERMOELASIC PLAES 363 we then have by Cauchy Schwarz, the trace etimate 38 for the Kirchoff plate above, the ue of the forcing data in 5, and the tandard Sobolev trace theory that 5 C ɛ ɛ ɛ ɛ ɛ = C ɛ C α ψ, h φ L Γ ψ φ H xx φ yy ɛ Ω L Γ φ x φ H y ɛ Ω H ɛ Ω ψ φ H ɛ Ω τ φ L Γ φ x φy H ɛ Ω H ɛ Ω c ψ c ψt c φ c3 φt c 4 φ L Γ L Γ H 3 ɛ Ω φ xy L Γ φ τ ψ L Γ H ɛ Ω ξ φ ξ φt L Γ φt φ φt L Γ H 3 ɛ Ω H ɛ Ω C φt c ψt c 4 φ lot φ, φt, ψ L Γ H 3 ɛ Ω o handle the term c ψ t c 4 φ H 3 ɛ Ω, we ue Propoition 44 in the appendix below and the fact that ψ t = ξ ψt e ξt ψ t t and φt =e ξt φt to have 53 = C C c ψt c 4 φ H 3 ɛ Ω ξc ψtc e ξt ψ t tc 4 e ξt φt H 3 ɛ Ω ψ L Ω φ H 3 ɛ Ω ψ H φ t ɛ Ω H φ t ɛ Ω φ t L Γ lot φ, φt, ψ L Γ

29 364 GEORGE AVALOS AN IRENA LASIECKA Collectively, etimate 5 and 53 then give ɛ α ψ, ɛ Γ ɛ h φ L Γ C φ t φ, lot φt, ψ 54 L Γ A6 In the ame way a in A5 we have 55 ɛ h ν φ φ x µ φ y x φ y µ φ dγ x y C φ t φ, lot φt, ψ L Γ A7 Finally, 56 ɛ h ν φ t φ t dγ C φ t φ, lot φt, ψ ɛ Γ L Γ Etimate 39 now come about by tringing together 36, 4 4, 5, and 56, and taking ɛ> mall enough Lemma 6 For > µ max x,y Ω hx, y, the olution φ, φ t, ψ of 5 atifie the following etimate: t A E φ E φ R ψ L Ω C ψ L Ω φ t lot φ, φt, ψ 57 L Γ Proof We have for any ɛ,, E φ t = ɛ ɛ ɛ ɛ E φ ɛ ɛ ɛ φ t ɛ t t E φ E φ ɛ ɛ t A E φ R ψt C ɛ lot φ, φ t, ψ L Γ after applying Lemma b twice C ɛ E φ ɛ E φ ɛ ɛ 58 ɛ A R ψ L Ω L Ω ɛ ɛ ɛ ɛ ɛ E φ ɛ C,ɛ φ t L Γ t E φ ɛ t E φ ɛ E φ t lot φ, φ t, ψ,

30 CONROLLABILIY OF HERMOELASIC PLAES 365 after applying Lemma 5 with C µ max x,y Ω hx, y, and ɛ ɛ ɛ therein Applying Lemma b twice more to the right-hand ide of 58 yield now 59 6 t ɛ E φ C ɛ ɛ E φ ɛɛ C ɛ ɛ ɛ C A ɛ R ψ C,ɛ L Ω lot φ, φ t, ψ Moreover, we have by 6 E φ t = E φ F,t, E φ t φ t L Γ where the function F i a defined in 34 Combining 59 and 6 yield 6 F,t ɛ E φ C ɛ ɛ E φ ɛɛ C t ɛ E φ ɛ ɛ C A ɛ R ψ C,ɛ φ t lot φ, φ t, ψ L Ω L Γ o ue thi inequality, we integrate both ide of 7 with = therein o a to have F,t ɛ t ɛ ɛ A E φ E φ R ψ L Ω 6 C,ɛ φ t lot φ, φ t, ψ L Γ Combining 6 and 6, we thu obtain 63 ɛ C ɛ ɛ E φ ɛ E φ ɛ C ɛ t ɛ E φ ɛ C ɛ A ɛ R ψ C,ɛ φ t L Ω L Γ lot φ, φ t, ψ aking now > ɛc ɛ ɛ, or what i the ame, >C for ɛ and ɛ mall enough, we then have 64 E φ C,ɛ φ t lot φ, φ t, ψ, L Γ ɛ C t A E φ R ψ L Ω

31 366 GEORGE AVALOS AN IRENA LASIECKA where throughout C will denote a contant independent of ɛ and ɛ mall enough In turn, applying thi to 59, we have ɛ ɛ C ɛ C t C E φ,ɛ φ t t L Γ ɛ E φ ɛ ɛ C ɛ C ɛ C A ɛ R ψ lot φ, φ t, ψ L Ω from which follow the etimate, for ɛ, ɛ > mall enough, 65 E φ t C,ɛ φ t ɛ C L Γ A R ψ lot φ, φ t, ψ L Ω Coupling together 64 and 65, we have the following preliminary inequality for the mechanical energy, again for >C : 66 t E φ E φ C,ɛ φ t L Γ ɛ C A R ψ lot φ, φ t, ψ L Ω It remain to etimate the thermal component o thi end, we can multiply by ψ, integrate in time and pace, ue the characterization 5 and 8 to have η e ξt ψt t= ξ β ψ α φ, ψ L Ω t= L Ω 67 A R ψ β = L Ω α φ t, ψ L Ω φ t, ψ Majorizing thi expreion reult in η A R ψ ψ L L Ω C ɛ Ω ɛ A R ψ lot φ, φ t, ψ, L Ω L Γ φ t L Γ t E φ and taking ɛ > mall enough above, thi become A R ψ C ψ L L Ω Ω φ t t L Γ E φ 68 lot φ, φ t, ψ, where C = C ɛ η ɛ Combining 66 and 68, we have 69

32 t E φ E φ C,ɛ φ t CONROLLABILIY OF HERMOELASIC PLAES 367 L Γ from which we obtain for ɛ> mall enough 7 t C E φ E φ,ɛ ɛ C C ψ L Ω t lot φ, E φ φ t, ψ, ψ L Ω φ t lot φ, φ t, ψ L Γ he final etimate 57 finally come about by combining 7 and 68 Concluion of the proof of heorem Aume initially that φ,φ,ψ A hrough the change of variable φt = e ξt φt and ψt = e ξt ψt, where again φ, φ t, ψ olve 5 and ξ α η >, we have for > hx, y µ max x,y Ω 7 E φ t A R ψt E φ L Ω Å = e ξt P φt e ξt φt tξe ξt A φt L Ω L R Ω eξt ψt Å e ξ P φ e ξ φt ξe φ L Ω L Ω C ψ L Ω φ t lot φ, φt, ψ L Γ after uing etimate 57 C ψ L Ω e ξt φ t t ξe ξt φt L Γ lot e ξt φ, e ξt φ t ξe ξt φ, e ξt ψ C ψ L Ω φ t t L Γ lot φ, φ t,ψ hi give the deired inequality 5 L Ω 3 Concluion of the proof of heorem For φ,φ L Π, we immediately have from heorem the following corollary Corollary 7 For φ,φ L Π and > µ max x,y Ω hx, y, the correponding olution φ, φ t,ψ of 6 atifie the following inequality: 7 E φ t E φ A R ψ L Ω C φ t L Γ lot φ, φ t,ψ

33 368 GEORGE AVALOS AN IRENA LASIECKA We will have the deired inequality upon the elimination of the tainting lower order term in 7 o thi end, we invoke a by now claical compactne uniquene argument ee, eg, 3 and, which make crucial ue of the new Holmgren-type uniquene reult for the thermoelatic ytem recently derived by Iakov in It i at thi point that the boundary trace ψ Γ, correponding to the control u 3, come into play Lemma 8 Let be a defined in 6 hen for > and initial data φ,φ L Π, there exit a C uch that the following etimate hold true for the olution of 6: φ L, ;H 3 φ t ɛ Ω L, ;H ɛ Ω ψ L, ;H ψ ɛ Ω H ɛ Ω 73 C φ t L Γ ψ H, Γ Proof If the propoition i fale, then there exit a equence {φ n,φn } n= L Π, and a correponding olution equence {φ n,φ n t,ψ n } n= to 6, which atifie φ n L, ;H 3 ɛ Ω φ n t ψ n L, ;H ɛ Ω L, ;H ɛ Ω ψ n 74 = n, H ɛ Ω lim φ n t ψ n 75 = n L Γ H, Γ A > µ max x,y Ω hx, y, we have the exitence of the inequality 7 hi and then imply the boundedne of the equence 76 φ n t φ n t t φ n φ n Å Å H Γ, Ω H Γ, Ω n= A R ψ n t L Ω here thu exit a ubequence, till denoted here a {φ n,φn } n=, and φ, φ Å HΓ, Ω, uch that φ n φ in Å weakly, φ n φ in HΓ,Ω weakly If we further denote φ, φ t, ψ a the olution to 6, correponding to initial data φ, φ,, then a fortiori, φ n,φ n t,ψ n φ, ψ 79 φt, in L,;H weak tar

34 CONROLLABILIY OF HERMOELASIC PLAES 369 From Propoition 43 of the appendix, we have that {φ n tt } n= i bounded in L,;Å P, inamuch a { φ n,φn Å HΓ, Ω} n= i bounded in Å HΓ, Ω Alo, from Propoition 44 we have that ψn t L,;H 3 ɛ Ω for all n, with the etimate 8 ψ n t C H 3 ɛ Ω φ n t lot φ, φ t,ψ, L Γ and thi combined with yield that {ψ n t } n= i bounded in L,;H 3 ɛ Ω hi boundedne of {φ n tt,ψ n t } n=, and that for the equence poted in 76, allow u to deduce through a compactne reult of Simon in 4 that φ n φ trongly in L,;H 3 ɛ Ω, φ n t φ t trongly in L,;H ɛ Ω, ψ n ψ trongly in L,;H ɛ Ω, ψ n ψ trongly in L,;H ɛ Ω hee convergence and 74 thu give φ φt ψ L L, ;H 3 ɛ Ω L, ;H ɛ Ω, ;H ɛ Ω ψ 8 = H ɛ Ω Moreover, the explicit repreentation of L Π in 59 and the convergence poted in 75 and give that φ, φ L Π, with 8 φ t ψ = H Γ H, Γ Now if we make the change of variable z = φ t, v = ψ t, then uing 8, z,v olve the ytem { ztt z tt zα v= on, Ω, βv t η v σv α z t = 83 z = z = on, Γ, z µb zαv = z µ B z z tt τ α v = on, Γ, v λv = on, Γ, v = on, Γ

35 37 GEORGE AVALOS AN IRENA LASIECKA Now by Iakov theorem in, p 3, Corollary, we have for > up d x, y, Γ x,y Ω that the uniquene property for the thermoelatic ytem i obtained, o that the olution z,v of 83 i necearily zero Conequently φ and ψ are each contant From the eential boundary condition on Γ in 6, we then have φ = on, Ω In turn, the free boundary condition on Γ give that ψ = on, Ω hu φ, ψ =,, which contradict the equality given in 8 hi conclude the proof of the lemma Corollary 7 and Lemma 8 in combination give inequality, the etablihment of which verifie the urjectivity of the control to partial tate map ΠL : L U Å HΓ, Ω hi complete the proof of heorem 3 he proof of heorem Given the pace C r Σ,, we conider ytem under the influence of boundary control in U r, a defined in 37 he controlled PE i then approximately controllable in U r for > up x,y Ω d x, y, Γ Indeed, if we take arbitrary φ,φ,ψ from the null pace of L, then uing the form of thi operator given in 55, we have necearily that φ t Γ = φ t Γ =, and ψ Γ =, where φ, φ t,ψ i the olution to 5 We can then ue the uniquene theorem of Iakov, in a fahion imilar to that employed in Lemma 8, to how that φ, φ t,ψ=,, on, Ω and, in particular, φ,φ,ψ =,, A preliminary tep a regularity property of L With the deignated control pace U r we then take > o a to enure both the approximate controllability of the entire ytem and the exact controllability with repect to the diplacement ee heorem In thi event, we have the obervability inequality, and therewith one can how in a manner identical to that done in 4, Appendix B that the operator 3 ΠL L Π i an iomorphim from L Π intol Π, where the projection Π onto Å HΓ, Ω i a defined in 46 Conequently, we have 3 ΠL L Π ΠL L Π Π L H,Å HΓ,Ω Moreover, if we denote the map L, L by t L u t = e At B u t L ut = e At B u u u, cf 4, then by a tandard energy method one can how that 33 L : L,;H Γ C,; H continuouly

36 CONROLLABILIY OF HERMOELASIC PLAES 37 o handle L on the other hand, one mut appeal to a new regularity reult in 7, which give L : L,;L Γ H Γ C,; H 3 Ω H Ω L Ω continuouly 34 Combining 33 and 34 at terminal time with 3, we thu deduce that the mapping 35 I Π Π L L Π ΠL L Π Π LH, where I : H H denote the identity Combining 3 and 35 thu give 36 L L Π ΠL L Π Π LH Step For arbitrary ɛ> we elect a u L U r, o that for arbitrary terminal tate ω,ω,θ H, the correponding olution ω t,ω t t,θ t to, with u,u,u 3 u and zero initial data, atifie ω ω ω ω t ω e A ω θ θ θ H ɛ 37 < I Π ΠL L Π ΠL L Π Π LH where the fact that I-Π ΠL L Π ΠL L Π Π i due to 36 Step We now elect u L to be the minimal norm teering control with repect to the partial terminal tate ω ω,ω ω t hat i to ay, u atifie ω ΠL u Πe A ω ω = ω 38 θ ω ω t and minimize the functional u U, over all u U, which atifie ω ΠL u = ω ω ω ω Πe A ω t θ By heorem we know there exit at leat one uch u By convex optimization theory and Lax Milgram, the minimizer u can be given explicitly by ω u = L Π ΠL L Π ω Π ω ω t e A ω 39 ω θ θ θ ee B of 4, p 88 With thi repreentation, we then have from 37 the norm bound I Π ΠL L I Π Π ΠL L Π Π ɛ LH 3 ΠL u H I Π ΠL L Π ΠL L Π Π LH

37 37 GEORGE AVALOS AN IRENA LASIECKA 3 Step 3 Set the control u = u u Conequently, there i the equality L u e A ω ω θ = L u L u e A I Π Π ω ω θ L u e A = ω ω θ ω ω θ Letting ω,ω t,θ denote the olution of correponding to the choen control u, we then have from 3 that ω,ω t=ω,ω Moreover, from 3, 37, and 3 we obtain the etimate θ θ L σλ Ω 3 θ θ I Π ΠL u H ɛ < I Π ΠL L Π ΠL L Π Π I Π Π e A ω ω θ LH H I Π ΠL u H <ɛ hu, the contructed control u =u,u,u 3 U r atifie the deired exact approximate controllability property Moreover, the Sobolev embedding theorem give that u 3 C r Σ, hi conclude the proof of heorem 4 Appendix Propoition 4 he operator A R σ η λåg ÅG i an element of LL Ω, Å and A R σ η λåg ÅG = A I a element of LÅ,L Ω Proof For every ϑ A R and ϖ Å, we have A R ση λåg ÅG ϑ, ϖ Å Å = ϑ, ϖ L Ω λåg ϑ, ϖ ÅG Å Å ϑ, ϖ Å Å ϑ = ϑ, ϖ L Ω,ϖ λ ϑ, G Åϖ L Γ ϑ, G Åϖ L Γ L Γ after the ue of Green formula and the taking of adjoint = ϑ, ϖ L Ω ϑ, G Åϖ =ϑ, ϖ L Γ L Ω after one more ue of Green theorem and the characterization 3 =ϑ, A I ϖ L Ω A A R i dene in L Ω, thi equality prove the aertion