Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, Germany

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1 Evoution variationa inequaities and mutidimensiona hysteresis operators. Pave Krej 27th August Mathematics Subject Cassication. 58E35, 47H3, 73E5. Keywords. Variationa inequaity, hysteresis operators. Weierstrass Institute for Appied Anaysis and Stochastics, Mohrenstr. 39, 7 Berin, Germany

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3 Contents Introduction 2 Physica motivation 3. Rheoogica eements Composition of rheoogica eements Linear kinematic hardening Isotropic and kinematic hardening....5 Noninear kinematic hardening Convex sets 4 2. Recession cone Tangent and norma cones Strict convexity The Minkowski functiona The pay and stop operators Absoutey continuous inputs Continuous inputs Uniformy stricty convex characteristics 33 5 Smooth characteristics Strict continuity Loca Lipschitz continuity in W ( T X ) Poyhedra characteristics 4 6. Lipschitz continuity in C ([ T] X ) Lipschitz continuity in W ( T X ) Second order variation 5 8 Integration of vector-vaued functions Bochner integra Functions of bounded variation Riemann-Stietjes integra References 6

4 Abstract We give an overview of the theory of mutidimensiona hysteresis operators de- ned as soution operators of rate-independent variationa inequaities in a Hibert space X with given convex constraints. Emphasis is put on anaytica properties of these operators in the space of functions of bounded variation with vaues in X,in Soboev spaces and in the space of continuous functions. We discuss in detai the inuence of the geometry of the convex constraint on the input-output behavior. It is shown how mutidimensiona hysteresis operators arise naturay in constitutive aws of rate-independent pasticity and concrete exampes of appication of the above theory in materia sciences are given. Introduction One may wonder why such a particuar probem ike the variationa inequaity (.) h _u(t) ; _x(t) x(t) ; ~xi 8~x 2 Z where Z is a convex cosed subset of a Hibert space X, u is a given X -vaued function of t 2 [ T], x is the unknown function with vaues in Z and dot denotes the derivative with respect to t, shoud draw an exceptiona attention. As in many anaogous cases, it has been extracted as a common feature of dierent physica modes. Its variationa character is typicay interpreted as a specia form of the maxima dissipation principe in evoution systems with convex constraints. It turns out that inequaities of the form (.) pay (expicity or impicity) a centra roe in modeing nonequiibrium processes with rate-independent memory in mechanics of eastopastic and thermoeastopastic materias incuding metas, poymers or for instance bread dough, as we as in ferromagnetism, piezoeectricity or phase transitions (see e.g. [DL, LC, A, LT, NH, BS, V, Be, KS, KS2, KS3, KS4, AGM]). They aso naturay arise in the anaysis of fatigue and damage accumuation, see [BDK, BS]. Another area of appication is reated to mathematica optimization, where inequaity (.) is known as a specia case of the Skorokhod probem, cf. [DI, DN], which consists in approximating a given function u :[ T]! X by a function of bounded tota variation in a given convex neighborhood of u in such away that _ (in a generaized sense) points in a prescribed direction. Equation (.) corresponds to the case where _ = _u; _x beongs to the outward norma cone to Z at the point x. On the other hand, (.) is a specia case of a `sweeping process', see [M]. If Z has nonempty interior, the decomposition u = x + dened by inequaity (.), where x is Z -vaued and has bounded variation, can be extended to every continuous function u. Moreover, there are some indications to conjecture that this decomposition is minima in the sense that among a decompositions of u of this form, the tota variation of is minima with respect to a suitabe norm in X. In the case dim X =,this observation has been made by V. Chernorutskii and a proof can be found in [K] in higher dimensions, this question seems to be open. The present textisdevoted to a discussion about the inuence of the geometry of the convex constraint Z (the characteristic) on anaytica properties of the mappings u 7! x 2

5 and u 7! (the so-caed stop and pay operators). They are hysteresis operators, that is, according to the cassication in [V], operators that are causa and rate-independent (see (.25), (.26) beow). This terminoogy is justied by the fact that in the scaar case dim X =,hysteresis operators are exacty those that admit a oca representation by means of superposition operators in each interva of monotonicity of the input, with a possibe branching when the input changes direction. Most of the materia coected here is taken from [KP, K] with some sma improvements. More recentcontributions ([BK, D, DT]) are referred to in the text, the resuts of Sections 5 and 7 are new to a arge extent. In Section we present sometypica issues of eastopasticity reated to inequaity (.). Basic eements of convex anaysis are recaed in Section 2. In Section 3 we construct the pay and stop operators in the space of continuous X -vaued functions of bounded variation and prove their continuity in W p ( T X) for every p< and every convex cosed set Z. A continuous extension to the space C([ T] X) of continuous functions is estabished provided Z has nonempty interior. The uniform continuity in C([ T] X) is proved in Section 4 under the hypothesis that the set Z is uniformy stricty convex. The oca Lipschitz continuity in W ( T X) is obtained in Section 5 when the boundary of Z is smooth. If Z is a convex poyhedron, the pay and stop are gobay Lipschitz in both C([ T] X) and W ( T X) a detaied proof is given in Section 6. In Section 7 we prove a maxima reguarity resut, namey that the tota variation of the derivative of the output can be estimated above by that of the input. Indeed, one cannot expect the output derivative tobecontinuous across the boundary of Z even if the input is arbitrariy smooth. The ast Section 8 gives a brief survey of the theory of Hibert space-vaued functions. Even in appication to eastopasticity, the investigation of the stop and pay operator is not just an academic question. Indeed, the theory of monotone operators provides a traditiona too for soving cassica probems ([DL, NH, A, LT]) without referring to hysteresis operators. The advantage of the hysteresis approach consists however in the fact that additiona geometrica considerations aow for soving aso nonmonotone probems. Typica exampes can be found in [K] and [BK]. We do not give an exhaustive ist of reated pubications and historica references here an interested reader may consut in particuar the pioneering monographs [KP] and [V], or a recent survey paper [Bro]. Physica motivation The equation of motion of a deformabe body R N for some N 2 N, wheren denotes the set of positive integers and R N is the N -dimensiona Eucidian space, is in cassica continuum mechanics ([LL]) considered in the form u 2 = NX j + g i i = ::: N where x 2, t > are the space and time variabes, respectivey, u = (u i ) is the dispacement vector, > is the density, =( ij ) is the symmetric stress tensor and 3

6 g =(g i ) is the voume force density, i j = ::: N. The meaningfu choice in appications is usuay N =3. Equation (.) has to be couped with initia and boundary conditions and with a constitutive aw between the stress tensor =( ij ) and, for exampe, the inearized strain tensor " =(" ij ) dened as the symmetric derivative of u (.2) " ij j i i j = ::: N: Whie (.) is a genera physica aw, the constitutive reation characterizes specic properties of a given materia, subject to time-dependent oading. In engineering appications, one has aways been searching for a mathematicay simpe phenomenoogica description of the strain - stress constitutive behavior for a possiby arge cass of dierent types of materia response incuding memory eects. Rheoogica modes pay a prominent roe here and oer one of the main toos in the theory of ineastic constitutive aws (see e.g. [LC], [A], [LT]). We reca here its main constituents.. Rheoogica eements Let T be the space of symmetric tensors = ( ij ), i j = ::: N, N 2 N, ij = P N ji, endowed with the scaar product : := i j= ij ij,andet be the we-known Kronecker tensor 2 T ij = if i 6= j if i = j: We spit the space T into the subspace T := span fg and its orthogona compement (the so-caed deviatoric space) T dev := T?.According to this decomposition, we denote by I := : the rst invariant (trace) of a symmetric tensor 2 T and by dev := ; =N I 2 T dev the deviator of. The strain and stress tensors " and, respectivey, are in genera functions of the space variabe x 2 R N and time variabe t with vaues in T. We consider here ony homogeneous media, where the constitutive aw is independent of the spatia variabe x which thus pays the roe of a parameter. Denition. A system consisting of (.3) (i) a constitutive reation between " and (ii) a potentia energy U is caed a rheoogica eement. A rheoogica eement is said to be thermodynamicay consistent, if the quantity (.4) _q := _" : ; _ U caed dissipation rate, where the dot denotes the time derivative, is nonnegative in the sense of distributions for a " U satisfying (.3). 4

7 Exampe.2 The eastic eement E. Eastic materias are characterized by ainear stress-strain reation and by thecompete reversibiity of dynamica processes. In mathematica terminoogy, it is assumed that there exists a matrix A =(A ijk`) over T such that (.5) = A" or equivaenty ij = NX k `= A ijk` " k` i j = ::: N: Reversibiity means that the potentia energy U invoves no memory and can be chosen in such away that the dissipation rate _q vanishes, i.e. the vaue of U(t) for each t> depends ony on the instantaneous vaue of "(t) and U _ = _" : A" amost everywhere for every absoutey continuous ". This necessariy impies that the matrix A is symmetric with respect to the scaar product ` : 'andu has the form (.6) U = 2 A" : " up to an additive constant. Indeed, for an arbitrary " 2 W ( T T) and t 2 ] T[ put ~"( ):="() + =t("(t) ; "()) for 2 [ t]. We can choose the initia vaue for U arbitrariy, for instance U() := =2 A"() : "(). We have by hypothesis (.7) U(t) =U() + Z t _~"( ) : A~"( ) d = 2 "(t) : A"(t)+ 2 "(t) : (A ; AT )"() where (A T ) ijk` = A k`ij, hence _U(t) = _"(t) : A"(t)+ 2 _"(t) : (A ; AT )("() ; "(t)) and we easiy concude that the matrix A is symmetric and (.6) hods. To guarantee that the stress-strain reation is one-to-one and the materia aw is deterministic, we assume that the matrix A is positive denite. The eastic eement is said to be isotropic, if the matrix A has the form (.8) A =2I + NP where are positive numbers caed Lam 's constants (see [Ra]), I is the identity matrix I = and P is the orthogona projection onto T,thatisP ==N I. Exampe.3 The viscous eement E. Modeing of rate-dependent reaxation eects makes often use of the concept of viscosity based on the hypothesis that there exist two coecients > > of proportionaity between the deviators and rst invariants of the strain rate and stress, that is (.9) dev = _" dev I = _" I : The assumption that no reversibe energy can be stored by the viscous eement (U =) ensures its thermodynamica consistency. 5

8 Exampe.4 The rigid-pastic eement R. The basic concept in pasticity is the yied surface in the stress space which can be described as the of a convex cosed set Z T. The rigid-pastic behavior consists of two dierent phases characterized by the instantaneous vaue of the stress tensor. The materia remains rigid as ong as 2 Int Z (the interior of Z ). In this case no deformation occurs and _" =. The materia becomes pastic if reaches the of Z. Pasticity is governed by three physica principes: the stress vaues remain conned to the set Z, no reversibe energy is stored, and the dissipation rate is maxima with respect to a admissibe stress vaues. Mathematicay, this means (.) (.) (.2) 2 Z U = _" : ( ; ~) 8~ 2 Z Geometricay, _" points in the direction of the outward norma cone, and condition (.2) is aso caed the normaity rue. We see that the variationa inequaity (.2) incudes the rigid behavior (for 2 Int Z it entais _" =). In order to ensure the thermodynamica consistency, we assume 2 Z. In fact, it is natura to assume that no deformation occurs for =. This is equivaent to the hypothesis 2 Int Z which, as we show in the next sections, has a considerabe impact on the reguarity in the mathematica setting. It has been observed that voume changes are negigibe during pastic deformation ([Ra]). Combining constitutive reation (.) - (.2) with the voume invariance condition (.3) _" I = we concude from Proposition 2.9 beow that Z has the form of a cyinder (.4) Z = Z + T where Z T dev is a convex cosed set. In appications, it is often assumed that Z is bounded. The cassica modes of Tresca and von Mises are specia cases of (.) (.4) with (von Mises) Z = B r () \ T dev (ba centered at with radius r) or (Tresca) Z :=f 2 T P N dev j k= kjrg for some r>, P where f k g are the eigenvaues of the N symmetric matrix =( ij ). Note that we have k= k =for 2 T dev. The Tresca set Z is usuay represented for N =3 by a hexagon in the pane =. Exampe.5 The rigid-pastic eement with isotropic hardening J. In many materias, the yied surface does not remain xed in time, but changes according to the oading history. This phenomenon is caed hardening (softening). We rst reca the concept of isotropic hardening, where the yied surface evoution is a simpe diation governed by a scaar hardening parameter. Foowing [NH] we assume anaogousy as in Exampe.4 that a bounded convex cosed set Z T dev is given such that 2 Int Z, and we denoteby M : T dev! [ [ the Minkowski functiona associated to Z by Denition 2.8 beow. Let further a concave nondecreasing function ' :[ [! [ [ be given, '() =. 6

9 We denote by T the space TR endowed with the natura scaar product h( ) ( )i := : + for 2 T, 2 R, and by Z the convex cosed subset of T (.5) Z := f( ) 2 T M ( dev ) '( + )g : The constitutive reations are anaogous to (.)-(.2), namey (.6) (.7) (.8) ( ) 2 Z U = () = h(_" ;(=c) _) ( ) ; (~ ~)i 8(~ ~) 2 Z where c> is a given physica constant. We immediatey observe that choosing ~ = in (.8), we obtain _(; ~) for every ~, hence _. Let Z := f 2 T ( ) 2 Z g be the domain of admissibe stresses for an instantaneous vaue of the hardening parameter. We see that Z increases without changing its shape with increasing..2 Composition of rheoogica eements A arge variety of modes for the materia behavior can be obtained by composing rheoogica eements from Exampes in series or in parae. Let G G 2 be two rheoogica eements and et " i i U i be the strain, stress and potentia energy, respectivey, corresponding to the eements G i, i = 2. The tota strain ", stress and potentia energy U for the combination in parae G jg 2 and in series G ; G 2 are dened by the foowing reations G jg 2 G ; G 2 " = " = " 2 " = " + " 2 = + 2 = = 2 U = U + U 2 U = U + U 2 in anaogy with the theory of eectrica circuits. It is easy to see that every combination of thermodynamicay consistent eements is thermodynamicay consistent. Exampe.6 Eastopastic modes E;R E=R. There are good reasons for rewriting constitutive variationa inequaities in pasticity in operator form. This enabes us to distinguish ceary between input and output quantities: whie the input can be controed, the output is determined by soving the constitutive equation. Let us compare the constitutive reations for two eastopastic modes EjR, E;R. We denote by " e e and " p p the strain and stress on the eastic and rigid-pastic eement, respectivey. 7

10 EjR E;R " = " e = " p " = " e + " p = e + p = e = p e = A" = A" e p 2 Z 2 Z _" : ( p ; ~) 8~ 2 Z _" p : ( ; ~) U = " : 2 e U = 2 "e : Reca that Z T is a given convex cosed set, 2 Int Z.We see that both modes are governed by a variationa inequaity of the same type, namey (.9) EjR : (A ; (_ ; _ p )) : ( p ; ~) E;R : (A ; (A _" ; _)) : ( ; ~) 8~ 2 Z: The sovabiity of such equations is ensured by the foowing theorem whose detaied proof (in a more genera setting) wi be given in Section 3. Denition and genera information about the space W ( T X) of absoutey continuous Hibert space vaued functions is given in Section 8. Theorem.7 Let X be area separabe Hibert space endowed with a scaar product : :. Let Z X be aconvex cosed set, 2 Z, and et x 2 Z be a given eement. Then for every function u 2 W ( T X) there exists a unique x 2 W ( T Z) satisfying the variationa inequaity _u(t) ; _x(t) x(t) ; ~x a.e. 8~x 2 Z (.2) and the initia condition (.2) x() = x : We dene the soution operators S P : ZW ( T X)! W ( T X) of the probem (.2), (.2) by the formua (.22) S(x u):=x P(x u):=u ; S(x u): According to [KP], the operators S P are caed stop and pay, respectivey. The set Z is caed the characteristic of S and P. The constitutive reations for the eastopastic modes above can be written in the form (.23) E=R : " = A ; P(p ) U = 2 (A; P( p )) : P( p ) E;R: = S( A") U = 2 (A; S( A")) : S( A") where S P are the stop and pay inx = T endowed with the scaar product := (A ; ) :, and p are given initia output vaues. 8

11 It is cear that the roes of input and output in the modes EjR and E;Rcannot be reversed. The denition immediatey suggests that the stop has the Semigroup property : For u 2 W ( T X), s 2 ] T[ u s (t) :=u(s + t). Then for every x 2 Z we have and t 2 [ T ; s] put (.24) S(x u)(t + s) =S(S(x u)(s) u s )(t): An operator F acting in some space R( T X) of functions [ T]! X is caed Rate-independent,ifforevery u 2 R( T X) and every nondecreasing mapping of [ T] onto [ T] such that u (t) :=u((t)) beongs to R( T X) we have (.25) F (u )(t) =F (u)((t)) for a t 2 [ T] Causa, if the impication (.26) u(t) =v(t) 8 t 2 [ t ] ) F (u)(t )=F (v)(t ) : Rate-independence and causaity characterize hysteresis operators according to the cassi- cation of [V]. By denition, the stop and pay arehysteresis operators in W ( T X) (we wi see in the next section that they can be extended to the space of continuous functions C([ T] X)). Indeed, the concept of `hysteresis branching' or `hysteresis oops' is meaningfu ony in the scaar case dim X =. However, the pay operator turns out to be the main buiding bock for a very arge famiy of scaar hysteresis modes used in eastopasticity (Prandt-Ishinskii mode), ferromagnetism (Preisach and Dea Torre modes), fatigue anaysis (the `rainow' method) and many others. A more compete information can be found in [BS] and [K]. Recent appications to thermopasticity ([KS, KS2]) and phase transitions ([KS3, KS4]) aso conrm its universa character. We remain here within the mutidimensiona framework and give some exampes of appication of hysteresis operators for modeing the kinematic and isotropic hardening in eastopastic materias..3 Linear kinematic hardening Let us consider the so-caed Prager mode E;(EjR). The genera rheoogica rues yied = e + p " = " e + " p e = A" p = B" e p 2 Z _" p : ( p ; ~) 8~ 2 Z U = 2 ("e : + " p : e ) 9

12 where A B are given constant symmetric positive denite matrices and Z T is a convex cosed set, 2 Int Z.For t 2 [ T] put (.27) Z(t) :=Z + e (t): Then (t) 2 Z(t) for a t 2 [ T]. Reation (.27) can be interpreted as a transation of Z in the stress space T driven by the eastic component e of the stress without changing shape and size. This phenomenon is caed kinematic hardening and is typica for metas, see [LC]. The word `inear' is reated to the inear dependence between e and " p. The evoution of e is governed by variationa inequaity of the form (.2), namey ; A ; _ e : ( p ; ~) 8~ 2 Z: (.28) Inequaity (.28) can be interpreted as a normaity condition for the hardening rate _ e with respect to the scaar product := (A; : ) both the hardening rate _ e A and the pastic strain rate _" p have the outward norma direction at the point, but with respect to dierent scaar products. With the intention to dea with severa scaar products in T we introduce the subscript A for the pay PA and stop SA corresponding to the scaar product A. Using (.23) we can express the constitutive awforthemodee ;(E j R) in the form (.29) " = B ; + A ; PA( p ) with input and output ". Wenowprove that the constitutive operator B ; + A ; PA is invertibe. Identity (.3) beow gives an equivaent expression for (.29) with input " and output. Lemma.8 Let p 2 Z be given and et A C be given constant matrices such that A CA are symmetric and positive denite. Put ^A := A + CA. Then for a 2 W ( T T) we have (.3) S ^A (p + C PA( p )) = SA( p ) : Proof. Put x := SA( p ), y := S ^A (p +C PA( p )). Then y = S ^A (p, (I+C);Cx), where I is the identity matrix. Putting ~ := (x + y)=2 in the variationa inequaities A ; (_ ; _x) : (x ; ~) ^A ; ((I + C)_ ; C _x ; _y) : (y ; ~) and using the identity ^A ; + ^A ; C = A ;,we concude _x; _y x; y, hencex = y. ^A We now appy Lemma.8 with C = BA ; to the constitutive equation (.29). We obtain S ^A (p B")= SA( p ) for ^A = A + B hence (I + BA ; ) = B" + BA ; S ^A (p B"), or equivaenty (.3) =(A ; + B ; ) ; " + B ^A ; S ^A (p B")=B"; B ^A ; P ^A (p B")

13 where " is the input and is the output. In the particuar case B = I A = I for some >, we obtain PA = P ^A = PI inversion formua (.32) (I + PI(x :)) ; = I; + PI (x :) and the hods for a x 2 Z, where I is the identity mapping in W ( T X). Exercise.9 Assume that the matrices A B commute, i.e. (.3) is the constitutive equation of the mode E j(e ; R) with e = A" ~ A ~ =(A ; + B ; ) ; p = B" ~ e B ~ = B 2 (A + B) ; " p : ( p ; ~) 8~ 2 ~ Z ~ Z = B(A + B) ; (Z) : AB = BA. Prove that Hint. Use the identity CSA(x u)= ~S CAC(Cx Cu) for each positive denite symmetric matrix C, where ~S is the stop with characteristic ~ Z = C(Z). The commutativityhypothesis AB = BA is satised for instance if both eastic eements are isotropic. In this case the modes E ;(E j R) and E j(e ; R) are equivaent..4 Isotropic and kinematic hardening Let us consider now the mode E ;(E j J ). With the notation taken from Exampe.5, the constitutive reations are anaogous to the mode E ;(E j R), namey (.33) (.34) (.35) ( ) = ( e ;) +( p ) (" ;(=c)) =(" e ) + (" p ;(=c)) h(_" p ;(=c) _) ( p ) ; (~ ~)i ( p ) 2 Z = B" e e = A" p 8(~ ~) 2 Z where A B are symmetric positive denite matrices. Let A B : T! T be the inear mappings dened by the identities A ( ) := (A c ), B ( ) :=(B c). Wehave A ; (( _ ) ; (_ p _)) ( p ) ; (~ ~) 8(~ ~) 2 Z hence ( p ) =S (( p ) ( )), (e ;) =P (( p ) ( )), where S P are the stop and pay int endowed with the scaar product A ;,withcharacteristic Z and with a given initia condition ( p ) 2 Z. The constitutive equation has the form (.36) (" ;(=c) ) =B ; ( ) + A; P (( p ) ( )) : We derivenow some consequences of the constitutive equation. For a function f :[ T]! R and t 2 [ T] we denote kfk [ t] := sup fjf( )j 2 [ t]g.

14 Lemma. Let 2 W ( T T) be given and assume () = p =. Let " be given by the equation (.36). Then we have (.37) '( + (t)) = max km ( p dev )k [ t] where ' M are as in (.5) and p dev is the deviator of the pastic stress p. Proof. We have ( p (t) (t)) 2 Z for a t 2 [ T], hencem ( p (t)) '( + (t)) dev by denition. The fact that is nondecreasing (cf. (.8)) entais km ( p )k dev [ t] '( + (t)). In the case km ( p )k dev [ t] < we obviousy have (t) = and (.37) hods. Let us assume now km ( p )k dev [ t] <'( + (t)) for some t 2 ] T[. Then there exists 2 ] t[ such that _( ) > and km ( p )k dev [ ] < '( + ( )), hence ( p ( ) ( )) 2 Int Z.From (.35) we concude _( )=, which isacontradiction. According to genera rheoogica principes, we associate to the mode E ;(E j J ) the potentia energy U = ; " e + " p e =2. The dissipated energy q(t) is then equa to the pastic work R t _"p ( ) : p ( ) d and is reated to (t) by the foowing identity. Proposition. Let the assumptions of Lemma. hod. Put r := inff > ' ( + ) =g2[ ]. For p 2 [ r] put (p) := Z p Then we have (t) 2 [ r] for a t 2 [ T] and '( + ) c' ( + ) d : (.38) q(t) =((t)) provided (t) 2 [ r[: Proof. Assume (t) >r for some t 2 ] T[. Then there exists <tsuch that _( ) > and ( ) > r. Putting ~ := p ( ) ~ = r we have '( + ~) = '( + ( )), hence (~ ~) 2 Z and (.35) yieds _( ), whichisacontradiction. Identity (.38) can be equivaenty written in the form (.39) '( + (t)) _q(t) = _(t) c' ( + (t)) a.e. provided (t) <r: To prove (.39) we distinguish two cases. a) _(t) =.We choose a> sucienty sma and b> sucienty arge such that ~ := (+a) p (t) and ~ := (t)+b satisfy M (~ dev );'(+~) = (+a) ; M ( p dev (t)); '( + (t)) +(+a)'( + (t)) ; '( + (t)+b), that is, (~ ~) 2 Z.From inequaity (.35) we infer a _" p : p, hence _q(t) =. b) _(t) >. We wi see in Section 3 that the pay depends continuousy on the characteristic with respect to the Hausdor distance. It therefore suces to assume that ' and M are smooth functions. We have ( p (t) (t)) and according to 2

15 (.35), the vector (_" p (t) ;(=c)_(t)) points in the direction of the outward norma vector to Z. In other words, we have (_e p (t) ;(=c)_(t)) = _(t) c' ( + (t)) (@M ( p dev (t)) ;' ( + (t))) is the gradient ofm. This yieds _q(t) = _" p (t) : p (t) = _" p (t) : p dev (t) = _(t) c' ( + ( p dev (t)) : p dev (t) : ( p ) : dev p = M dev ( p ) by Lemma 2.2 and M dev ( p (t)) = '(+(t)) dev by hypothesis, hence identity (.39) hods. As a consequence of Proposition., we see that the isotropic hardening can be equivaenty characterized by the pastic work (or dissipation) q. For this reason it is sometimes referred to as work hardening, see [NH], [LC]..5 Noninear kinematic hardening In order to account for the phenomenon of ratchetting which is manifested by the accumuation of the pastic strain under cycic stress oading, Armstrong and Frederick proposed in [AF] a modication of the Prager mode from Exampe.3, repacing the inear reation between e and " p by a noninear one, namey (.4) _ e = (R _" p ; e j _" p j) with given positive constants, R. It is assumed that the convex cosed set Z is the von Mises cyinder of radius r>, (.4) Z =(B r () \ T dev )+T : The normaity rue here impies _" p = p dev j _" p j=r, hence (.4) is equivaent to (.42) _ e = ((R + r) _" p ; dev j _" p j) : Introducing an auxiiary function u by the formua (.43) u := (R + r) " p + p dev we see that the variationa inequaity (.44) can be rewritten as (.45) _" p : ( p ; ~) (_u ; _ p dev ) : (p dev ; ~) 8~ 2 Z 8~ 2 B r () \ T dev 3

16 hence p = dev S( p u), dev "p =(=((R +r))) P( p u) according to the above notation dev for a given initia pastic stress p. The function u is to be determined as the soution of the operator - dierentia equation (.46) _u =_ dev + dev R + r d dt P ( p u) dev for each given stress input and with an appropriate initia condition. The constitutive operator of the stress-controed Armstrong-Frederick mode thus contains a superposition of the pay operator to the soution operator 7! u of the equation (.46). It is shown in [BK] that the mode is we posed. The strain-controed case eads to simiar considerations. We do not give the detais here et us just point out that the sovabiityof equation (.46) is cosey reated to the oca Lipschitz estimate (5.6) for the pay operator in W ( T T dev ). 2 Convex sets The aim of this section is to reca some basic eements of convex anaysis in Hibert spaces. Most of the resuts are we-known. We present them in order to x the notation and to keep the presentation consistent (for more information we refer the reader to the monographs [Ro] and [AE]). The compementary function of a convex set (Denition 2.4 beow) has been introduced in [K]. Throughout the section, X denotes a rea separabe Hibert space endowed with a scaar product and norm jxj := x x =2. By Z we denote a convex cosed subset of X such that 2 Z.Wexthenumber (2.) m := dist z It is cear that m> if and ony if 2 Int Z. We start with a simpe emma. Lemma 2. For each x 2 X there exists a unique z 2 Z such that jx;zj =dist(x Z) = minfjx ; yj y 2 Zg. Proof. Let x 2 X be given. Put p := inf fjx ; yj y 2 Zg and et fy n g be a sequence in Z such that jx ; y n j!p. From the identity (2.2) ju ; vj 2 + ju + vj 2 =2(juj 2 + jvj 2 ) for u = x ; y n, v = x ; y k, it foows 2 jy n ; y k j 2 = jx ; y n j 2 + jx ; y k j 2 ; 2 x ; y n + y k 2 jx ; yn j 2 + jx ; y k j 2 ; 2p 2 2 hence fy n g is a convergent sequence and it suces to put z := im y n. Uniqueness is n! obtained anaogousy. 4

17 Using Lemma 2. we can dene the projection Q : X! Z onto Z and its compement P := I ; Q (I is the identity) by the formuae (2.3) Qx 2 Z jpxj = dist (x Z) for x 2 X: In the seque, we ca (P Q) the projection pair associated to Z.Wemake extensive use of the foowing emma. Lemma 2.2 For every x y 2 X we have (i) Px Qx; z 8z 2 Z, (ii) Px; Py Qx; Qy, (iii) Px x mjpxj + jpxj 2 with m given by (2.), (iv) Q(x + P x) =Qx 8 ;. Proof. (i) For z 2 Z, z 6= Qx and 2 ] [ we have jx ; z ; ( ; )Qxj 2 > jpxj 2, hence 2 Px Qx; z + jqx ; zj 2 > and the assertion foows easiy. Statement (ii) is an obvious consequence of (i). We obtain (iii) from (i) by putting z := mpx=jpxj if x =2 Z, the case x 2 Z is trivia. To prove (iv)we notice that for a z 2 Z we have jx + P x ; zj 2 = jqx ; zj 2 +(+) 2 jpxj 2 + 2( + ) Px Qx; z, hence the minimum of jx + P x ; zj is attained for z = Qx. 2. Recession cone Denition 2.3 A nonempty cosed convex set C X is caed a cone, if the impication x 2 C ) x 2 C hods for a x 2 X and. Denition 2.4 Let Z X be aconvex cosed set, 2 Z. The set (2.4) C Z := fx 2 Z x 2 Z 8 g is caed therecession cone of Z and the function K Z :[ [! [ [ dened by the formua (2.5) K Z (r) :=supfdist (x C Z ) x 2 Z \ B r ()g for r is caed the compementary function of Z, where (2.6) B r (x ):=fx 2 X jx ; x jrg denotes the ba centered atx with radius r. The foowing properties of the compementary function are proved in [K]. 5

18 Proposition 2.5 Let Z X be aconvex cosed set with 2 Int Z and with the recession cone C Z and compementary function K Z. Then (i) x + y 2 Z 8x 2 C Z 8y 2 B m (), where m is given by (2.), (ii) K Z is nondecreasing in [ [, K Z (s)=s K Z (r)=r for <s<r, (iii) if dim X<, then (2.7) K Z (r) im r! r =: Let us note that by Proposition 2.5 (ii) we have K Z (r) ; K Z (s) (r ; s) K Z (r)=r for r>s>, hence K Z is Lipschitz. Property (2.7) is crucia for the extension of hysteresis operators to the space of continuous functions. We therefore introduce the foowing terminoogy. Denition 2.6 Aconvex cosed set Z X is caed arecession set if 2 Int Z and the compementary function K Z satises (2.7). Indeed, every convex cosed set Z X with 2 Int Z is a recession set if dim X<. This is not true for innitey dimensiona spaces, where there exist unbounded sets Z with C Z = fg, but condition (2.7) hods for instance for a sets of the form Z = C + Z B, where C is a cone and Z B is bounded, 2 Int Z B. 2.2 Tangent and norma cones A natura generaization of norma vectors and tangent hyperpanes which in genera are not uniquey determined, is the concept of norma cone N Z (x) and tangent cone T Z (x) to a convex cosed set Z X at a point x 2 Z. They are dened by the formua (2.8) ( N Z (x) :=fy 2 X y x ; z T Z (x) :=fw 2 X w y 8z 2 Zg 8y 2 N Z (x)g: Every eement u 2 X admits a unique orthogona decomposition into the sum u = v + w of the norma component v 2 N Z (x) and the tangentia component w 2 T Z (x), namey v = Q N (u), w = P N (u), where (P N Q N ) is the projection pair associated to N Z (x). Indeed, by Lemma 2.2 (i) we have w ( ; ) v for a, hence w v = and w y for every y 2 NZ (x). Uniqueness is easy: assume v + w = v 2 + w 2 for some v i 2 N Z (x), w i 2 T Z (x), hw i v i i =, i = 2. Then hw ;w 2 v ;v 2 i;jw ;w 2 j 2, hence w = w 2, v = v 2. For x 2 Int Z we obviousy have N Z (x) =fg, T Z (x) = X. One might expect that for x the norma cone shoud contain nonzero eements. The exampe Z := fx 2 X j x e k j =k 8k 2 Ng, wherefek g is an orthonorma basis, shows that this conjecture is fase, since and N Z () = fg. In reguar cases this cannot happen. 6

19 Proposition 2.7 Assume Int Z 6=. Then for every x we have N Z (x) nfg6=. Proof. Let fz n n 2 Ng be a sequence in X n Z such thatim n! jz n ; xj =. Put " n := jpz n j >, y n := z n +=" n Pz n.wehave " n jz n ; xj and Lemma 2.2 (iv) yieds Qy n = Qz n, Py n =(+=" n ) Pz n. By Lemma 2.2 (i) we further have jqy n ; xj 2 = jqz n ; xj 2 = jz n ; xj 2 ;jpz n j 2 ; 2 Pz n Qz n ; x jz n ; xj 2 and Pyn Qy n ; z 8z 2 Z 8n 2 N: (2.9) Passing to subsequences we can assume that fpy n g converges weaky to an eement which beongs to N Z (x) by (2.9). It remains to verify that 6=. We x an arbitrary ba B (x ) Int Z. Putting z := x + =( + " n ) Py n in (2.9) we obtain x; x, hence 6=. Let us mention the important particuar case of cyinders in X. Denition 2.8 Let Y X be a cosed subspace ofx, et Y? be its orthogona compement and et ~ Z Y be aconvex cosed set. Then the set Z := ~ Z + Y? is caed aconvex cyinder in X. Proposition 2.9 Aconvex cosed set Z X is a convex cyinder of the form Z = ~ Z+Y? with ~ Z Y if and ony if N Z (x) Y for a x 2 Z. Proof. The `ony if' part is trivia. To prove the converse we put ~ Z := Z \ Y and choose arbitrariy u 2 ~ Z and w 2 Y?.From Lemma 2.2 (i) we infer P (u+w) Q(u+w);u, hence jp (u + w)j 2 P (u + w) w. On the other hand, we have P (u + w) 2 N Z (Q(u + w)) Y,andwe concude P (u + w) w = jp (u + w)j 2 =. Consequenty, Z ~ + Y? Z and equaity foows from the convexity of Z. Remark 2. Cyinders of the form Z = ~ Z + Y? with ~ Z Y are characterized by the condition Px 2 Y for a x 2 X. Denoting by ( ~ P ~ Q) the projection pair associated to ~Z in Y,we obtain for every x 2 X of the form x = u + w, u 2 Y, w 2 Y? the identities Px = ~ Pu, Qx = ~ Qu + w. 2.3 Strict convexity In genera, the of a convex cosed set Z X can contain straight segments. We reca two criteria for their existence. It is easy to verify contains a segment of ength r> if one of the foowing conditions is satised. Interna criterion There exist x y such thatjx ; yj = r, (x + y)=2 Externa criterion There exists a point z and a sequence fw n n 2 Ng in X nt Z (z) such thatjw n j =, im n! w n = w, z + rw 7

20 The terminoogy is justied by the fact that we aways have (x + y)=2 2 Z for x y 2 Z and z + rw =2 Z for z w 2 X n T Z (z) and r>. According to these criteria we introduce the functions :[ [! [ [ by the formuae (2.) ( (r) :=inf dist ; 2 (x + x y 2 Z jx ; yj =2r (r) := inf fjp (z + rw)j z w 2 X n T Z (z) jwj =g where P is dened by (2.3). We naturay have (r) = + if 2r > diam Z := sup fjx ; yj x y 2 Zg (the diameter of Z )and(r) j(x + y)=2 ; xj = r for r< =2 diam Z. Choosing an arbitrary x 2 XnZ we obtain (r) P (Qx+rPx=jPxj) = r by Lemma 2.2. The case dim X = is trivia (then (r) = r for r < =2 diam Z, (r) = r for a r ), as we as the case Int Z = (then (r) =(r) = for r<=2 diam Z ). Proposition 2. Let Z X be aconvex cosed set, Int Z 6=. Then for a p<r we have (i) (p)=p (r)=r, (ii) (p)=p (r)=r, (iii) (r) (r). Proof. (i) Let p<r and "> be given. Put := p=r. Wexz and w 2 X n T Z (z), jwj = such that jp (z + rw)j <(r) +". For v := ( ; )z + Q(z + rw) 2 Z we have hence (i) hods. (p) jp (z + pw)jjz + pw ; vj = jp (z + rw)j < p ((r) +") r (ii) It suces to assume (r) <. Wendx y 2 Z and z such thatjx ; yj =2r and (2.) x + y 2 ; z ; " x + y 2 dist (r) + " 2 : Put ^x := x+(; )z, ^y := y+(; )z with as above. Then ^x ^y 2 Z, j^x; ^yj =2p and (p) (^x +^y)=2 ; z = (x + y)=2 ; z ((r) +") p=r. (iii) Let x y z " be as in (ii). We x an arbitrary 2 N Z (z), j j = and assume x;y (otherwise we interchange x and y ). Put v" := (x;y)=2+" 2 X nt Z (z). Then (r) ; P z+rv" =jv " j z+rv" =jv " j;x z;(x+y)=2 + " ; ; ;r=jv " j v " z ; (x + y)=2 + ". Letting " tend to we obtain (iii) from (2.). We see that both, are nondecreasing in their domains. One can derive by eementary means further interesting properties of these functions. Detais are eft to the reader as an exercise. 8

21 Exercise 2.2 Let Z X be a cosed convex domain with a nonempty interior. Prove that (i) (r) (2r +2(r))=2 for r 2 [ =2 diamz[, (ii) (r) ; (p) r ; p for p<r, (iii) if dim X 2, then for every x 2 Int Z, c := dist and r 2 [ c] we have 2c(r) r (r) (iv) if (r) > for some r 2 ] =2 diamz[,then diam Z r 2 (r) (r2 + 2 (r)) : Hint. (i) Assume (2r +2(r)) < 2(r) " for some r>, ">. Find x w ; () \ ; X n T Z (x) such that ; P x +(2r +2(r))w < 2(r) and put z := Q x +(2r +2(r))w.Thenz2Z, jx ; zj > 2r, x +(r + (r))w 62 Z,hence x +(r + (r))w ; (x + z)=2 >(r) which is a contradiction. (ii) Use the Lipschitz continuity of P which foows from Lemma 2.2 (ii). (iii) Let z " be such that jz " ;xjc+". Find w " 2 p B () such that w " z " ;x = and put u := x + c 2 ; r 2 (z " ; x)=jz " ; xjrw ".Thenu 2 B c (x) Z, ju + ; u ; j =2r and (r) z" ; (u + + u ; )=2. (iv) Assume s := jx ; yj=2 >r=(2 2 (r)) ; r (r) for some x y 2 Z. Then s>r, hence (s) s(r)=r > ; r (r) =(2(r)) r. By (iii) we have 2(s)(r) r (r) which is a contradiction. The upper bound for diam Z in Exercise 2.2 (iv) does not seem to be optima. If Z is a ba, then we obtain for instance diam Z =(r (r))=(r). We can nevertheess concude that Z is unbounded if and ony if (r) = for a r. Let us consider now the opposite situation. Denition 2.3 (i) A convex cosed set Z X is said to be stricty convex, if (x + y)=2 2 Int Z for a x y 2 Z, x 6= y. (ii) A convex cosed set Z X is said to be uniformy stricty convex, if (r) > for a r>. Proposition 2.4 Let Z be a uniformy stricty convex subset of X, dim X 2, B m (x) Z for some x 2 Int Z. Then ; : [ [! [ [ is ocay Lipschitz in ] [, im s! ; (s)=s =, ; (s) p ms for a s. Proof. Proposition 2. (i) entais (r) ; (p) (r ; p) (p)=p for a r>p>, hence ; is ocay Lipschitz in ] [. We obviousy have r (r) r ; diam Z, hence im s! ( ; (s))=s =.To concude, notice that Exercise 2.2 (iii) and Proposition 2. (iii) yied m(r) r 2 for r 2 [ m] and the trivia inequaity (r) r<r 2 =m for r>mcompetes the proof. 9

22 2.4 The Minkowski functiona Denition 2.5 Let A X be a given set. Then (2.2) is caed thepoar of A. A := fy 2 X hy xi 8 x 2 Ag We immediatey see that A is convex and cosed, 2 A. The foowing duaity statement hods. Lemma 2.6 Let A be as in Denition 2.5, et A be the poar of A and et conv denote the cosure of the convex hu. Then A = conv (A [fg) : Proof. Put ^A := conv (A [fg).wehave by denition (2.3) A = fz 2 X hy zi 8 y 2 A g hence 2 A and A A. Since A is convex and cosed, we necessariy have ^A A.Toprove the incusion A ^A, we x an arbitrary z 2 A and appy Lemma 2. with the projection pair ( ^P ^Q) associated to ^A. This yieds (2.4) h ^Pz z ; ^Pz ; xi 8 x 2 ^A: For every k> we have in particuar (2.5) hk ^Pz zik j ^Pzj 2 +sup n hk ^Pz xi x 2 Ao : Put (2.6) := inf n k> k ^Pz =2 A o : From (2.5) it foows > and we distinguish two cases. (i) =+ : Putting x := in (2.4), we obtain (2.7) k j ^Pzj 2 hk ^Pz zi 8 k> : (ii) <+ : Then ^Pz hence sup n h ^Pz xi x 2 Ao = and (2.5) yieds (2.8) + j ^Pzj 2 h ^Pz zi : In both cases (2.7) and (2.8), we concude ^Pz =, hence z 2 ^A. Lemma 2.6 is proved. 2

23 Lemma 2.7 Let A, A be as in Denition 2.5 and et R> be given. Then (2.9) A B R () () B =R () A : Proof. Assume A B R () and x y 2 B =R (). Then for x 2 A we have hy xi jyjjxj, hence y 2 A. Conversey, etb =R () A and x x 2 A. Then jxj = sup fhx wi w 2 B ()g = R sup fhx yi y 2 B =R ()gr. Denition 2.8 Let Z X be aconvex cosed set, 2 Z. The functiona M : X! R + [f+g dened by the formua s > s x 2 Z (2.2) M(x) := inf is caed the Minkowski functiona of Z. for x 2 X: The functiona M is sometimes caed gauge, cf. [Ro]. We ist without proof some of its basic properties. Proposition 2.9 In the situation of Denition 2.8, we have (i) Z = fx 2 X M(x) g (ii) C Z := fx 2 X M(x) = g (iii) M(tx) =tm(x) 8 x 2 X 8 t (iv) M(x + y) M(x) +M(y) 8 x y 2 X: As an immediate consequence of the above considerations, we have the foowing Proposition 2.2 Let Z X be aconvex cosed set and et R>r> be given numbers such that (2.2) B r () Z B R () : Then (2.22) B =R () Z B =r () (2.23) R jxj M(x) r jxj 8 x 2 X where Z is the poar and M is the Minkowski functiona of Z. According to (2.23) and Proposition 2.9, the Minkowski functiona of a convex set Z satisfying the hypotheses of Proposition 2.2 is convex and Lipschitz continuous. Its subdierentia has the foowing properties. 2

24 Lemma 2.2 Let Z satisfy the hypotheses of Proposition 2.2, et M M be the Minkowski functionas of Z Z,respectivey, and be the subdierentia of M. Then 6= 8 x 2 X, =@M(x) 8 x 2 X 8 t>, (iii) hw xi = M(x) hw yi M(y) 8 x y 2 X 8 w (iv) M (w) = 8 w 8 x 6=. Proof. (i) We have for a x 2 X (2.24) w () hw x ; yi M(x) ; M(y) 8 y 2 X hence For x 6=,wechoose a sequence <t n % M(x) n= 2 :::, and put x n := x=t n, x := x=m(x). Let (P Q) be the projection pair associated to Z by (2.3). Then x n 62 Z for n, hence Px n 6= and (2.25) hpx n Qx n ; zi 8 z 2 Z: On the other hand, we have Qx = x,andjqx n ; x jjx n ; x j! as n!. Seecting a subsequence, if necessary, wemay assume that Px n =jpx n j converge weaky to some w 2 B (). Then (2.25) yieds (2.26) hw x ; zi 8 z 2 Z: Putting z := rpx n =jpx n j in (2.25) and passing to the imit as n!,we obtain (2.27) hw x ir> : Inequaity (2.26) impies x (2.28) w ; M(x) or equivaenty, y M(y) 8 y 2 X nfg (2.29) hw x ; yi (M(x) ; M(y)) hw x i 8 y 2 X: According to (2.24) and (2.27), we have w := w =hw x i2@m(x) and (i) is proved. Using Proposition 2.9 (iii) we obtain (ii) triviay from (2.24), part (iii) foows from (2.24) by putting successivey y := and y := 2x and part (iv) foows from (iii). Remark 2.22 Lemma 2.2 does not hod for genera convex cosed sets Z.To see this, we rst notice that by (2.24), for every x with M(x) > and every w we have (2.3) (2.3) w w 6= x ; M(x) y 8 y 2 Z: 22

25 As an exampe, we choose X := L 2 ( ), Z := fz 2 X ; z(t) a.e. g, x(t) :=t for t 2 [ ]. Then Z is convex and cosed, 2 Z, M(x) =. Assume is nonempty and et w be arbitrary. By (2.3), we have Z w(t) tdt sup = Z Z jw(t)j dt hence w =, which contradicts (2.3). The main resut of this section reads as foows. w(t) y(t) dt y 2 X ; y(t) a.e. Theorem 2.23 Let Z satisfy the hypotheses of Proposition 2.2. Let Z be the poar of Z and et M M be the Minkowski functionas of Z Z,respectivey. For x 2 X put J(x) J (x) :=M (x). Then (i) hw ; z x ; yi (M(x) ; M(y)) 2 8 x y 2 X w 2 J(x) z 2 J(y) (ii) hw ; z x ; yi (M (x) ; M (y)) 2 8 x y 2 X w 2 J (x) z 2 J (y) (iii) y 2 J(x) () x 2 J (y) 8 x y 2 X (iv) where J(Z) := Z = J(Z) Z = J (Z ) [ x2z J(x) J (Z ):= [ y2z J (y). Before proving Theorem 2.23, we state an auxiiary Lemma. Lemma 2.24 Let the hypotheses of Theorem 2.23 hod. Then for a x y 2 X nfg we have (2.32) hy xi M(x) M (y) (2.33) hy xi = M (y) M(x) () x M(x) (y) () y M (y) : Proof of Lemma Inequaity (2.32) foows immediatey from the denition of Z and Lemma 2.2 (iii) yieds the impications Assume now x 2 (y) ) hy xi = M (y) M(x) y M (y) ) hy xi = M (y) M(x) : (2.34) hx yi = M(x) M (y) for some x y 2 X nfg : 23

26 Then, by (2.32) wehave x M(x) y ; z y M (y) x ; z M (y) ; M (z) 8 z 2 X M(x) ; M(z) 8 z 2 X and the assertion foows. Proof of Theorem Inequaities (i), (ii) foow from (2.24) (and the corresponding inequaity for M ). To prove (iii), it suces to x x 2 X and y 2 J(x) and prove that x 2 J (y). The other impication then foows from the duaity Z = Z and J = J. The denition of J immediatey entais J() = fg J () = fg, hence it suces to assume x 6=. By Lemma 2.2 (iii), (iv) we have (2.35) hy xi = M 2 (x) M (y) =M(x) : and Lemma 2.24 (ii) yieds the assertion. To prove (iv), it suces to use (iii) and (2.35). We ca J the duaity mapping induced by Z. It can be interpreted geometricay by means of the norma cone N Z (x) in the foowing way. Proposition 2.25 Let the hypotheses of Theorem 2.23 hod. Then for every x we have J(x) N Z (x). Conversey, for each y 2 N Z (x), y 6=, we have hy xi = M (y) and y=hy xi2j(x). Proof. The incusion J(x) N Z (x) foows immediatey from the denition. Let now y 2 N Z (x), y 6= be given. Then hy xi hy zi for a z 2 Z, hence y=hy xi 2Z. We have in particuar M (y) hy xi and from (2.32) (note that M(x) =)we obtain hy xi = M (y). Lemma 2.24 then competes the proof. Exercise 2.26 Prove that M 2 =2 is the conjugate function to M 2 =2 in the sense of [AE], that is, (2.36) 2 M 2 (y) = sup hy xi; 2 M 2 (x) x 2 X for every y 2 X: `Smooth' convex domains Z X are those where N Z (x) reduces to a haf-ine for each x By Proposition 2.25, this is equivaent tosaying that J is a singe-vaued mapping. We have the foowing duacharacterization of such domains. Theorem 2.27 In the situation of Theorem 2.23, the foowing conditions are equivaent. (i) J is singe-vaued, (ii) Z is stricty convex according to Denition

27 Proof. (ii) ) (i) : Let x 2 X and y y 2 J(x) be given. For x =we have y = y =, otherwise we put y := (y + y )=2. Then y 2 J(x) and M (y) =M (y )=M (y )= M(x). Consequenty, ay =M(x), y =M(x), y=m(x) beong hence y = y. non (ii) ) non (i) : Assume that there exist y 6= y 2 Z such that y := (y + y )=2 Let x 2 J (y) be arbitrariy chosen. Then M(x) =M (y) = and =hx yi = 2 (hx y i + hx y i) : This yieds hx y i = hx y i = = M (y ) =M (y ) and from Lemma 2.24 (ii), we concude y y 2 J(x) and Theorem 2.27 is proved. Exampe 2.28 If Z = fx 2 X hx n i i i i = ::: pg is a poyhedron with a system fn i i = ::: pg of unit vectors and with i >, then Z is the poyhedron Z = conv (f n = ::: n p = p g). 3 The pay and stop operators The eementary hysteresis operators caed stop and pay have aready been introduced in Section. The rigorous construction presented here foows the exposition in [K] and is sighty dierent from the approach of[kp] and[v]. We admit the innitey dimensiona case and start with nonsmooth input functions. More precisey, we dene the inputs and outputs in the space CBV ( T X) of continuous functions of bounded variation with vaues in a Hibert space X. We further prove that the restriction of the pay and stop operators to Soboev spaces W p ( T X) is continuous and bounded if p < and discontinuous for p =+. If the convex constraint Z has nonempty interior, the extension of these operators is shown to be continuous (but not necessariy bounded) from C([ T] X) to C([ T] X), together with an interesting smoothening property of the pay, namey that it maps C([ T] X) into CBV ( T X). A brief survey of the functiona framework used here can be found in Section 8. The rst step consists in proving the foowing generaization of Theorem.7. Theorem 3. Let a rea separabe Hibert space X,a convex cosed set Z X with 2 Z, an eement x 2 Z and a function u 2 CBV ( T X) be given. Then there exist uniquey determined 2 CBV ( T X), x 2 CBV ( T Z) such that (3.) (i) x(t) +(t) =u(t) 8t 2 [ T] (ii) (iii) x() = x Z T x(t) ; '(t) d(t) 8' 2 C([ T] Z) : We rewrite the Riemann-Stietjes integra in (iii) in an equivaent, but more convenient form. 25

28 Lemma 3.2 Let x 2 C([ T] Z) and 2 NBV ( T X) satisfy (3.) (iii). Then (3.2) Z t s x( ) ; ( ) d( ) 8 2 C([s t] Z) for a s<t T: Proof. Let <s<t T and 2 C([s t] Z) be given (the case s =is anaogous). For <<minfs t ; sg put ' ( ) := 8 >< >: x( ) for 2 [ s; [ [ ]t T ] x(s ; )+ ;s+ ( (s) ; x(s ; )) for 2 [s ; s[ ( ) for 2 [s t ; ] ( (t ; ) ; x(t)) for 2 ]t ; t] : x(t) + t; Then (3.) (iii) and (8.26) yied = + + Z T Z t s Z s s; Z t x( ) ; ' ( ) d( ) x( ) ; ( ) d( ) + Z s x( ) ; x(s ; ) d( ) + t; s; Z t (s) ; ( ) x(s ; ) ; (s) d t; (t) ; ( ) (t ; ) ; x(t) d : ( ) ; (t ; ) d( ) Using (8.22) and (8.7) we can pass to the imit as! and the proof is compete. Let us note that if two variationa inequaities of the form (3.2) are satised, that is, (3.3) Z t s xi ( ) ; ( ) d i ( ) 8 2 C([s t] Z) i = 2 with u i = x i + i, x i 2 C([s t] Z), i 2 CBV (s t X), then putting := (x + x 2 )=2, we obtain from (3.2), (8.24) and (8.22) (3.4) j (t) ; 2 (t)j 2 j (s) ; 2 (s)j 2 +2ju ; u 2 j Var [s t] +Var [s t] 2 : Proof of Theorem 3.. Uniqueness foows immediatey from the inequaity (3.4). The existence proof is carried out by a simpe time-discretization scheme. For a xed n 2 N we dene (3.5) jt u j := u n j = ::: n: Let (P Q) be the projection pair dened by formua (2.3). We construct the sequences (3.6) ( x j := Q(x j; + u j ; u j; ) j := u j ; x j j = ::: n: j = ::: n 26

29 We have j ; j; = P (x j; + u j ; u j; ) and Lemma 2.2 (i) yieds j ; j; x j ; z 8z 2 Z 8j 2f ::: ng: (3.7) Putting z := x j; (3.8) and V u := Var u, we immediatey obtain from (3.7) [ T ] nx j= j j ; j; jv u : We now dene piecewise inear functions u (n) (n) x (n) 2 W ( T X) by the formua (3.9) 8 >< >: u (n) (t) :=u j; + n ; t ; j; T n ; (n) t (t) := j; + n ; j; T n ; x (n) t (t) :=x j; + n ; j; T n (uj ; u j; ) (j ; j; ) (xj ; x j; ) for t 2 [(j ; )T =n jt =n[ and j = ::: n,continuousy extended to t = T. Let u : R +! R + (3.) be the continuity moduus of u, thatis u () := sup fju(t) ; u(s)j jt ; sjg for >: For every 2 ](j ; )T =n jt =n[ and z 2 Z we have by (3.7) and Lemma 2.2 (i) _ (n) ( ) x (n) ( ) ; z ; n T j ; j; x j ; x j; and estimate (3.8) yieds (3.) Z t ; n T ; n T u j ; j; u j ; u j; T n j j ; j; j x (n) ( ) ; '( ) d (n) ( ) ;V u u T n for a n 2 N, t 2 [ T] and ' 2 C([ T] Z). The proof of Theorem 3. wi be compete if we prove that (3.2) f (n) n 2 Ng is a uniformy convergent sequence. Indeed, in this case it suces to use inequaity (3.) and Theorem 8.6, since the sequence fu (n) g is uniformy convergent and Var (n) V u by (3.8). [ T ] ; To prove (3.2), we put '( ) := x (n) ( )+x (`) ( ) =2 for two dierent vaues of n in (3.), say n `. Then (3.3) Z t _ (n) ( ) ; _ (`) ( ) x (n) ( ) ; x (`) ( ) d ;V u u T n + u T` 27

30 hence, by inequaity (8.22), 2 (n) ; (`) 2 u (n) ; u (`) ; T ; Var (n) +Var (`) + V u u + u [ T ] [ T ] n T` The sequence f (n) g is therefore fundamenta in C([ T] X), hence (3.2) hods and Theorem 3. is proved. Denition 3.3 Let Z X be aconvex cosed set, 2 Z and et u 2 CBV ( T X), x 2 Z be given. Let (x ) be the soution of (3.). We dene the vaue P(x u) S(x u) of the pay and stop operators P S : Z CBV ( T X)! CBV ( T X), respectivey, by the formua : (3.4) P(x u):= S(x u):=x: Remark 3.4 The initiay unperturbed state is characterized by thechoice x = Qu() of the initia condition (3.) (ii). In this case we use the simpied notation (3.5) P(u) :=P(Qu() u) S(u) :=S(Qu() u): 3. Absoutey continuous inputs It is natura to expect that pay and stop operators act in Soboev spaces W p ( T X). Before passing to the continuity statement, we give in Proposition 3.5 beow a precise meaning to the normaity rue mentioned in Section. It aso yieds the unique orthogona decomposition of _u(t) into the components (t) _ 2 N Z (x(t)) and _x(t) 2 T Z (x(t)), see Subsection 2.2. This can be used as an aternative denition of the pay and stop operators, see [KP]. Proposition 3.5 Let Z X be aconvex cosed set with 2 Z,etx 2 Z be a given initia vaue and et u 2 W ( T X) be given. beong to W ( T X) and satisfy Then := P(x u), x := S(x u) (3.6) (i) (ii) (t) _ x(t) ; z a.e. 8z 2 Z (t) x(t) = a.e. Proof. For arbitrary s<t T and 2 [s t] put ( ):=x(s) in (3.2). Then (8.24) and (8.26) yied (3.7) hence (3.8) 2 j(t) ; (s)j2 j(t) ; (s)j 2 Z t Z t s hu( ) ; u(s) d( )i = Z max fj(t) ; ( )jg t j _u( )j d st Z t s s j _u( )j d 8 s<t T: s h(t) ; ( ) _u( )i d 28