ON APPROXIMATION OF STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM IN COEFFICIENTS FOR p-BIHARMONIC EQUATION

We study a Dirichlet-Navier optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. The coe cient of the p-biharmonic operator we take as a design variable in BV (Ω)∩L∞(Ω). In order to handle the inherent degeneracy of the p-Laplacian and the pointwise state constraints, we use regularization and relaxation approaches. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted p-biharmonic operator and Henig approximation of the ordering cone. Further we discuss the asymptotic behaviour of the solutions to regularized problem on each (ε, k)-level as the parameters tend to zero and in nity, respectively.


Introduction
he im of this rtile is to study stte onstrined hirihletExvier optiml ontrol prolem @ygA for qusiEliner monotone ellipti equtionD the soElled weighted pEihrmoni prolemF he oe0ient of the pEihrmoni opertorD the weight uD we tke s ontrol in BV (Ω) ∩ L ∞ (Ω)F ine n importnt mtter for pplitions is to otin solution to given oundry vlue prolem with desired propertiesD it leds to the resonle questionsX n we de(ne n pproprite oe0ient of pEihrmoni opertor to minimize the disrepny etween given displement y d nd n expeted solution to suh prolemF wore preiselyD we nlyse the following optiml design prolemD whih n e regrded s n optiml ontrol prolemD for qusiEliner prtil di'erentil eqution @hiA with mixed oundry onditions winimize I(u, y) = ˆΩ |y − y d | p dx + ˆΩ |Du| @IFIA sujet to the qusiEliner eqution ∆(u|∆y| p−2 ∆y) = f in Ω, @IFPA y = ∂y ∂ν = 0 on Γ D , y = ∆y = 0 on Γ S , @IFQA the pointwise stte onstrints 0 ∂y(s) ∂ν ζ max (s) FeF on Γ S , @IFRA nd the design @ontrolA onstrints u ∈ BV (Ω) nd 0 < α ξ 1 (x) u(x) ξ 2 (x) FeF in Ω. @IFSA rereD Γ D nd Γ S re the disjoint prt of the oundry ∂Ω @∂Ω = Γ D ∪ Γ S AD BV (Ω) ∩ L ∞ (Ω) stnds for the ontrol speD re given distriutionsF rolems of this type pper for p−powerElike elsti isotropi )t pltes of uniform thiknessD where the design vrile u is to e hosen suh tht the de)etion of the plte mthes given pro(leF he model extends the lssil weighted ihrmoni equtionD where the weight u = a 3 involves the thikness a of the plteD see eFgF VD PID PSD PTD or u n e regrded s rigidity prmeterF he yg @IFIA!@IFRA n e onsidered s prototype of design prolems for qusiliner stte equtionsF por n interesting exposure to this sujet we n refer to the monogrphs VD ITD IUF e prtiulr feture of yg @IFIA!@IFRA is the restrition y the pointwise onstrints @IFRA in L p (Γ S )EspeF sn ftD the ordering one of positive elements in L p Espes is typilly nonEsolidD iFeF it hs n empty topologil interiorF pollowing the stndrd multiplier ruleD whih gives neessry optimlity ondition for lol solutions to stte onstrined ygsD the onstrint quli(tions suh s the lter ondition or the oinson ondition should e pplied in this seF roweverD these onditions nnot e veri(ed for ones suh s due to the ft tht int L p + (Γ S ) = ∅D where int (A) stnds for the topologiE l interior of the set AF hereforeD it would e resonle to propose suitle relxtion of the pointwise stte onstrints in the form of some inequlity ondiE tions involving soElled renig pproximtion L p + (Γ S ) ε (B) of the ordering one of positive elements L p + (Γ S )F rereD B is (xed losed se of L p + (Γ S )F es it ws shown in our reent pulition IPD due to ft tht L p + (Γ S ) ⊂ L p + (Γ S ) ε (B) for ll ε > 0D we n reple the one L p + (Γ S ) y its pproximtion L p + (Γ S ) ε (B)F es resultD it leds to some relxtion of the inequlity onstrints of the onsidered prolemD ndD heneD to the pproximtion of the fesile set to the originl ygF reneD the solvility of given lss of ygs n e hrterized y solving the orresponding renig relxed prolems in the limit ε → 0 @for the detilsD we refer to IPD IQAF he ones more hrteristi feture of the yg @IFIA!@IFRA is relted with degenery of qusiliner di'erentil opertor ∆(u|∆y| p−2 ∆y) s ∆y tends to zero nd lso if u pprohes zeroF woreoverD when the term u|∆y| p−2 is regrded s the oe0ient of the hrmoni opertorD we lso hve the se of unounded oe0ientsF sn spite of the ft tht the gontrol in the oe0ients of ellipti prolems hs long history of its own strting with work of wurt IWD PH nd rtr PU @see lso gss RD where the onstrined optiml ontrol prolem in the oe0ients of the leding order di'erentil expressions ws (rst disussed in detilsAD nlogous results for the se of weighted pEihrmoni equtions of the type ∆(u|∆y| p−2 ∆y) remined openF sn this pperD in order to void degenery with respet to the ontrol uD we ssume tht u is ounded wy from zeroF por the preise sttements see the next setionF e leve the se of potentilly degenerting ontrols to future ontriutionF snstedD in this rtileD we fous on the degeneries relted to the nonlinerityF e numer of regulriztions hve een suggested in the litertureF ee PP for disussion for wht hs ome to e known s εEpEvple prolemD suh s ∆ u,ε,p y := div(u(ε + |∇y| 2 ) p−2 2 )∇yF hile the εEpEvplen regulrizes the degenery s the grdients tend to zeroD the term u|∇y| p−2 D viewed gin s oe0ient for the otherwise liner prolemD my grow lrgeF hereforeD we introdue yet nother regulriztion tht leds to sequene of monotone nd ounded pproximtion F k (|∆y| 2 ) of |∆y| 2 @see our reent pulition TD where this pproh ws developed for pEvple prolemAF por (xed prmeter p ∈ [2, ∞)D nd ontrol uD we rrive t twoEprmeter prolem governed y 2 )∆y.
pinllyD we hve to del with twoEprmeter fmily of optiml ontrol prolems in the oe0ients for monotone nonliner di'erentil equtions nd renig relxtion of the the inequlity stte onstrintsF e onsequently provide the wellEposedness nlysis for the underlying prtil di'erentil equtions s well s for the optiml ontrol prolemsF efter tht we pss to the limits s k → ∞ nd ε → 0F he pproximtions nd renig relxtion re not only onsidered to e useful for the mthemtil nlysisD ut lso for the purpose of numeril simultionsF

Setting of the Optimal Control Problem
vet ξ 1 D ξ 2 e (xed elements of L ∞ (Ω) ∩ BV (Ω) stisfying the onditions ) e given distriutionsF he optiml ontrol prolemD we onsider in this pperD is to minimize the disreE pny etween y d nd the solutions of the following homogeneous hirihletExvier oundry vlued prolem st is ler tht A ad is nonempty onvex suset of L 1 (Ω) with n empty topologil interiorF wore preiselyD we re onerned with the following optiml ontrol prolem winimize I(u, y) = ˆΩ |y − y d | p dx + ˆΩ |Du| sujet to the onstrints @QFPA!@QFSAF @QFTA Denition 3.1.e sy tht n element y ∈ W p (Ω) is the wek solution @in the sense of wintyA to the oundry vlue prolem @QFPA!@QFQAD if stnds for the dulity piring etween (W p (Ω)) * nd W p (Ω) ndD in the sequelD we will omit this index when it is from the ontextF he existene of unique solution to the oundry vlue prolem @QFPA!@QFQA follows from n strt theorem on monotone opertorsY seeD for instneD IR or PRD ssFPF Theorem 3.1.Let V be a reexive separable Banach space.Let V * be the dual space, and let A : V → V * be a bounded, semicontinuous, coercive and strictly monotone operator.Then the equation Ay = f has a unique solution for each f ∈ V * .Moreover, Ay = f if and only if Aϕ, ϕ − y f, ϕ − y for all ϕ ∈ V * .
rereD the ove mentioned properties of the strit monotoniityD semiontinuityD nd oerivity of the opertor A hve respetively the following meningX sn our seD we n de(ne the opertor A s mpping Remark QFI. he reson of suh representtion omes from the following oE servtionX hving pplied qreen9s formul twie to the opertor ∆(u|∆y| p−2 ∆y) hen it is esy to show tht A stis(es ll ssumptions of heorem QFI @for the detils we refer to IRD PPAF es onsequene of this theoremD we lso know tht y ∈ W p (Ω) stis(es @QFUA if nd only if the reltions @QFPA!@QFQA re ful(lled s follows @for the detilsD we refer to PPD etion PFRFR nd VD etion PFRFPA sn prtiulrD tking ϕ = y in @QFIPAD this yields the reltion ˆΩ u|∆y| p dx = ˆΩ f y dx, @QFIQA whih is usully referred to s the energy equlityF es resultD onditions @QFIAD @QFSAD priedrih9s inequlityD nd identity @QFIQA led us to the following priori estimte king this ft into ountD we dopt the following notionF Denition 3.2.e sy tht (u, y) is fesile pir to the yg @QFTA if u ∈ where L p + (Γ S ) stnds for the nturl ordering one of positive elements in e denote y Ξ the set of ll fesile pirs for the yg @QFTAF Remark QFP.fefore we proeed furtherD we need to mke sure tht minimiztion prolem @QFTA is onsistentD iFeF there exists t lest one pir (u, y) suh tht (u, y) stisfying the ontrol nd stte onstrints @QFQA!@QFSAD nd (u, y) would e physilly relevnt solution to the oundry vlue prolem @QFPA!@QFQAF sn ftD one needs the set of fesile solutions to e nonemptyF fut even if we re wre tht Ξ = ∅D this set must e su0iently rih in some senseD otherwise the yg @QFTA eomes trivilF prom mthemtil point of viewD to del diretly with the ontrol nd espeilly stte onstrints is typilly very di0ult RD IID PQF husD the onsisteny of ygs with ontrol nd stte onstrints is n open question even for the simplest situtionF sn view of this remrkD it is resonly now to mke use of the following rypothesisF @H 1 A yg @QFTA is regulr in the following sense " there exists t lest one pir vet τ e the topology on the set Ξ ⊂ L 1 (Ω) × W p (Ω) whih we de(ne s the produt of the norm topology of L 1 (Ω) nd the wek topology of W 2,p 0 (Ω; Γ D )F e sy tht pir (u 0 , y ith this nottionD the ontrol prolem @QFTA n e written s follows

Existence of Optimal Solutions
sn this setion we fous on the solvility of optiml ontrol prolem @QFPA!@QFTAF rereinfterD we suppose tht the spe F reneD it is immedite to pss to the limit nd to dedue @RFIAF es onsequeneD we hve the following propertyF Corollary 4.1.Let {(u k , y k ) ∈ Ξ} k∈N and {ζ k ∈ W p (Ω)} k∈N be sequences such yur next step onerns the study of topologil properties of the set of fesile solutions Ξ to prolem @QFTAF he following result is ruil for our further nlysisF Theorem 4.
Then there is a pair Proof.fy heorem PFI nd re)exivity of the spe W p (Ω)D there exists suseE quene of hen y vemm RFID we hve st remins to show tht the limit pir (u, y) is relted y inequlity @QFUA nd stis(es the stte onstrints @QFISAF ith tht in mind we write down the winty reltion for @RFQA sn view of @RFPA nd vemm RFID we hve husD pssing in reltion @RFQA to the limit s k → ∞D we rrive t the inequlity @QFUA whih mens tht y ∈ W 2 (Ω) is wek solution to the oundry vlue prolem @QFPA!@QFQA in the sense of wintyF ine the injetions @PFIA re ompt nd the one L p + (Γ S ) is losed with respet to the strong onvergene in L p (Γ S )D it follows tht ∂y k ∂ν → ∂y ∂ν strongly in L p (Γ S ) ndD heneD his ft together with u ∈ A ad leds us to the onlusionX (u, y) ∈ ΞD iFeF the limit pir (u, y) is fesile to optiml ontrol prolem @QFTAF he proof is ompleteF sn onlusion of this setionD we give the existene result for optiml pirs to the prolem @QFTAF Theorem 4.2.Assume that, for given distributions f ∈ L p (Ω), y d ∈ L p (Ω), and ζ max ∈ L p (∂Ω), the Hypothesis (H 1 ) is valid.Then optimal control problem @QFTA admits at least one solution (u opt , y opt ) ∈ BV (Ω) × W p (Ω).
Proof.ine the set of fesile pirs Ξ is nonempty nd the ost funtionl is ounded from elow on ΞD it follows tht there exists minimizing sequene {(u k , y k ) ∈ Ξ} k∈N to prolem @QFTAF hen the inequlity reneD in view of the de(nition of the lss of dmissile ontrols A ad nd priori estimte @QFIRAD the sequene es ws pointed out in PPD the pEvplin ∆ p (u, y) provides n exmple of qusiEliner ellipti opertor with soElled degenerte nonlinerity for p > 2F sn this ontext we hve nonEdi'erentiility of the stte y with respet to the ontrol uF es follows from heorem RFPD this ft is not n ostle to prove existene of optiml ontrols in the oe0ientsD ut it uses ertin di0ulties when deriving the optimlity onditions for the onsidered prolemF yn the other hndD the ordering one of positive elements L p + (Γ S ) is nonEsolidD iFeF it hs n empty topologil interior in L p EspeF hereforeD it is resonly to pply suitle relxtion of the pointwise stte onstrints in the form of some inequlity onditions involving the soElled renig pproximtion for ll ε > 0D it llows us to reple the one L p + (Γ S ) y its pproximtion L p + (Γ S ) ε (B)F sn ftD it leds to some relxtion of the inequlity onstrints of the onsidered prolemD ndD heneD to the pproximtion of the fesile set to the originl ygF es resultD we introdue the following fmily of pproximting ontrol prolems @seeD for omprisonD the pproh of gss nd pernndez S for qusiE liner ellipti equtions with distriuted ontrol in the right hnd side nd the pproh of uogut nd veugering IPD where the renig regulriztion of pointwise stte onstrints hve een proposedAF winimize sujet to the onstrints rereD k ∈ ND ε is smll prmeterD whih vries within stritly deresing sequene of positive numers onverging to 0D 1 is the losed unit ll in L p (Γ S ) entered t the originF es for the funtion F k : R + → R + D it n eFgF e de(ned y e diret lultion shows tht in this se δ = 4/27F st is ler tht the e'et of suh perturtions of ∆ 2 p (u, y) is its regulriztion round ritil points where |∆y(x)| vnishes or eomes unoundedF sn prtiulrD shows tht the veesgue mesure of the set Ω k (y) stis(es the estimte ) is essentil on sets with smll veesgue mesureF he min gol of this setion is to show tht for eh ε > 0 nd k ∈ ND the pertured optiml ontrol prolem @SFIA!@SFSA is well posed nd its solutions n e onsidered s resonle pproximtion of optiml pirs to the originl prolem @QFTAF o egin withD we estlish few uxiliry results onerning monotoniity nd growth onditions for the regulrized pEhrmoni opertor ∆ 2 ε,k,p F por our further nlysisD we mke use of the following the nottion Remark SFI.por n ritrry element y * ∈ W 2,2 0 (Ω) let us onsider the level set reneD the veesgue mesure of the set Ω k (y * ) stis(es the estimte xowD we estlish the following resultsF Proposition 5.1.por every u ∈ A ad D k ∈ ND nd ε > 0D the opertor Proof.prom the ssumptions on F k nd the oundedness of u we otin whih onludes the proofF Proposition 5.2.por every u ∈ A ad D k ∈ ND nd ε > 0D the opertor A ε,k,u is stritly monotoneF Proof.o egin withD we mke use of the following lgeri inequlityX @SFIIA sn order to prove itD we note tht the left hnd side of @SFIIA n e rewritten s follows 2 |a − b| 2 nd we rrive t the inequlity @SFIIAF ith this we otin ine the reltion implies tht y = v lmost everywhere in ΩD it follows tht the strit monotoniity property @QFWA holds in this seF Proposition 5.3.por every u ∈ A ad D k ∈ ND nd ε > 0D the opertor A ε,k,u is oerive @in the sense of reltion @QFIIAAF Proof.sn order to hek this property it is enough to oserve tht for ny y ∈ e re now in position to pply the strt theorem on monotone opertors @see heorem QFIA to the eqution A ε,k,u y = f with f ∈ L p (Ω)F glosely following the rguments of etion QD we rrive t the following ssertionF Theorem 5.1.For each ε > 0, k ∈ N, u ∈ A ad , and f ∈ L p (Ω), the boundary value problem @SFPA@SFQA admits a unique weak solution y ε,k ∈ W 2 (Ω), i.e.
ˆΩ u(ε @SFIRA por every ε > 0 nd k ∈ ND we denote the set of fesile pirs to the prolem @SFIA!@SFSA s follows (u, y) re relted y equlity @SFIPA, ∂y ∂ν stis(es the inlusions @SFRA. @SFISA st is worth to notie tht rypothesis @H 1 A out regulrity of the originl yg @QFTA n e hrterized y the nonEemptiness properties of the sets of fesile solutions Ξ ε,k for pproximting ontrol prolem @SFIA!@SFSAF sndeedD we hve the following result @see IPD heorem VAF Theorem 5.2.Let {ε k } k∈N ⊂ (0, δ) be a monotonically decreasing sequence converging to 0 as k → ∞.Then, for given distributions f ∈ L p (Ω), y d ∈ L p (Ω), and ζ max ∈ L p (Γ S ), the Hypothesis (H 1 ) implies that the approximating control problem @SFIA@SFSA has a nonempty set of feasible solutions Ξ ε,k for all ε = ε k , k ∈ N.And vice versa, if there exists a sequence (u k , y k ) k∈N satisfying conditions then the sequence (u k , y k ) k∈N is τ -compact and each of its τ -cluster pairs is a feasible solution to the original OCP @QFTA.
husD in view of heorem SFP nd rypothesis @H 1 AD we n suppose tht the sets Ξ ε,k re lwys nonempty ndD thereforeD the pproximting ontrol prolem is onsistentF enlogously to prolem (P)D we n prove the following theorem Theorem 5.3.For every positive value ε > 0 and integer k ∈ N, the optimal control problem (P ε,k ) has at least one solution.