On Henig Regularization of Material Design Problems for Quasi-Linear p-biharmonic Equation

Transcription

1 Alied Mathematics Published Online August 6 in cires htt://wwwsciog/jounal/am htt://dxdoiog/436/am67434 On Henig Regulaization of Mateial esign Poblems fo Quasi-Linea -Bihamonic Euation Pete Kogut Günte Leugeing Ralh chiel eatment of iffeential Euations nioetovs National Univesity nio Uaine eatment Mathemati Lehstuhl II Univesität Elangen-Nünbeg Cauest Elangen Gemany Received 5 June 6 acceted August 6 ublished 4 August 6 Coyight 6 by authos and cientific Reseach Publishing Inc This wo is licensed unde the Ceative Commons Attibution Intenational License (CC BY htt://ceativecommonsog/licenses/by/4/ Abstact We study a iichlet otimal design oblem fo a uasi-linea monotone -bihamonic euation with contol and state constaints We tae the coefficient of the -bihamonic oeato as a design vaiable in BV ( In this aticle we discuss the elaxation of such oblem Keywods -Bihamonic Poblem Otimal esign Relaxation Henig ilating Cone Existence Result Intoduction The aim of this aticle is to analyze the following otimal design oblem (OCP which can be egaded as an otimal contol oblem fo uasi-linea atial diffeential euation (PE with mixed bounday conditions subject to the uasi-linea euation the ointwise state constaints { I ( u y = y yd x + u ( Minimize d uy y+ F xy = f in ( y y = = on Γ y = y = on Γ ν (3 How to cite this ae: Kogut P Leugeing G and chiel R (6 On Henig Regulaization of Mateial esign Poblems fo Quasi-Linea -Bihamonic Euation Alied Mathematics htt://dxdoiog/436/am67434

2 and the design (contol constaints y s max ζ ( s ae on Γ ν ξ ξ u BV and x u x x ae in (5 Hee Γ and Γ ae the disjoint at of the bounday ( BV stands fo the max contol sace y d f and ζ ae given distibutions Poblems of this tye aea fo -owe-lie elastic isotoic flat lates of unifom thicness whee the design vaiable u is to be chosen such that the deflection of the late matches a given ofile The model extends the classical weighted bihamonic euation whee the 3 weight u = a involves the thicness a of the late see eg []-[3] o u can be egaded as a igidity aamete The OCP (-(4 can be consideed as a ototye of design oblems fo uasilinea state euations Fo an inteesting exosue to this subject we can efe to the monogahs [4]-[6] A aticula featue of OCP (-(4 is the estiction by the ointwise constaints (4 in L ( Γ -sace In fact the odeing cone of ositive elements in L -saces is tyically non-solid ie it has an emty toological inteio Following the standad multilie ule which gives a necessay otimality condition fo local solutions to state constained OCPs the constaint ualifications such as the late condition o the Robinson condition should be alied in this case Howeve these conditions cannot be veified fo cones such as L+ ( Γ = { v L ( Γ v ae in due to the fact that ( int L + Γ = whee int ( A stands fo the toological inteio of the set A Theefoe ou main intention in this aticle is to oose a suitable elaxation of the ointwise state constaints in the fom of some ineuality conditions involving a so-called Henig aoximation ( L+ ( Γ ( B of the odeing cone of ositive elements L + ( Γ Hee B is a fixed closed base of L + ( Γ ue to fact that L+ ( Γ ( L+ ( Γ ( B fo all > we can elace the cone L + Γ by its aoximation ( L + ( Γ ( B As a esult it les to some elaxation of the ineuality constaints of the consideed oblem and hence to the aoximation of the feasible set to the oiginal OCP Hence the solvability of a given class of OCPs can be chaacteized by solving the coesonding Henig elaxed oblems in the limit As was shown in the ecent ublication [7] the oosed aoach is numeically viable fo state-constained otimal contol oblems with the state euation given by linea atial diffeential euations In aticula using the finite element discetization of the Henig dilating cone of ositive functions it has been shown in [7] that the above aoximation scheme called conical egulaization whee the egulaization is done by elacing the odeing cone with a family of dilating cones les to a finite-dimensional otimization oblem which can conveniently be teated by nown numeical techniues The non-emtiness of the feasible set fo the stateconstained OCPs is an oen uestion even fo the simlest situation Theefoe we conside a moe flexible notion of solution to the bounday value oblem (-(3 With that in mind we discuss a vaiant of the enalization aoach called the vaiational ineuality (VI method Following this aoach we weaen the euiements on missible solutions to the oiginal OCP and conside inste the family of enalized OCPs fo aoiate vaiational ineualities ( = Γ Γ u y y + F xy ζ y f ζ y ζ K ( whee the sets K ae defined in a secial way As a esult we show that each of new enalized OCP is solvable fo each > and thei solutions can be used fo aoximation of otimal ais to the oiginal oblem The outline of the ae is the following In ection we eot some eliminaies and notation we need in the seuel In ections 3 we give a ecise statement of the state constained otimal contol (o design oblem and descibe the main assumtions on the initial data and contol functions In ection 4 we ovide the esults concening solvability of the oiginal oblem with contol and state constaints We show that this oblem mits at least one solution if and only if the coesonding set of feasible solutions is nonemty In ection 5 we show that the ointwise state constaints can be elaced by the weaened conditions coming fom Henig elaxation of odeing cones As a esult we give a ecise definition of the elaxed otimization oblems and show that the solvability of the oiginal OCP can be chaacteized by the associated elaxed oblems In aticula we ove that the otimal solution to the oiginal oblem can be attained in the limit by the otimal solution of the elaxed oblem We conside in ection 6 the vaiational ineuality method as an aoximation of the OCPs Following this aoach we weaen the euiements on feasible solutions to the oiginal OCP In (4 548

3 contast to the Henig elaxation aoach the enalized otimal contol oblem fo indicated vaiational ineuality has a non-emty feasible set and this oblem is always solvable In conclusion we show that some of the otimal solutions to the oiginal oblem can be attained in the limit by otimal solutions of the enalized oblem Howeve it is unnown whethe the entie set of the otimal solutions can be attained in such way efinitions and Basic Poeties N Let be a bounded oen connected subset of ( N We assume that the bounday is Lischitzian so that the unit outwad nomal ν = ν ( x is well-defined fo ae x whee the abbeviation ae should be inteeted hee with esect to the ( N -dimensional Hausdoff measue We also assume that the bounday consists of two disjoint ats = Γ Γ whee the sets Γ and Γ have ositive ( N -dimensional measues and Γ is of C Let be a eal numbe such that < By W ( we denote the obolev sace as the subsace of L ( of functions y having genealized deivatives y u to ode = in L ( We note that W is a Banach sace with esect to the nom thans to inteolation theoy see ([8] Theoem 44 whee Fo any y C ( = + = W L L + y y y y y d x N y v y= ( yy and yv = x x x x we define the taces γ = y and = ν ( y y γ ( y i i= i i i i By ([9] Theoem 83 these linea oeatos can be extended continuously to the whole of sace W ( We set : : W = γ W W = γ W as closed subsaces of W ( and L esectively Moeove the injections and W W W L (6 ae comact N N ϕ Let C ( Γ = ϕ C ( : ϕ = and = on Γ We define the Banach sace W ( Γ ν as the closue of C ( N Γ with esect to the nom y W ( Let W ( Γ be the dual sace to W ( Γ whee = ( is the conjugate of We also define the sace W ( as the closue of C ( with esect to the nom y = ( y dx W Thoughout this ae we use the notation : = W Γ W Let us notice that euied with the nom : ( d N y d y = y = y x = x L i = xi (7 is a unifomly convex Banach sace [] Moeove the nom is euivalent on nom of W ( Indeed since the Lalace oeato acts fom in L bounday value oblem to the usual and the iichlet y = f in y = on (8 549

4 is uniuely solvable in fo all f L ( it follows that the invese oeato T = ( L W W ( is well defined and satisfies the following ellitic egulaity estimate [] Tf C f W L ( This allows us to conclude the following If f L ( and y W and y is a solution of (8 then y L Hence fo a suitable ositive constant : : y ae such that = on Γ ν y y = on the bounday and theefoe y = T y C y = C y (9 W W L C indeendent of f On the othe hand it is easy to see that y y W Thus by the Closed Gah Theoem we can conclude that is euivalent to the nom induced by W (fo the details we efe to [] [3] By BV ( we denote the sace of all functions in is finite f = f + f BV L L L fo which the nom = f + su f div d x: C x fo x N ϕ ϕ ( ϕ We ecall that a seuence { f = conveges wealy-* to f in conditions hold (see [4]: f f stongly in L ( and f f measues ie lim ϕf = ϕf ϕ C BV if and only if the two following wealy-* in the sace of Ron It is well-nown also the following comactness esult fo BV-saces (Helly s selection theoem see [5] Theoem If { f BV ( and su = f BV ( < + then thee exists a subseuence of { f = stongly conveging in L ( to some f BV ( such that f f wealy- in the sace of Ron measues Moeove if { f BV ( stongly conveges to some f in L ( and satisfies = su f < + then (i f BV and f liminf f (ii f f in BV 3 etting of the Otimal Contol Poblem Let ξ ξ be fixed elements of L BV satisfying the conditions ( x ( x Ca of Caathéodoy func- whee α is a given ositive value Let F : tions on ie the function F( x is continuous in fo almost all x F y is measuable fo each y the function ( < α ξ ξ ae in ( be a nonlinea maing such that F is in the sace In dition the following conditions of subcitical gowth monotonicity and non-negativity ae fulfilled: F x η C η fo ae x and all η ( 55

5 fo some ( ( F x η F x η ηη > fo ae x and all ηη η η (3 F x ηη fo ae x and all η (4 whee N N < N = + N and C > In aticula conditions F x = fo almost all x d ζ be given distibutions The otimal contol oblem we conside in this ae is to minimize the disceancy between y d and the solutions of the following state-constained bounday valued oblem is the citical exonent fo the obolev imbedding W L (3 - (4 imly that F ( x is monotonically inceasing on and Let f W max ( Γ y L ( and L by choosing an aoiate function + = in (5 uy F xy f y y = = on Γ y = y = on Γ ν y s max ζ ( s ae on Γ ν u A as contol Hee ( : = u y u y y is the oeato of fouth ode called the genealized -bihamonic oeato and the class of missible contols A we define as follows { ξ ξ (6 (7 A = u BV x u x x ae in (8 It is clea that L with an emty toological inteio Moe ecisely we ae concened with the following otimal contol oblem A is a nonemty convex subset of { I ( u y = y yd x + u (9 Minimize d subject to the constaints (5-(8 Befoe we will discuss the uestion of existence of missible ais to the oblem (9 we note that the F Ca can be associated with oeato F : defined by the ule function = ( F y v F xyvx d v ( Moeove taing into account the gowth condition ( and the comactness of the obolev imbedding W ( Γ L ( fo < it is easy to show that oeato F : ( is comact efinition 3 We say that an element y is the wea solution (in the sense of Minty to the bounday value oblem (5 - (6 fo a given missible contol u A if d ϕ Γ W Γ u ϕ ϕ ϕ y x+ F ϕ ϕ y f ϕ y C Γ W Rema 3 ince the set C ( N Γ an abitay is dense in it follows that the element ϕ = y + tw with w and t > can be taen as a test function in ( As a esult ( imlies that W w Γ W Γ u y+ tw y+ tw wd x+ F x y+ tw wdx f w ( 55

6 t (because F Ca ( we get u y y w x+ F ( x y w x f w w ( W Γ W Γ Passing to the limit as Hence d d u y y wd x+ F( x y wd x = f w W ( ( Γ W ( Γ and we aive at the standad definition of wea solution to the bounday value oblem (5-(6 Howeve in ode to avoid some mathematical difficulties we will mainly use the Minty ineuality in ou futhe analysis It is woth to note that having alied Geen s fomula twice to oeato ( u y y we aive at the identity u y y vdx ( N N tested by v C Γ = u y y v dx+ u y y vd ν v N v N = u y yvdx u y y d u y y d Γ ν Γ ν v N = u y yvdx u y y d v C ( Γ Γ ν Hence if y as an element of : = W Γ W is the wea solution of the bounday value oblem (5 - (6 in the sense of efinition 3 then elations (5-(6 ae fulfilled as follows (fo the details we efe to ([6] ection 44 and ([4] ection 4 γ γ ( y W ( y in W ( ( u y + F( y = f in C ( Γ γ = in = Γ = in Γ : = ( Γ ( y W W In aticula taing w= y in ( this yields the elation u y d x+ F( xy yx d = f y (3 W ( Γ W ( Γ As a esult conditions ( (8 and ineualities (4 and (9 le us to the following a ioi estimate ( α W ( Γ y : = y d x C f u A (4 The existence of a uniue wea solution to the bounday value oblem (5-(6 in the sense of efinition 3 follows fom an abstact theoem on monotone oeatos Theoem ([7] Let V be a eflexive seaable Banach sace Let V * be the dual sace and let A: V V be a bounded hemicontinuous coecive and stictly monotone oeato Then the euation Ay = f has a uniue solution fo each f V Hee the above mentioned oeties of the stict monotonicity hemicontinuity and coecivity of the oeato A have esectively the following meaning: Ay Av y v y v V (5 V V Ay Av y v = y = v (6 V V V the function t A y + tv w is continuous fo all y v w V (7 V Ay y V V lim = + (8 V y y V 55

7 ( In ou case we can define the oeato Au ( as a maing by A( u y w : = u y y wd x+ F ( ( x y wd x In view of the oeties (-(4 and comactness of the obolev imbedding W ( L < it is easy to show that A( u y = ( u y + F( y and ( Γ fo Au satisfies all assumtions of Theoem (fo the details we efe to [6] [7] Hence the vaiational oblem Fo a given u A find y such that Auy ϕ = f ϕ ϕ ( We note that the duali- Γ because ( ( It emains to show that the solution y of (3 satisfies the Minty elation ( Indeed in view of the monotonicity of A we have fo which Auy ( = f is its oeato fom has a uniue solution y = y( u ty aiing in the ight hand side of (3 maes a sense fo any distibution f W ( W ( Γ : = W ( Γ Thus Auv Auy v y ( ( ( ( = Auv v y Auy v y ( by (3 = Auv v y f ϕ ( ( ϕ ( W Γ W Γ Auv v y f v and hence in view of Rema 3 the Minty elation ( holds tue Taing this fact into account we ot the following notion efinition 3 We say that ( I u y < + and u y is a feasible ai to the OCP (9 if u A BV y the ai ( u y is elated by the Minty ineuality ( max γ ( y L ζ γ ( y L + + (9 (3 Γ Γ (3 whee L + ( Γ stands fo the natual odeing cone of ositive elements in L Γ ie N L Γ : = v L Γ + v -ae on Γ (3 We denote by Ξ the set of all feasible ais fo the OCP (9 We say that a ai ( u y BV ( is an otimal solution to oblem (9 if Ξ = ( uy u y and I u y inf I u y Ξ Rema 3 Befoe we oceed futhe we need to mae sue that minimization oblem (9 is meaningful ie thee exists at least one ai ( u y such that ( u y satisfying the contol and state constaints (6-(8 I( u y < + and ( u y would be a hysically elevant solution to the bounday value oblem (5-(6 In fact one needs the feasible set Ξ to be nonemty But even if we ae awae that Ξ this set must be sufficiently ich in some sense othewise the OCP (9 becomes tivial Fom a mathematical oint of view to deal diectly with the contol and esecially state constaints is tyically vey difficult [8]-[] Thus the nonemtiness of feasible set fo OCPs with contol and state constaints is an oen uestion even fo the simlest situation It is easonably now to mae use of the following Hyothesis u y BV u y Ξ (H Thee exists at least one ai such that 4 Existence of Otimal olutions In this section we focus on the solvability of otimal contol oblem (5-(9 Heeinafte we suose that the 553

8 sace BV ( is endowed with the nom ( u y : = u BV BV + y ology on the set Ξ BV ( which we define as the oduct of the wea- toology of BV ( and the wea toology of W ( Γ We begin with a coule of auxiliay esults Lemma Let {( u y we have Let τ be the to- τ Ξ be a seuence such that ( u y ( u y in BV ( Then lim u y ϕdx u y ϕd x ϕ C ( = Γ (33 u in L ( and { u is bounded in L we get that u u stongly in L ( u u in y ϕ y ϕ in L ( Poof ince u fo evey < + In aticula we have that Hence it is immediate to ass to the limit and to deduce (33 As a conseuence we have the following oety Coollay Let {( u y Ξ and ζ W ( Γ in BV ( ζ ζ in W ( Γ Then and lim u y ζ dx = u y ζd x L and be seuences such that ( τ u y ( u y Ou next ste concens the study of toological oeties of the feasible set Ξ to oblem (9 The following esult is cucial fo ou futhe analysis Theoem 3 Let {( u y Ξ be a bounded seuence in BV ( Then thee is a ai τ u y BV u y u y u y Ξ ( such that u to a subseuence ( and Poof By Theoem and comactness oeties of the sace {( u y Ξ still denoted by the same indices and functions u BV ( and y u u in L y yin W ( Γ and theefoe y yin W Then by Lemma we have thee exists a subseuence of such that (34 lim u ϕ y dx = uϕyd x ϕ C Γ It emains to show that the limit ai ( (3 With that in mind we wite down the Minty elation fo ( In view of (34 and Lemma we have u y is elated by ineuality ( and satisfies the state constaints u y : d ( d ϕ Γ W Γ ϕ ϕ ϕ + ϕ ϕ u y x F x y x f ϕy C Γ W lim ϕ u dx = ϕ ud x lim u ϕ ϕ y dx = u ϕ ϕyd x Moeove due to the comactness of the obolev imbedding W ( L F( x ϕ( ϕ y d x = F( x ϕ( ϕ y d x+ J whee Hölde s ineuality yields by ( ( ϕ ( ϕ Γ fo < we have J : = F x y y dx C dx y y as L We thus can ass to the limit in elation (35 as and aive at the ineuality ( which means that y is a wea solution to the bounday value oblem (5-(6 ince the injections (6 ae comact y y L + Γ is closed with esect to the stong convegence in L ( Γ it follows that ν ν and the cone (35 554

9 stongly in L Γ and hence ( y = ( y L ( Γ ( y max L ( Γ lim γ γ and γ ζ + + P Kogut et al This fact togethe with u y is feasible to otimal contol oblem (9 The oof is comlete In conclusion of this section we give the existence esult fo otimal ais to oblem (9 Theoem 4 Assume that fo given distibutions f W max ( Γ yd L ( and ζ L ( the Hyothesis (H is valid Then otimal contol oblem (9 mits at least one solution ( ot ot u y BV ( Poof ince the set Ξ is nonemty and the cost functional is bounded fom below on Ξ it follows that u y Ξ to oblem (9 Then the ineuality thee exists a minimizing seuence u A les us to the conclusion: ( u y Ξ ie the limit ai ( uy Ξ imlies the existence of a constant C > such that inf I u y = lim y x y x d d x u + < + su u Hence in view of the definition of the class of missible contols A and a ioi estimate (4 the seu y BV Theefoe by Theoem 3 thee exist functions uence {( Ξ is bounded in * * * * u A and y such that ( * and y y wealy in * u y Ξ and u to a subseuence u u wealy- in BV( W Γ To conclude the oof it is enough to show that the cost functional I is * lowe semicontinuous with esect to the τ-convegence ince y y stongly in L ( by obolev embedding theoem it follows that Thus * * Hence ( C * = lim y x y x dx y x y x dxand d d * lim inf u u by ( * * ( ( u y Ξ I u y liminf I u y = inf I u y u y is an otimal ai and we aive at the euied conclusion 5 Henig Relaxation of tate-constainted OCP (9 The main goal of this section is to ovide a egulaization of the ointwise state constaints by elacing the odeing cone Λ= : L + ( Γ (see (3 by its solid Henig aoximation ( Λ (see []-[4] and show that the conical egulaization aoach les to a family of otimization oblems such that thei solutions can be obtained by solving the coesonding otimality system and the egulaized solution τ-convege in the limit as to a solution of the oiginal oblem We begin with some fomal descitions and abstact esults Let be a eal nomed sace and let Λ be a closed odeing cone in efinition 5 A nonemty convex subset B of a nontivial odeing cone Λ (ie Λ { whee is the zeo element in is called base of Λ if fo each element z Λ \{ thee is a uniue eesentation z = µ b whee µ > and b B In what follows we always assume that the odeing cone Λ has a closed base B Λ We note that in geneal bases ae not uniue We denote the nom of by and fo abitay elements z z we define Λ Λ z z z z Λ as well as z < z z Λ \ In ode to intoduce a eesentation fo a base of Λ let * be the toological dual sace of and let 555

10 * be the dual aiing Moeove by and { z z z z * * * * * Λ = Λ : { z z z z * : \ # * * * Λ = > Λ we define the dual cone and the uasi-inteio of the dual cone of Λ esectively Using the definition of the dual cone the odeing cone Λ can be chaacteized as follows (see [5] Lemma 3: { z z * * z z * * Λ= Λ ue to Lemma 8 in [5] we can give the following esult Lemma Let Λ be a nontivial odeing cone in a Banach sace Then the set * { * B : = z Λ z z = is a base of Λ fo evey z * Λ # Moeove if Λ is eoducing in ie if ΛΛ= and if B is a base of Λ then thee is an element Rema 53 As follows fom Lemma the set N B: = { ξ L+ ( Γ ξd = Γ is a closed base of odeing cone Λ= : ( Γ L + * # * z Λ satisfying B : { z z z * = Λ = (36 Now we ae eaed to intoduce the definition of a so-called Henig dilating cone (see huang [4] which is based on the existence of a closed base of odeing cone Λ efinition 5 Let be a nomed sace and let Λ be a closed odeing cone with a closed base B Choosing > abitaily the coesonding Henig dilating cone is defined by ({ µ µ ( ( ( Λ B : = cl cone B+ B : = cl z z B+ B whee B ( : = { v v is the closed unit ball in centeed at the oigin It is clea that Λ ( B deends on the aticula choice of B As follows fom this definition int ( Λ ( B fo evey > ie Henig dilating cone is oe solid Moeove we have the following oeties of such cones (see [4] [6] Poosition 5 Let be a nomed sace and let Λ be a closed odeing cone with a closed base B δ whee Choosing the following statements hold tue Λ ( B is ointed ie Λ ( B ( Λ ( B = { Λ ( B Λ ( B > + γ γ ( 3 ( B cone cl ( B B ( Λ = + 4 Λ= Λ ( B < < δ 5 the imlication holds tue with κ su { ξ : ζ B = { b b B δ : = inf > (37 ξ Λ ξ ξ κ + + Λ ( B / ie ξ ξ Λ κ + (38 556

11 In the context of constaint ualifications oblem the following esult lays an imotant ole Poosition 6 Let be a nomed sace and let Λ be a closed odeing cone with a closed base B δ abitaily whee δ is defined by (37 the inclusion Choosing { int ( ( B Λ Λ (39 holds tue Poof Let z Λ \{ be chosen abitaily By the definition of a base thee is a uniue eesentation z = λb with λ > and b B Obviously holds tue Let s assume fo a moment that Then we obtain ({ λ λ ( ( λ ( λ z int b + B = int B b ( ( B λb cone b + B (4 λ ( ( ( ( ( ( z int cone b + B int cone B+ B = int Λ B which comletes the oof In ode to show (4 let x B ( λb Then yields be chosen abitaily ie λ xλb λ x λ b = xλb = λ λ λ { µ µ ( ( x y yb = cone b + B As a esult (4 is satisfied Rema 5 The following oety coming fom Poosition 6 tuns out athe useful: in ode to ove z int ( Λ ( B it is sufficient to chec whethe z Λ \{ The following esult shows that Henig dilating cones Λ ( B ossess good aoximation oeties Poosition 7 Let Λ be a closed odeing cone in a nomed sace and let B be an abitay closed base δ be a monotonically deceasing seu- of Λ Let aamete δ be defined as in (37 and let ( ence such that lim = Then the seuence of cones Λ ( B { with esect to the nom toology of as tends to infinity that is whee K liminf Λ B =Λ= K limsu Λ B { K lim inf Λ B : = z fo all neighbohoods N of z thee is a { conveges to Λ in Kuatowsi sense such that N Λ B K lim su Λ B : = z fo all neighbohoods N of z and evey thee is a such that N Λ B Poof Let z Λ be chosen abitaily Then N Λ holds tue fo evey neighbohood N of z and due to the inclusions Λ Λ we see that Λ fo all Hence Taing into account the inclusion (4 and the fact that N ( B Λ K liminf Λ (4 557

12 we get To show that the seuence Λ ( B K liminf Λ B K limsu Λ B Λ K liminf Λ B K limsu Λ B (4 { Howeve the inclusion (43 is euivalent to conveges to Λ in Kuatowsi sense it emains to show ( B K lim su Λ Λ (43 \ \ lim su ( Λ K Λ ( B Let z \ Λ be an abitaily element ince Λ is closed thee is an oen neighbohood N of z with esect to the nom toology of such that N Λ= By Poosition 5 (see item (4 thee is a sufficiently lage index such that This imlies N Λ B = z \ lim su Λ B Combining (4 (43 and (44 we aive at the elation Thus K lim ( B Λ K liminf Λ B K limsu Λ B Λ Λ= Λ and the oof is comlete Taing these esults into account we associate with OCP (9 the following family of Henig elaxed oblems subject to the constaints (44 { I ( u y = y yd x + u (45 Minimize d ( y = W ( ( y = in W ( Γ ( y W ( ( y L B ( + ( y ( L+ ( ( B ( ( uy + F( y = f in C ( Γ γ in γ γ = in Γ γ ζ max Γ u A Γ o in a moe comact fom each of these oblems can be stated as follows whee γ ( uy Ξ ( δ (46 inf Iuy (47 { B L ( Γ δ = inf ξ : ξ (48 the base B taes the fom (36 and the feasible set Ξ BV ( we define as follows: ( u y if and only if u A I( u y < + y the ai ( and Ξ u y is elated by the Minty ineuality ( 558

13 ( + ( B Hee L max ( + ( + ( γ y L Γ B ζ γ y L Γ B (49 Γ stands fo the coesonding Henig dilating cone ince by Poosition 6 the inclusion Ξ Ξ holds tue fo all > it is easonable to call the OCP (47 a Henig elaxation of OCP (9 Moeove as obviously follows fom Poosition 7 the convegence Ξ Ξ in Kuatowsi sense holds tue with esect to the τ-toology on BV ( We ae now in a osition to show that using the elaxation aoach we can educe the main suositions of Theoem 4 In aticula we can chaacteize Hyothesis ( H by the non-emtiness oeties of feasible sets Ξ fo the coesonding Henig elaxed oblems Theoem 8 Let { ( δ be a monotonically deceasing seuence conveging to as Then f W max y L ζ L the Hyothesis (H imlies that fo given distibutions ( Γ ( and the Henig elaxed oblem (47 has a nonemty set of feasible solutions u y satisfying conditions vesa if thee exists a seuence ( u y I( u y then the seuence s ( u y OCP (9 d Ξ fo all = And vice Ξ fo all and su < + (5 is τ-comact and each of its τ-cluste ais is a feasible solution to the oiginal Poof ince the imlication ( fo all on the oof of the invese statement oety (5 imlies the existence of at least one ai ( ( u y Ξ Let ( u y Ξ Ξ > is obvious by Poosition 7 we concentate u y such that be an abitay seuence with oety: ( and a ioy estimate (4 do not deend on aamete u y Ξ fo all ince the set A and the condition (5 imlies su u u y it follows by comactness aguments (see the oof of Theoem 4 that thee exist a subseuence of (still denoted by the same index and a ai ( u y A such that τ ( u y ( u y as Closely following the oof of Theoem 3 it can be shown that the limit ai ( u y ( y < + and function y is such that u A J u is a wea solution to the bounday value oblem (5 - (6 Moeove in view of the comactness oeties of injections (6 we may suose that It emains to establish the inclusions γ ( γ y y stongly in L Γ as (5 max γ( y L+ ( ζ γ( y L+ ( By contaosition let us assume that : max \ ξ ζ γ y L L+ ( closed it follows that thee is a neighbohood ( ξ of ξ in L ( Γ such that ( ξ L + ( Using the fact that Γ Γ (5 L + Γ is Γ = = Γ Γ ince the cone ( L Γ L Γ B L Γ B l by Poosition 7 and definition of the Kuatowsi limit it is easy to conclude the existence of an index such that l ( ξ ( L+ Γ ( B = (53 Howeve in view of the stong convegence oety (5 thee is an index satisfying Combining (53 and (54 we finally obtain ( ξ ξ (54 ( y L ( ( L ( ( B { ξ = ζ γ Γ \ + Γ max max 559

14 This howeve is a contiction to ( ( max ζ γ y L+ Γ B max Thus ζ γ ( y L+ ( Γ In the same manne it can be shown that γ ( y L+ ( ( u y is feasible fo OCP (9 Γ Hence the ai As an obvious conseuence of this Theoem and Theoem 4 we have the following notewothy oety of the Henig elaxed oblems (47 Coollay Let f W max ( Γ yd L ( and ζ L ( be given distibution Then the Henig elaxed oblem (47 is solvable fo each ( δ ovided Hyothesis ( H is satisfied The next esult is cucial in this section We show that some otimal solutions fo the oiginal OCP (9 can be attained by solving the coesonding Henig elaxed oblems (45-(46 Howeve we do not claim that the entie set of the solutions to OCP (9 can be estoed in such way ζ be given distibutions Let be a monotonically deceasing seuence such that as whee δ > is de fined by (48 Let {( u y Ξ be a seuence of otimal solutions to the Henig elaxed oblems (45- (46 such that Theoem 9 Let f W max ( Γ yd L ( and L { ( δ i i Then thee is a subseuence ( u y i u su BV ( of ( u y i i τ < + and a ai ( u y such that (55 u y u y as i (56 ( uy u y Ξ and I u y = inf I u y (57 Poof In view of a ioy estimate (4 the unifom boundedness of otimal contols with esect to BV-nom (55 imlies the fulfilment of condition (5 Hence the comactness oety (56 and the inclusion ( u y Ξ ae a diect conseuence of Theoem 8 It emains to show that the limit ai ( u y is a solution to OCP (9 Indeed the condition ( u y Ξ imlies the fulfilment of Hyothesis ( H Hence by Theoem 4 the oiginal OCP (9 has a nonemty set of solutions Let ( u y be one of them Then the following ineuality is obvious I( u y I( u y On the othe hand by Poosition 5 (see oety (4 we have ( i ( i u y i i i I( u y I( u y I( u y i Ξ (58 u y Ξ i fo evey i ince ae the solutions to the coesonding elaxed oblems (47 it follows that inf = (59 ( uy Ξ i As a esult taing into account the elations (58 and (59 and the lowe semicontinuity oety of the cost functional I with esect to the τ-convegence we finally get Thus by (59 i i inf I( u y = I( u y liminf I( u y ( uy Ξ i ( uy Ξ by (58 i i liminf I u y I u y I u y i i i = = inf I u y lim I u y I u y i and we aive at the desied oety (57 The oof is comlete Rema 53 It is woth to note that condition (55 can be omitted if the oiginal OCP (9 is egula that is when Hyothesis ( H is valid Indeed let us assume that Ξ and ( uˆ yˆ Ξ is an abitay ai Then uˆ y ˆ is feasible to each Henig elaxed oblems (45-(46 and hence 56

15 ( uy Ξ ( ˆ ˆ inf I u y = I u y I u y (6 ince by Poosition 6 the inclusion Ξ Ξ holds tue fo all monotone in the following sense (because of the oety ( of Poosition 5 it follows that Ξ Ξ Ξ Ξ ( uy Ξ ( uy Ξ ( uy Ξ As a esult (6 les to the estimate > and the seuence ( ˆ ˆ inf I u y inf I u y inf I u y I u y u u x I( u y ξ I ( uˆ yˆ + ( uy Ξ L + < + su su d inf BV Ξ is As was mentioned at the beginning of this section the main benefit of the elaxed otimal contol oblems (45-(46 comes fom the fact that the Henig dilating cone ( L + ( Γ ( B has a nonemty toological inteio Hence it gives a ossibility to aly the late condition o the Robinson condition in ode to chaacteize the otimal solutions fo the state constained OCP (9 On the othe hand this aoach ovides nice convegence oeties fo the solutions of elaxed oblems (45-(46 Howeve as follows fom Theoems 8 and 9 (see also Rema 55 the most estictive assumtion deals with the egulaity of the elaxed oblems (45-(46 fo all ( δ o if we eject the Hyothesis ( H it becomes unclea in geneal whethe the elaxed sets of feasible solutions Ξ ae nonemty fo all In this case it maes sense to ovide futhe elaxation fo each of Henig oblems (45-(46 In aticula using the methods of vaiational ineualities we show in the next section that oiginal OCP (9 may mit the existence of the so-called weaened aoximate solution which can be inteeted as an otimal solution to some otimization oblem of a secial fom 6 Vaiational Ineuality Aoach to Regulaization of OCP (9 As follows fom Theoem 4 the existence of otimal solutions to the oblem (9 can be obtained by using comactness aguments and the Hyothesis ( H Howeve because of the state constaints (7 the fulfilment of Hyothesis ( H is an oen uestion even fo the simlest situation Nevetheless in many alications it is an imotant tas to find a feasible (o at least an aoximately missible in a sense to be me ecise solution when both contol and state constaints fo the OCP ae given Thus if the set of feasible solutions is athe thin it is easonable to weaen the euiements on feasible solutions to the oiginal OCP In aticula it would be easonable to assume that we may satisfy the state euation and the coesonding state constaint Auy ϕ = f ϕ ϕ max y K : = { v γ( v L+ ( Γ ζ γ( v L+ ( Γ with some accuacy Hee the oeato A( : BV ( ( of elation (9 Fo this uose we mae use of the following obsevation: If a ai ( oiginal oblem ie ( u y Ξ then this ai satisfies the elation Auy ζ y f ζ y ζ K fo each > whee K is defined as follows Hee L is defined by the left-hand side u y is feasible to the ( ( max { γ ( + ( ζ γ ( + ( K : = v v L Γ B v L Γ B (6 ( + ( B Γ is the coesonding Henig dilating cone Note that the evese statement is not tue in geneal In fact we discuss a vaiant of the enalization aoach u y A K u y satisfy the oeato called the vaiational ineuality (VI method This idea was fist studied in [7] Thus if a ai is elated by vaiational ineuality (6 then it is not necessay to suose that (6 56

16 = f In view of this we can use the enalized tem Auy ( f ( measue in an associated cost functional As a esult we aive at the following enalized OCP: euation Auy ( subject to the constaints Minimize I ˆ u y y y d x u A u y f ( = d + + u A y K Auy ( ζ y f ζ y ( ( ζ K as a deviation o in a moe comact fom this oblem can be stated as follows inf Iˆ ( u y ( δ (65 ( uy Ξˆ whee δ > is given by (48 the set K is defined in (6 and the set of feasible solutions Ξˆ BV we descibe as follows: {( u y u A ˆ K I ( u y ( u y Ξ ˆ : = : < + and satisfies VI (6 In this section we show that enalized OCP (65 is solvable fo each > without any assumtion about fulfilment of Hyothesis (H We also study the asymtotic oeties of seuences of otimal ais {( u y > to oblem (65 when the small aamete > vaies in a stictly deceasing seuence of ositive numbes conveging to zeo We begin with the following esult Lemma 3 Unde assumtions (-(4 fo evey fixed u U and ( δ the vaiational ineuality (6 mits at least one solution y = y( u such that y K Poof Let ( δ be a fixed value As follows fom definition of the set K (see (6 and Rema 5 K is a nonemty convex closed subset of with esect to the -nom toology ue to the assumtions (-(4 we have the following estimates = ( su Auy su su Auy v ( y ρ y ρ v y ρ v y ρ v y ρ v = su su u y y vd x+ F x y vdx su su ξ + y v C y v L L L su su ξ + ξ ρ + CN ρ < + y v C L N y v L Auy y uy x α y ( whee N is the nom of the embedding oeato W L oeato ( d Hence fo evey fixed u A the Au : ( is bounded and coecive Moeove it is shown in [6 Poosition 4] the oeties (-(4 ensue the following imlication ( Γ y yin W lim su Auy ( y y ( v W ( Γ Auy ( yv liminf Auy ( ( y v ( ( Thus the oeato Au ( : ( is seudo-monotone fo each u A the well-now existence esult (see fo instance [8] [9] thee exists at least one solution y y( u (63 (64 (66 Hence following = of vai- 56

17 ational ineuality (6 such that y K As an obvious conseuence of Lemma 3 we have the following notewothy oety of enalized OCP (63 - (64 Coollay 3 Fo each ( δ the feasible set Ξ ˆ is nonemty To oceed futhe we intoduce the following notion efinition 6 An oeato A : {( u y = such that { u τ A and ( u y ( u y in BV W ( lim su Au ( y y y imlies the elation fo all v W Γ A is said to be uasi-monotone if fo any seuence ( ( liminf ( ( Auy y v Au y y v ( Γ the condition efinition 6 We say that an oeato A : A ossesses the oety any seuence {( u y = such that { u τ A and ( u y ( u y in BV W ( conditions Au y din limsu Au y y ( dy ( imly the elation d Auy ( = Ou next intention is to ove the following cucial esult A : A vided assumtions (-(4 hold tue Theoem The oeato Poof Let {( u y = W be a seuence such that { u (67 (68 M if fo Γ the given by fomula (9 is uasi-monotone o- τ A and ( u y ( u y in BV Γ We assume that ineuality (67 holds tue Ou aim is to establish the elation (68 With that in mind we set = + ( u A and yvw W ( B u v y w : u y y wd x F x v wd x Γ and divide ou oof onto seveal stes v W Γ te We show that fo each ( lim B u y v y y ( ( ( ( Indeed since y y in s L ( fo all s ( F( x y( y y dx C y y y dx : = lim u v v y y dx+ lim F x y y y dx = W Γ it follows by the obolev embedding theoem that y Hence maing use of the subcitical gowth condition ( we get C su y y y as L L As fo the fist tem in (7 we note that u u in L ( fo evey fo all by the initial assumtions Hence v u u dx v W Γ (69 (7 y in (7 < + because uu L ( (7 by the Lebesgue ominated Theoem ince the seuence { ζ : = u v v is bounded in L and 563

18 L stongly in ζ u v v = v u u d x it follows fom (7 that u v v u v v L Theefoe the fist tem in (7 tends to zeo as as the oduct of stongly and wealy convegent seuences Combining this fact with (7 we aive at the desied oety (7 te Let us show that ( lim B u y v w ( = lim u v v wdx+ lim F x y wdx = u v v wd x+ F( x y wdx = Buyv ( w vw W ( ( Γ u u in L ( fo evey uϕ stongly in L ( ϕ L ( In view of this we infe lim u wϕdx = uwϕd x ϕ L w W ( Γ By analogy with the evious ste we note that yields u ϕ This means that u w uw L in But we also have that the seuence { u w is bounded in fo each w W ( Γ ince v v L ( fo any v W by definition of the wea convegence in L show that (73 < + In aticula this L Hence uw uw Γ it follows that in L ( lim u w v vdx u w v vdx = (74 Thus in ode to conclude the euality (73 it emains to F( xy wx= F( xywx w W ( Γ (75 lim d d In view of the subcitical gowth condition ( we have the following estimate ( L F y : = F xy dx C y dx ( ( L C y C N su y whee N is the nom of the embedding oeato W L uence { F( xy exists an element ψ L is comact with esect to the wea convegence in L such that u to a subseuence Hence we may suose that the se- and theefoe thee F xy ψ in L as (76 Thus to conclude this ste we have to show that ψ = F( xy that fo evey z and evey ositive function φ C ( we have φ ( x ( F( x y F( x z ( y z dx o taing into account (76 and the fact that y y stongly in L we can ass to the limit in this ineuality as As a esult we get By monotonicity oety (3 it follows by obolev embedding theoem 564

19 fo all ositive φ C ( ψ φ x F xz yzdx z Afte localization we have ince the function F : is a diect conseuence of the convegence (76 ( ψ F xz yz z is stictly monotone it follows that F( xy ψ = Thus the elation (75 te 3 This is the final ste of ou oof As follows fom (69 fo evey element v W each index we have the estimate ( ( ( ( B u y y B u y v y v : = u y y v v y v dx y v dx y vin α > Let v W ( Γ be a fixed element We ut y = ( y+ v fo all σ [ ] the monotonicity condition (77 we see that ince A( u y B( u y y σ σ σ ( ( B u y y B u y y y σ yσ ( = it follows fom (78 that Γ and (77 Taing into account (78 ( σ Au y y v ( ( ( σ B( u y yσ y v A u y y y + B u y y y y ( + σ Passing to the limit in (79 as we obtain whee ( ( σ liminf Au y y v ( ( limsu Au y y y ( ( σ + liminf B u y y y y ( ( σ + σ liminf B u y y yv ( ( σ liminf B u y y y y ( (79 (8 and ( σ by (7 = lim B u y y y y ( = ( by (67 lim su Au y y y ( by (73 ( σ = ( σ lim B u y y y v B u y y y v ( Hence fo each σ [ ] we have the ineuality ( ( σ liminf A u y y v B u y y y v ( (8 565

20 ince the convegence yσ y is stong in ( stongly in L ( and theefoe B( u y yσ y v ( W Γ it follows that σ σ y y y y σ u y y y v d x + F x y y v d x = : A u y y v ( As a esult we deduce fom (8 and (8 that ( liminf Au y y v ( ( ( ( ( ( ( ( liminf Au y y y + liminf Au y yv ( liminf Au y y y + Auy yv liminf B u y y B u y y y y ( ( + Auy ( yv + liminf B u y y y y ( by (78 ( ( ( liminf B u y y y y + A u y yv by (7 = Auy y v ( that is the ineuality (68 is valid Rema 6 In fact (see [9] Rema 33 we have the following imlication: ( M Ais uasi-monotone Aossesses the oety which is de- Hence in view of Theoem we can claim that the oeato A : A fined by elation (9 ossesses the oety ( M (8 We ae now in a osition to show that the enalized otimal contol oblem in the coefficient of vaiational ineuality (63-(64 is solvable fo each value ( δ Lemma 4 If the assumtions (-(4 ae valid then the OCP (63-(64 mits at least one solution ( u y ˆ Ξ fo evey fixed ( δ and any f W max ( Γ yd L ( and ζ L ( Poof Let {( u y U K be a minimizing seuence to oblem (63-(64 The coeciveness o- = ety (66 and estimate ζ immediately imly that the seuence V = and V = we have ( Au y y f y ( ζ ( y = f W ( Γ y ζ is bounded in (83 W Γ Indeed using the notations Ay y ζ Ay y ζ Ay y ζ V V V V V V = + as y V y ζ y + ζ y ζ V V V V V + y On the othe hand fom (83 it follows that Ay y ζ f y ζ V V V V f = V yζ yζ V V f V W ( Γ o comaing these two chains of elations we aive at the existence of a constant C > such that C is indeendent of u A and y C as fa as y K V is a solution to (63 ince 566

21 su u su Iˆ u y < + and the set A K is seuentially closed with esect to the τ-convegence we may assume by Theoem τ u y K u y u y Then assing to the limit in that thee exists a ai as we obtain Having ut hee Hence A such that ( ( Au y ζ y f ζ y ( ( lim su Au y y ζ f ζ y ( ζ K ( ζ = y K we aive at the ineuality ( lim su Au y y y ( ( liminf Au y y ζ Au ( y y ζ K ( ζ by the uasi-monotonicity oety of the oeato A Combining this ineuality with (84 we come to the elation Thus ( u y K Let us show that ( u y { Au ( y is bounded in ( ubstituting Au ( y ζ y fζ y ζ K A is a feasible ai to the oblem (63-(64 ( is an otimal ai to this oblem As follows fom (83 the seuence as Then Let d be its wea limit in ( lim su Au y y ( f y ζ + d ζ ( = d y > + d f ζ y ζ K ( ( y fo ζ in the last ineuality we get ( limsu Au y y dy ( ince the uasi-monotone oeato ossesses the (84 M -oety (see Rema 66 it follows that d Au ( y As a esult using the τ-lowe semicontinuity oety of the cost functional (63 we finally obtain ˆ ˆ inf I ˆ ( ˆ u y = liminf I u y I u y + A u y f = I u y uy Ξ ( Thus ( u y is an otimal ai to the enalized oblem (63-(64 The next ste of ou analysis is to conside a seuence of otimal ais ( u y K = A in the limit > as tends to Theoem Let {( u y be a seuence of otimal ais to enalized oblems (63 - (64 In dition > to the assumtions of Lemma 4 assume that thee exists a constant C > such that Then the seuence ( u y τ-cluste ai ( > su Iˆ u y C (85 is elatively comact with esect to the τ-convegence and each of its u y > is such that (u to a subseuence ( τ u y ( u y as (86 ( uy u y Ξ and I u y = inf I u y (87 Ξ 567

22 ie ( Poof Let ( u y u y is an otimal ai to the oiginal OCP (9 > the set K W be a given seuence of otimal ais to enalized oblems (63-(64 ince each of Γ contains zeo we have d W Γ = α y u y x Au y y ( by (6 Hence the following estimate fo the otimal states taes lace ( f y f y ( W ( Γ ( α ( W Γ y f u A (88 Let us show that the seuence of coesonding otimal contols { u > estimate (85 the numeical seuence Iˆ ( u y of the stuctue of the cost functional (63 we deduce > is BV-bounded Indeed due to the is unifomly bounded with esect to Hence in view A u y f C u C su u Fom this we immediately conclude that > < + and hence due to Theoem Poosition 7 BV ( and estimate (88 we may assume that thee exists a ai ( u y A K such that ( u y τ ( u y as in BV ( W ( Γ (hee we have used the fact that the sets K convege in Kuatowsi sense to K see the oof of Theoem 8 Let us show that the ai ( u y is feasible to the oiginal oblem (9 Using the aguments of the oof Au y d d= Au y Then as follows fom (89 we have of Lemma 4 we have ( in ( and Au ( y f Au ( y f Au ( y f ( ( liminf = lim = Thus Au ( y = f as elements of ( and hence ( u y Ξ It emains to ove that ( ( u y Ξ such that I( u y < I( u y then I( u y I( u y A( u y f I( u y ( (89 u y is an otimal ai If on the contay we assume that the exists a ai + > Theefoe assing to the limit in this ineuality as and using the w-lowe semicontinuity oety of the cost functional we finally get I( u y liminf I( u y I( u y This contiction immediately les us to the conclusion: The ( u y is an otimal ai to the OCP (9 Rema 6 As follows fom the oof of Theoem whateve the seuence of otimal solutions {( u y to the enalized oblems (63-(64 has been chosen if this seuence satisfies condition (85 then it always gives in the limit as some otimal ai to the oiginal OCP (9 Howeve it is unnown whethe the entie set of the solutions to OCP (9 can be attained in such way Rema 63 It is easy to see that in the case if the feasible set to the oiginal OCP is nonemty it suffices to guaantee the fulfilment of assumtion (85 Indeed let ( u y Ξ be any feasible ai to the oiginal OCP (9 Then ˆ I ( u y = I( u y fo each > ince ( u y is an otimal ai to oblem (63-(64 this yields and we aive at the ineuality (85 ˆ = su Iˆ u y su I u y I u y > > 568

23 Acnowledgements Reseach is funded by FG-Excellence Cluste Engineeing fo Advanced Mateials Refeences [] eels J and Tiba (5 Otimal esign of Mechanical tuctues In: Imanuvilov O Leugeing G Tiggiani R et al Eds Contol Theoy of Patial iffeential Euations Chaman& Hall/CRC Boca Raton 59-7 [] eels J and Tiba (9 Otimization Poblems fo Thin Elastic tuctues Numbe 58 in Intenational eies of Numeical Mathematics Bihause Velag Basel [3] Neitaanmai P eles J and Tiba (6 Otimization of Ellitic ystems Theoy and Alications inge Monogahs in Mathematics inge-velag Belin [4] Gazzola F Gunau H-C and wees G ( Polyhamonic Bounday Value Poblems: Positivity Peseving and Nonlinea Highe Ode Ellitic Euations in Bounded omains inge-velag Belin htt://dxdoiog/7/ [5] Litvinov VG (987 Otimization in Ellitic Bounday Value Poblems with Alications to Machanics MIR Moscow [6] Luie KA (993 Alied Otimal Contol Theoy of istibuted ystems Planum Pess [7] Khan A and ama M (3 A New Conical Regulaization fo ome Otimization and Otimal Contol Poblems: Convegence Analysis and Finite Element iscetization Numeical Functional Analysis and Otimization htt://dxdoiog/8/ [8] Adams R (975 obolv aces Acemic Pess Cambidge [9] Lions J-L and Magenes E (968 Poblemes aux limites non homogenes et alications Vol unod Pais [] Bucu and Gazzola F ( The Fist Bihamonic telov Eigenvalue: Positivity Peseving and hae Otimization Milan Jounal of Mathematics htt://dxdoiog/7/s3--43-x [] Gilbag and Tudinge N ( Ellitic Patial iffeential Euations of econd Ode inge-velag Belin [] Colasuonno F and Pucci P ( Multilicity of olutions fo (x-polyhamonic Ellitic Kichoff Euations Nonlinea Analysis Theoy Methods and Alications htt://dxdoiog/6/jna573 [3] Lubyshev VF ( Multile olutions of an Even-Ode Nonlinea Poblems with Convex-Concave Nonlineaity Nonlinea Analysis Theoy Methods and Alications htt://dxdoiog/6/jna7 [4] Giusti E (984 Minimal ufaces and Functions of Bounded Vaiation Bihause Velag Basel htt://dxdoiog/7/ [5] Attouch H Buttazzo G and Michaille G (6 Vaiational Analysis in obolev and BV aces: Alication to PE and Otimization IAM Philelhia [6] Roubice T (3 Nonlinea Patial iffeential Euations with Alications Bihause Velag Basel htt://dxdoiog/7/ [7] Lions J-L (969 ome Methods of olving Non-Linea Bounday Value Poblems unod-gauthie-villas Pais [8] Casas E (99 Otimal Contol in the Coefficients of Ellitic Euations with tate Constaints Alied Mathematics & Otimization 6-37 htt://dxdoiog/7/bf8394 [9] Kogut PI and Leugeing G ( Otimal Contol Poblems fo Patial iffeential Euations on Reticulated omains: Aoximation and Asymtotic Analysis Bihause Velag Basel htt://dxdoiog/7/ [] Roubice T (997 Relaxation in Otimization Theoy and Vaiational Calculus e Guyte Belin New Yo htt://dxdoiog/55/ [] Bowein JM and huang M (993 ue Efficiency in Vecto Otimization Tansactions of the Ameican Mathematical ociety htt://dxdoiog/9/ [] Henig MI (98 Poe Efficiency with Resect to Cones Jounal of Otimization Theoy and Alications htt://dxdoiog/7/bf [3] Maaov EK and Racovsi NN (999 Unified Reesentation of Poe Efficiency by Means of ilating Cones Jounal of Otimization Theoy and Alications 4-65 htt://dxdoiog/3/a:7759 [4] huang M (994 ensity Result fo Poe Efficiencies IAM Jounal on Contol and Otimization htt://dxdoiog/37/

24 [5] Jahn J (4 Vecto Otimization: Theoy Alications and Extensions inge-velag Belin htt://dxdoiog/7/ [6] chiel R (4 Vecto Otimization and Contol with PEs and Pointwise tate Constaints Phd Thesis Fiedich-Alexande-Univesity Elangen-Nunbeg Nunbeg [7] Mel ni V (986 Method of Monotone Oeatos in the Theoy of Constained Otimal ystem Re Uain Ac ci [8] Babu V (993 Analysis and Contol of Infinite imensional ystems Acemic Pess Cambidge [9] Eeland I and Temam R (976 Convex Analysis and Vaiational Poblems Elsevie Amstedam ubmit o ecommend next manuscit to CIRP and we will ovide best sevice fo you: Acceting e-submission inuiies though Faceboo LinedIn Twitte etc A wide selection of jounals (inclusive of 9 subjects moe than jounals Poviding 4-hou high-uality sevice Use-fiendly online submission system Fai and swift ee-eview system Efficient tyesetting and oofeing ocedue islay of the esult of downlos and visits as well as the numbe of cited aticles Maximum dissemination of you eseach wo ubmit you manuscit at: htt://aesubmissionsciog/ 57