Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not G\^ateaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

Q is the sum of the solution operator of linear elasticity for given stress distribution, the elasticity tensor, and a coercive operator modeling hardening e ects. Finally, A is the convex subdi erential of the indicator functional of the closed and convex set of feasible stresses de ned by a suitable yield condition. Details on models in elastoplasticity can be found in [ ]. Another model which is covered by the state equation in (P) is the system of homogenized elastoplasticity, which we will study in detail in Section below.
Let us put our work into perspective. Assume for a moment that A is the convex subdi erential of a proper, convex, and lower semicontinuous functional ϕ and that Q is self adjoint. Then, by convex duality, the state equation is equivalent to ( . ) ∈ ∂ϕ * ( where E is the quadratic energy functional given by Systems of this type have been intensively studied concerning existence of solutions and their numerical approximation, and we only refer to [ ] and the references therein. In contrast to this, the literature on optimization problems governed by ( . ) is rather scarce. The research on optimal control of equations of type ( . ) probably started with the sweeping process, where ϕ = I −C(t ) is the indicator functional of a moving convex set C(t), see [ ]. In the optimal control setting, C(t) is most frequently set to C(t) = (t) − Z with a convex set Z and a driving force . This ts into the setting of ( . ) by de ning ϕ I Z and Q = R = id (identity). Optimal control problems of this type are investigated in [ , , -, -], where the underlying Hilbert space is mostly nite dimensional. Problems in an in nite dimensional Hilbert space are investigated in [ , ]. To be more precise, in these contributions, H is the Sobolev space H (Ω) and E is the Dirichlet energy. Moreover, ϕ * is set to ϕ * (z) = z L (Ω) and its viscous regularization, respectively, i.e., ϕ * δ (z) = z L (Ω) + δ z H (Ω) . Optimal control problems governed by quasi-static elastoplasticity with linear kinematic hardening and von Mises yield conditions are treated in [ , , ]. As already indicated above, ϕ is the indicator functional of the convex set of feasible stresses in this case. All mentioned problems t into our framework and can thus be seen as special cases of our non-smooth evolution. Our analysis therefore represents a generalization of existing results on optimal control of non-smooth evolution problems and can also be applied to application problems that were not treated in the literature so far such as optimal control of homogenized plasticity, which is investigated in Section . We emphasize that problems with non-convex energies such as damage evolution are not covered by our analysis. Optimal control problems governed by ( . ) with non-convex energy are investigated in [ , ].
Our strategy to analyze (P) is as follows: After showing well-posedness of the state equation and the optimal control problem, we employ the Yosida-regularization with an additional smoothing to obtain a smooth (i.e., Fréchet di erentiable) control-to-state map. We will prove that accumulation points of global minimizers of the regularized optimal control problems for vanishing regularization are solutions of the original non-smooth problem (P). Moreover, rst-order necessary and second-order su cient optimality conditions for the regularized problems are derived. The passage to the limit to establish optimality conditions for the original problem goes beyond the scope of this paper and is subject to future work. The results of [ , ] indicate that the optimality conditions obtained in this way are rather weak and we expect the same all the more for our general setting. Let us underline that regularization is a widely used approach to treat optimal control problems governed by non-smooth evolutions. We only refer to [ , , ] and the references therein.
The paper is organized as follows. After stating our standing assumptions in Section , we investigate the state equation and its regularization in Section . In Section , we then turn to the optimal control problem and show that it admits an optimal solution under our standing assumptions. Moreover, we where A is as de ned in ( . ).
Proof. The proof essentially follows the lines of [ , Theorem . ] and [ , Proposition . ]. For convenience of the reader, we sketch the main arguments. At rst, one employs the transformation H ( ,T ; H ) z → q R − Qz ∈ H ( ,T ; H ) with its inverse H ( ,T ; H ) q → z Q − (R − q) ∈ H ( ,T ; H ) to see that ( . ) is equivalent to Since Q is coercive the operator,Ã: H → H , h → QA(h) is maximal monotone with respect to the scalar product Therefore, [ , Proposition . ] yields the existence of a unique solution q ∈ H ( ,T ; H ) of ( . ). To verify the estimate in ( . ), we employ [ , Lemme . ], which gives whereq is the unique solution of A (R (τ ) − Qz(τ )) H , which gives the second inequality.
Remark . . In order to prove Theorem . , it is su cient to require that A is bounded on compact subsets (in addition to the closedness of D(A)), cf. [ , Proposition . ]. However, the boundedness on bounded sets of A is needed to prove Theorem . below and therefore, we impose it as a standing assumption.
Based on Theorem . , we may introduce the solution operator associated with ( . ) and reduce (P) to an optimization problem in the control variable only, see De nition . below. Due to the set-valued operator A, this solution operator will in general be non-smooth, which complicates the derivation of rst-and second-order optimality conditions. A prominent way to overcome this issue is to regularize the state equation in order to obtain a smooth solution operator. This is frequently done by means of the Yosida-approximation, see e.g. [ ], and we will pursue the same approach. For this purpose, we will investigate the Yosida-approximation and its convergence properties in the next subsection.
. For the rest of this section, we x z ∈ H and ∈ U(z , D(A)) and denote the unique solution of ( . ) by z. We start with a convergence result of the Yosida-approximation for xed data z and and then turn to perturbation of the data. Proof. The proof in principle follows the lines of [ , Proposition . ], since our assumptions and assertions however are slightly di erent, we provide the arguments in detail. First of all, since z → A λ (R − Qz) is Lipschitz-continuous by [ , Proposition . (b)], the existence of a unique solution of ( . ) follows from Banach's contraction principle by standard arguments, cf. e.g. [ ]. Moreover, [ , Proposition . (a)] and the de nition of A λ give The operators A n, λ are uniformly Lipschitz continuous with Lipschitz constant λ − . Thus, thanks to assumption ( . ), we can apply Lemma . with M (R − Qz λ )[ ,T ], N H , G n A n, λ and G A λ . Together with the assumptions on n and z n, this gives that the right side in the inequality above converges to zero as n → ∞. Using this, Proposition . , and Remark . (with A = A n ), we conclude Now, since λ was arbitrary, ( . ) holds for every λ > . Therefore, as . z n is bounded in L ( ,T ; H ) by assumption, we obtain z n → z in C([ ,T ]; H ). Moreover, again due to the boundedness assumption on . z n , there is a weakly converging subsequence in H ( ,T ; H ). Due to z n → z in C([ ,T ]; H ), the weak limit is unique and hence, the whole sequence z n converges weakly to z in H ( ,T ; H ).
Lemma . . Let {λ n } n ∈N ⊂ ( , ∞) be a sequence converging towards zero. Then the sequence A n A λ n , n ∈ N, of maximal monotone operators ful lls ( . ) for all λ > and all h ∈ H .
Proof. At rst we prove that, for all h ∈ H and λ > µ > , the following inequality holds For this purpose, let h ∈ H be arbitrary and set y R λ (h) and y R λ+µ (h). Then we have h ∈ y + λA(y ), hence, h−y λ ∈ A(y ) and analogously h−y λ+µ ∈ A(y ). The monotonicity of A thus implies which yields ( . ). With this inequality and [ , Proposition . (d)] at hand, we obtain which completes the proof. Now, we are in the position to state our main convergence results in Theorem . and Theorem . , where the loads and initial data are no longer xed. The rst theorem addresses the continuity properties of the solution operator to the original equation ( . ), whereas Theorem . deals with the Yosida-approximation. In order to sharpen these convergence results and prove strong convergence in H ( ,T ; H ), we additionally need the following Assumption . . The maximal monotone operator A is given as a subdi erential of a proper, convex and lower semicontinuous function ϕ : H → (−∞, ∞], that is, A = ∂ϕ. Theorem . (Continuity of the solution operator). Let {z n, } n ∈N ⊂ H and { n } n ∈N ⊂ U(z n, , D(A)) be sequences such that z n, → z in H , n in H ( ,T ; X) and n → in L ( ,T ; X). Moreover, denote the solution of . z n ∈ A(R n − Qz n ), z n ( ) = z n, by z n ∈ H ( ,T ; H ) (whose existence is guaranteed by Theorem . ). Then z n z in H ( ,T ; H ) and z n → z in C([ ,T ]; H ).
Proof. Thanks to Theorem . , to be more precise ( . ), {z n } is bounded in C([ ,T ]; H ). Since A is bounded on bounded sets by assumption, ( . ) then gives that { . z n } is bounded in L ( ,T ; H ). Therefore, we can apply Lemma . with A n A for all n ∈ N to obtain z n z in H ( ,T ; H ) and z n → z in C([ ,T ]; H ).
Theorem . (Convergence of the Yosida-approximation). The statement of Theorem . holds true when z n is, for every n ∈ N, the solution of where {λ n } n ∈N ⊂ ( , ∞) is a sequence converging to zero.
Proof. According to Lemma . , the sequence of maximal monotone operators A n A λ n ful lls ( . ) so that it only remains to prove that . z n is bounded in L ( ,T ; H ) to apply again Lemma . . To this end, let n ∈ H ( ,T ; H ) be the solution of . n ∈ A(R n − Q n ), n ( ) = z n, , whose existence is guaranteed by Theorem . (note that n ∈ U(z n, , D(A)) by assumption). Thanks to Theorem . , it holds n z in H ( ,T ; H ). From Proposition . , it follows If additionally n → in H ( ,T ; X), A ful lls Assumption . , and ϕ(R n ( ) − Qz n, ) → ϕ(R ( ) − Qz ), then Theorem . implies n → z in H ( ,T ; H ) so that [ , Proposition . ] gives the strong convergence z n → z in H ( ,T ; H ) because of Remark . . The assertions of Theorem . and Theorem . are remarkable due to the following: As a rst approach to prove the (strong) convergence of the states in H ( ,T ; H ), one is tempted to follow the lines of the proofs of [ , Lemme . ] and Proposition . , respectively. This would however require the strong convergence of the derivatives of the given loads, which we want to avoid in order to enable less regular controls. The detour via the Yosida-regularization in Lemma . allows to overcome this issue. z n in L ∞ ( ,T ; H ) in this case. Thus, Lemma . is again applicable and we can argue similar as we did in the proof of Theorem . to verify the previous convergence results. In this setting, we would not have any restrictions on A (except monotonicity), but would need more regular loads, which is not favorable, as the latter serve as control variables in our optimization problem. Moreover, the boundedness assumption on A is ful lled for our concrete application problem in Section . . Therefore, we decided to choose the present setting and to impose the additional boundedness assumption on A. Remark . . It is to be noted that most of the above results can also be shown in more general Bochner-Sobolev spaces, that is, when loads are contained in W ,r ( ,T ; X) and states in W ,r ( ,T ; H ) for some r ∈ [ , ∞). However, since a Hilbert space setting is advantageous when it comes to the derivation of optimality conditions, we focus on the case r = .
Unfortunately, the Yosida-approximation is frequently not su cient for the derivation of optimality conditions by means of the standard adjoint approach, since the solution operator associated with ( . ) is in general still not Gâteaux di erentiable. Therefore, we apply a second regularization turning the Yosida-approximation of A into a smooth operator. The properties needed to ensure convergence of this second regularization are investigated in the following Lemma . (Convergence of the Regularized Yosida-Approximation). Consider a sequence {λ n } n ∈N ⊂ ( , ∞) and a sequence of Lipschitz continuous operators A n : H → H , n ∈ N, such that Let moreover { n } n ∈N ⊂ C([ ,T ]; X) be given and denote by z n , z λ n ∈ C ([ ,T ]; H ) the solutions of and Proof. Again, thanks to the Lipschitz continuity of A n and A λ n , the existence and uniqueness of z n and z n, λ follows from Banach's contraction principle by classical arguments. Moreover, the continuity of n carries over to the continuity of . z n and . z n, λ . Let us abbreviate c n sup h ∈H A n (h) − A λ n (h) H . Then, in light of ( . ) and ( . ), we nd which completes the proof.
Now we turn to the optimal control problem (P). We rst address the existence of optimal solutions and afterwards discuss the approximation of (P) in Section . . Based on Theorem . , we reduce the optimal control problem (P) into a problem in the control variable only. Recall that the control space X c embeds compactly in X. This operator will be frequently called control-to-state map.
With the de nition above, problem (P) is equivalent to the reduced problem: Recall our standing assumptions on the objective, namely that (z, ) = Ψ(z, ) + Φ( ), where Ψ : H ( ,T ; W) × H ( ,T ; X c ) → R is weakly lower semicontinuous and bounded from below and Φ : H ( ,T ; X c ) → R is weakly lower semicontinuous and coercive. These assumptions allow us to show the existence of (globally) optimal solutions: Theorem . (Existence of Optimal Solutions). Let z ∈ H and M be a closed subset of D(A). Then there exists a global solution of (P).
Proof. Based on Theorem . , the proof follows the standard direct method of the calculus of variations. First of all, since Ψ is bounded from below and Φ is coercive, every in mal sequence of controls is bounded in H ( ,T ; X c ) and thus admits a weakly converging subsequence. Due to the compact embedding of X c in X, this sequence converges strongly in C([ ,T ]; X) so that the weak limit belongs to U(z ; M), due the closeness of M. Moreover, thanks to weak convergence in H ( ,T ; X) and strong convergence in C([ ,T ]; X), Theorem . gives weak convergence of the associated states in H ( ,T ; H ). The weak lower semicontinuity of Ψ and Φ together with H → W then implies the optimality of the weak limit.
Clearly, in view of the nonlinear state equation, one cannot expect the optimal solution to be unique. Note that, since D(A) is closed by our standing assumptions, the choice M = D(A) is feasible.
. While the existence of optimal solutions for (P) can be shown by well-established techniques as seen above, the derivation of optimality conditions is all but standard because of the lack of di erentiability of the control-to-state map. We therefore apply a regularization of A built upon the Yosida-approximation in order to obtain a smooth control-to-state mapping. In view of Lemma . , this regularization is assumed to satisfy the following Assumption . . Let {A n } n ∈N be a sequence of Lipschitz continuous operators from H to H such that, together with a sequence {λ n } n ∈N ⊂ ( , ∞), it holds i.e., the requirements in Lemma . are ful lled.
In Section . , we show how to construct such a regularization for a concrete application problem. Given the regularization of A, we de ne the corresponding optimal control problem: Since A n is Lipschitz continuous, the equation admits a unique solution for every z ∈ H and every ∈ L ( ,T ; X). Similar to De nition . , we denote the associated solution operator by Moreover, the solution operator associated with the Yosida-approximation, i.e., the solution operator of Proposition . (Existence of Optimal Solutions of the Regularized Problems). Let n ∈ N, z ∈ H , and M a closed subset of D(A). Then, under Assumption . , there exists a global solution of (P n ).
Proof. Let , ∈ L ( ,T ; X) be arbitrary and de ne z i S n ( i ), i = , . Then, due to the Lipschitz continuity of A n , we have for almost all t ∈ [ ,T ] which yields, thanks to Gronwall's inequality , the Lipschitz continuity of S n . Using this together with the fact that X c is compactly embedded into X, one can argue as in the proof of Theorem . to obtain the existence of a global solution of (P n ) for all n ∈ N.
Theorem . (Weak Approximation of Global Minimizers). Let z ∈ H and M be a closed subset of D(A). Suppose moreover that Assumption . holds and let { n } n ∈N be a sequence of globally optimal controls of (P n ), n ∈ N. Then there exists a weak accumulation point and every weak accumulation point is a global solution of (P). Hence, by virtue of the boundedness of Ψ from below and the radial unboundedness of Φ, { n } is bounded and therefore admits a weak accumulation point in H ( ,T ; X c ).
Let us now assume that a given subsequence of { n } n ∈N , denoted by the same symbol for simplicity, converges weakly to˜ in H ( ,T ; X c ). Since X c is compactly embedded in X, we obtain n →˜ in C([ ,T ]; X) and consequently,˜ ∈ U(z ; M). In addition, the strong convergence in C([ ,T ]; X) in combination with Theorem . and Lemma . yields weak convergence of the states, i.e., S n ( n ) S(˜ ) in H ( ,T ; H ) and thus also in H ( ,T ; W). Now, let be a global solution of (P). We can again H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . . ), page of use Proposition . and Lemma . to obtain S n ( ) → S( ) in H ( ,T ; H ). This, together with the weak lower semicontinuity of Ψ and Φ, implies giving in turn the optimality of the weak limit.
Corollary . (Strong Approximation of Global Minimizers). In addition to Assumption . , assume that Φ : H ( ,T ; X c ) → R is such that, if a sequence { n } n ∈N satis es n in H ( ,T ; X c ) and Φ( n ) → Φ( ), then n → in H ( ,T ; X c ). Then every weak accumulation point of a sequence of globally optimal controls of (P n ) is also a strong one.
Moreover, if in addition, at least one of the following holds • Assumption . is satis ed, that is A = ∂ϕ, and ϕ is continuous on M or Proof. Consider an arbitrary accumulation point˜ of a sequence of global minimizers of (P n ), i.e., n ˜ in H ( ,T ; X c ). From the previous proof, we know that then ( . ) holds, giving in turn Since S n ( n ) S(˜ ), as seen in the previous proof, and both, Ψ and Φ, are weakly lower semicontinuous by assumption, this implies Φ( n ) → Φ(˜ ) and Ψ(S n ( n ), n ) → Ψ(S(˜ ),˜ ). The hypothesis on Φ thus yields n →˜ in H ( ,T ; X c ) so that˜ is indeed a strong accumulation point as claimed.
Due to X c → X, the strong convergence carries over to H ( ,T ; X) and therefore, we deduce from Theorem . that S λ n ( n ) → S(˜ ) in H ( ,T ; H ), provided that Assumption . is ful lled and ϕ(R n ( ) − Qz ) → ϕ(R˜ ( ) − Qz ) holds. If the additional requirements on Ψ are ful lled, we also obtain the strong convergence S λ n ( n ) → S(˜ ) in H ( ,T ; H ), since we already showed n →˜ in H ( ,T ; X c ). Thus, in both cases, Lemma . gives S n ( n ) → S(˜ ) in H ( ,T ; H ), which is the second assertion.
Example . . Let us assume that X c is a Hilbert space. Then a possible objective functional ful lling the requirements on Φ in Corollary . reads as follows: i.e., Φ( ) α/ H ( ,T ;X c ) . Herein, Ψ : H ( ,T ; H )×H ( ,T ; X c ) → R is again lower semicontinuous and bounded from below and α > is a given constant. Since H ( ,T ; X c ) is a Hilbert space, too, weak convergence and norm convergence give strong convergence and consequently, this speci c choice of Φ ful lls the condition in Corollary . .

Remark . (Approximation of Local Minimizers).
By standard localization arguments, the above convergence analysis can be adapted to approximate local minimizers. Following the lines of, for instance, [ ], one can show that, under the assumptions of Corollary . , every strict local minimum of (P) can be approximated by a sequence of local minima of (P n ). A local minimizer of (P), which is not necessarily strict, can be approximated by replacing the objective in (P n ) by (z, l) Since these results and their proofs are standard, we omitted them. Now that we answered the question of approximation of optimal controls via regularization, we turn to the regularized problems and derive optimality conditions for these in the next two sections.

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In the following, we consider a single element of the sequence of regularized problems. The associated regularized operator is denoted by A s so that the regularized optimal control problems reads as follows: Beside our standing assumption and the Lipschitz continuity required in Assumption . , we need the following additional assumptions for the derivation of rst-order necessary optimality conditions for (P s ). Recall the continuous embeddings Y → Z → H from Section . Assumption . .
can be extended to elements of L(Z; Z) and L(H ; H ), respectively, denoted by the same symbol. There exists a constant C such that these extensions satisfy Remark . . It is well known that a norm gap is often indispensable to ensure Fréchet di erentiability.
This is also the case in our application example in Section . . This is the reason for considering two di erent spaces Y and Z in context of the Fréchet di erentiability of A s in Assumption . .
We start the derivation of optimality conditions for (P s ) with the Fréchet-derivative of the associated control-to-state mapping.
. As A s : Y → Y is supposed to be Lipschitz continuous and R and Q are not only linear and continuous as operators with values in H , but also in Y according to our standing assumptions, Banach's xed point theorem immediately implies that the state equation in (P s ), i.e., admits a unique solution z ∈ H ( ,T ; Y) for every right hand side ∈ L ( ,T ; X), provided that z ∈ Y. Therefore, similar to above, we can de ne the associated solution operator S s : L ( ,T ; X) → H ( ,T ; Y) (for xed z ∈ Y). We will frequently consider this operator with di erent domains, e.g. H ( ,T ; X), and ranges, in particular H ( ,T ; Z). With a little abuse of notation, these operators are denoted by the same symbol. Lemma . (Lipschitz Continuity of S s ). The solution operator S s is globally Lipschitz continuous from Proof. This can be proven completely analogously to the Lipschitz continuity of S n from L ( ,T ; X) to H ( ,T ; Y) in Proposition . .
Lemma . . Assume that Assumption . (ii) is ful lled and let y ∈ L ( ,T ; Y) and w ∈ L ( ,T ; Z) be given. Then there exists a unique solution η ∈ H ( ,T ; Z) of ( . ) H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . .

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Proof. Let us de ne ) ∈ Z is Bochner measurable as a pointwise limit of Bochner measurable functions. Furthermore, Assumption . (ii) implies for almost all t ∈ [ ,T ] and all η , η ∈ Z that B(t, η ) − B(t, η ) Z ≤ C η − η Z . Therefore, we can apply Banach's xed point argument to the integral equation associated with ( . ), which gives the assertion.
Proof. Let , h ∈ H ( ,T ; X) be arbitrary and abbreviate z h S s ( + h). Thanks to Lemma . , there exists a unique solution η ∈ H ( ,T ; Z) of ( . ). Clearly, the solution operator of ( . ) is linear with , the continuity of the solution operator of ( . ). This also proves the asserted inequality (after having proved that η = S s ( )h, which we do next). It remains to verify the remainder term property. For this purpose, let us denote the remainder term of A s by r , i.e., Moreover, we abbreviate Then, in view of the de nition of z, z h , and η (as solution of ( . )), we nd for almost all t ∈ [ ,T ] Hence, Assumption . (ii) and Gronwall's inequality yield (note that r (y; ζ ) ∈ L ( ,T ; Z) by its de nition as remainder term). Furthermore, thanks to Lemma . and the de nition of ζ , we obtain H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . .
; Y) and the remainder term property of r thus give for almost all t ∈ ( ,T ) that In combination with ( . ) and Lebesgue's dominated convergence theorem, this yields , which, in view of ( . ) nishes the proof.
Remark . . It is to be noted that we did not employ the implicit function theorem to show the di erentiability of S s . The reason is that ), which would be the right spaces for the existence result from Lemma . . The same observation for a more abstract setting was already made in [ ].
. Now that we know that the (regularized) control-to-state map is Gâteaux di erentiable, we can apply the standard adjoint approach to derive rst-order necessary optimality conditions in form of a Karush-Kuhn-Tucker (KKT) system. To keep the discussion concise, we restrict our analysis to the case without further control constraints. To be more precise, we require the following: Assumption . . Let z ∈ Y such that −Qz ∈ D(A). The set M in the de nition of the set of admissible controls is given by the singleton Note that U is a linear subspace of H ( ,T ; X). Remark . (Additional Control Constraints). One could allow for additional control constraints in our analysis, even more complex ones than the ones covered by U(z ; M) such as for instance box constraints over the whole time interval or vanishing initial and nal loading, i.e., ( ) = (T ) = , which is certainly meaningful for many practically relevant problems. However, since the di erentiability of the control-to-state map is the essential issue in the derivation of optimality conditions and additional (convex and closed) control constraints can be incorporated by standard argument, we decided to leave them out in order to keep the discussion as concise as possible.
However, without any further assumptions, the existence of solutions to the unregularized state equation ( . ) cannot be guaranteed. To be more precise, one needs that R ( ) − Qz ∈ D(A), see Theorem . , which holds in case of U, provided that −Qz ∈ D(A). This is the reason for considering the set H ( ,T ; X C ) ∩ U as set of admissible controls in the rest of the paper.
Note moreover that, if the operator R is injective (which is the case in Section ), then U = { ∈ H ( ,T ; X) : ( ) = }. The chain rule immediately gives that the reduced objective de ned by is Fréchet di erentiable, too. Thus, by standard arguments, one derives the following Lemma . (Purely Primal Necessary Optimality Conditions). Let Assumption . and Assumption . hold. Then, if a control ∈ H ( ,T ; X c ) ∩ U with associated state z = S s ( ) is locally optimal for (P s ), then Next, we reformulate ( . ) in terms of a KKT-system by introducing an adjoint equation, which formally reads Depending on the regularity of the right hand side , we de ne di erent notions of solutions: If takes the form with some ∈ L ( ,T ; H ) and ∈ H , then we call φ ∈ H ( ,T ; H ) strong solution of ( . ), if, for .
In the following, we will-as usual-identify ∈ L ( ,T ; H ) with an element of H ( ,T ; H ) * via ( , · ) L ( ,T ;H) and denote this element with a slight abuse of notation by the same symbol. Lemma . . Let y ∈ L ( ,T ; Y) and ∈ H ( ,T ; H ) * . Then there is a unique weak solution of ( . ), which is given by Moreover, if is of the form ( . ), then there exists a unique strong solution of ( . ), and the weak and the strong solution coincide.
Proof. At rst note that the existence of a solution of ( . ) can be proven exactly as in Lemma . .

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To prove uniqueness, letφ ∈ L ( ,T ; H ) be another weak solution. Then, we choose an arbitrary w ∈ L ( ,T ; H ) and set η S y (w) to see that and therefore φ =φ. Now we turn to the strong solution and suppose that is as given in ( . ). Existence and uniqueness of a strong solution can again be shown by means of Banach's xed point theorem. To this end, let us consider the a ne-linear operator Since H is separable by our standing assumptions, we can apply [ , Chap. IV, Thm. . ] to obtain that, for every φ ∈ L ( ,T ; H ), the mapping ( ,T ) → B(t, φ(t)) is Bochner measureable. Moreover, since A s (y) * L(H;H) = A s (y) L(H;H) , Assumption . (ii) yields that B is also Lipschitz continuous w.r.t. the second variable for almost all t ∈ ( ,T ). Therefore, similarly to the proof of Lemma . , one can apply Banach's xed point theorem to the integral equation associated with ( . ) to establish the existence of a unique strong solution.
Finally, if we test ( . ) with an arbitrary η ∈ H ( ,T ; H ) with η( ) = and integrate by parts, then we see that every strong solution is also a weak solution. Since the latter one is unique, as seen above, we deduce that weak and strong solution coincide.
Theorem . (KKT-Conditions for (P s )). Assume that Assumption . and Assumption . hold and let ∈ H ( ,T ; X c ) ∩ U be a locally optimal control for (P s ) with associated state z = S s ( ). Then there exists a unique adjoint state φ ∈ L ( ,T ; H ) such that the following optimality system is ful lled If enjoys extra regularity, namely Remark . . The exemplary objective functionals in Section are precisely of the form in ( . ).
Proof of Theorem . .
This together with Lemma . shows that (z, , φ) ful lls the optimality system ( . Corollary . . Let Assumption . and Assumption . hold. Then¯ ∈ H ( ,T ; X c ) ∩ U with associated state z = S s ( ) ful lls ( . ) if and only if there exists an adjoint state φ ∈ L ( ,T ; H ) such that (z, , φ) satis es the optimality system ( . ).
Proof. The proof of Theorem . already shows that ( . ) implies the optimality system in ( . ).
To prove the reverse implication, assume that (z, , φ) ful lls the optimality system ( . ). Then choose an arbitrary h ∈ H ( ,T ; H ), de ne η S s ( )h, and use the fact that φ is the weak solution of ( . b) to obtain ( . ). This together with ( . c) nally give ( . ).
Example . . Under suitable additional assumptions, it is possible to further simplify the gradient This type of objective will also appear in the application problem in Section . Then ( . c) becomes Note that X c as a Hilbert space satis es the Radon-Nikodým-property. Since X c → X, we may identify R * A s (R − Qz) * φ with an element of L ( ,T ; X c ), too, which we denote by the same symbol. Then, if we choose h(t) = ψ (t) ξ with ψ ∈ C ∞ c ( ,T ) and ξ ∈ X c arbitrary, we obtain Now, since ξ ∈ X c was arbitrary, we nd that the second distributional time derivative of is a regular distribution in L ( ,T ; X c ), i.e., ∈ H ( ,T ; X c ), satisfying for almost all t ∈ ( ,T ) Since X c is supposed to be a Hilbert space, we can apply integration by parts to ( . ). Together with ∈ U = { ∈ H ( ,T ; X) : ( ) = } and ( . ), this implies the following boundary conditions: where we again identi ed ∂ Ψ (z(T ), (T )) ∈ X * c with its Riesz representative. In summary, we have thus seen that the gradient equation in ( . c) becomes an operator boundary value problem in X c , namely ( . )-( . ).

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The next section is devoted to the derivation of second-order su cient optimality conditions for the regularized problem (P s ). As it was the case for the rst-order conditions, the main part concerns the di erentiability properties of the control-to-state map S s and the reduced objective, to be more precise to show that these are twice continuously Fréchet di erentiable. For this purpose, we need the following sharpened assumptions on the objective and the regularized operator A s : Assumption . .
(ii) The Fréchet-derivative A s is Lipschitz continuous from Y to L(Z; Z). Moreover, for every y ∈ Y, A s (y) can be extended to an element of L(W; W). The mapping arising in this way is Lipschitz continuous from Y to L(W; W). Furthermore, there is a constant C > such that A s (y)w W ≤ C w W hold for all y ∈ Y and all w ∈ W.
(iii) A s is Fréchet di erentiable from Y to L(Z; W). For all y ∈ Y, its derivative A s (y) can be extended to an element of L(Z; L(Z; W)) and the mapping y → A s (y) is continuous in these spaces. Moreover, there exists a constant C such that A s (y)[z , z ] W ≤ C z Z z Z for all y ∈ Y and all z , z ∈ Z.
Remark . . We point out that a second norm gap arises in Assumption . , since A s is only Fréchet di erentiable as an operator with values in W and not in Z → W. This assumption is again motivated by the application problem in Section . The example given there demonstrates that such as second norm gap is indeed necessary in general, since, given a concrete application, one cannot expect A s to be twice Fréchet di erentiable in Y, and even not as an operator from Y to Z.
The following proposition addresses the second derivative of the solution operator under the above assumptions. Its proof is in principle completely along the lines of the proof of Theorem . on the rst derivative of S. We therefore postpone it to Appendix .
where z S s ( ) ∈ H ( ,T ; Y) and η i S s ( )h i ∈ H ( ,T ; Z), i = , . Moreover, there exists a constant C such that Proof. Let , , h , h ∈ H ( ,T ; X) be arbitrary. We again abbreviate

With the help of
and Gronwall's inequality, we thus arrive at where we used the estimate in Theorem . , ( . ), and the Lipschitz continuity of S s by Lemma . in the Appendix.
If A s were Lipschitz continuous from Y to L(Z; L(Z; W)), then Lemma . would immediately imply the Lipschitz continuity of S s . However, to obtain the continuity of the second derivative, this additional assumption is not necessary as the following theorem shows: , then the solutions of all three equations can be shown to be continuously di erentiable in time with values in the respective spaces (Y, Z, and W, respectively). Moreover, the time derivatives of z and η are absolutely continuous and the same would hold for ξ , if A s were Lipschitz continuous. However, we did not exploit this additional regularity, since the original unregularized problem (P) does not provide this property in general.
With the above di erentiability result at hand, it is now standard to derive the following: Theorem . (Second-Order Su icient Optimality Conditions for (P s )). Assume that Assumption . , Assumption . , and Assumption . hold. Let (z, , φ) ∈ H ( ,T ; Y) × (H ( ,T ; X c ) ∩ U) × L ( ,T ; H ) be a solution of the optimality system ( . ). Moreover, suppose that there is a δ > such that where F is the reduced objective from ( . ). Then (z, ) is locally optimal for (P s ) and there exist ε > and τ > such that the following quadratic growth condition Proof. Thanks to the assumptions on and Theorem . , the chain rule implies that the reduced objective function F (·) = (S s (·), ·) : H ( ,T ; X c ) → R is twice continuously Fréchet di erentiable and, according to Corollary . , the equation in ( . ) holds for all h ∈ H ( ,T ; X c ) ∩ U. Since U is a linear subspace, the claim then follows from standard arguments, see e.g. [ , Satz . ].
Remark . . As already mentioned in Remark . , one could also account for additional control constraints. In this case, a critical cone would arise in the second-order conditions, cf. e.g. the survey article [ ].
Using the adjoint equation, the second derivative of the reduced objective in (SSC) can be reformulated as follows: Corollary . . Assume in addition to the hypotheses of Assumption . (iii) that A s (y)[z , z ] H ≤ C z Z z Z for all y ∈ Y and z , z ∈ Z, i.e., the last inequality in Assumption . holds in H instead of the weaker space W. Then it holds for all , h ∈ H ( ,T ; H ) that where z = S s ( ), η = S s ( )h, and φ solves the adjoint equation in ( . b).
Proof. Let us again abbreviate y = R − Qz. According to the chain rule, the second derivative of the reduced objective is given by is a bilinear form on H by assumption, we obtain that ξ ∈ H ( ,T ; H ). Therefore, we are allowed to test the adjoint equation in ( . b) (in its weak form) with ξ , which results in In the upcoming sections, we apply the analysis from the previous sections to an optimal control problem governed by a system of equations that arise as homogenization limit in elastoplasticity and was derived in [ , Theorem. . ]. It describes the evolution of plastic deformation in a material with periodic microstructure and formally (i.e., in its strong form) reads as follows: , one recovers the plastic strain from the internal variables z. The evolution of the internal variables is determined by a maximal monotone operator A: V → V . In Section . below, we present a concrete example for such an operator, namely the case of linear kinematic hardening with von Mises yield condition. Finally, z is a given initial state and π is the averaging over the unit cell, i.e., ( . ) π : Σ → Y Σ(·, y) dy |Y |ˆY Σ(·, y) dy .
The precise assumptions on these data as well as the precise notion of solutions to ( . ) are given below.
The volume force f : ( ,T ) × Ω → R d and the boundary loads : ( ,T ) × Γ N → R d , serve as control variables. In the following, we will frequently write for the tuple (f , ). Possible objectives could include a desired displacement or stress distribution at end time, i.e., where u d : Ω → R d and σ d : Ω → R d ×d sym are given desired displacement and stress eld, respectively, α, β ≥ , and Φ is a regularization term depending on the choice of the control space that will be speci ed below, see Remark . .

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Before discussing the optimal control problem, we rst have to introduce the precise notion of solution for homogenized elastoplasticity system in ( . ). For this purpose, we need several assumptions and de nitions. We start with the following Assumption . (Hypotheses on the data in ( . )).

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• Regularity of the domain: The domain Ω ⊂ R d , d ∈ { , }, is bounded with Lipschitz boundary Γ. The boundary consists of two disjoint measurable parts Γ N and Γ D such that Γ = Γ N ∪ Γ D .
While Γ N is a relatively open subset, Γ D is a relatively closed subset of Γ with positive measure. In addition, the set Ω ∪ Γ N is regular in the sense of Gröger, cf. [ ].
• Assumptions on the coe cients: The elasticity tensor and the hardening parameter satisfy C ∈ L ∞ (Ω × Y ; L(R d ×d sym )) and B ∈ L ∞ (Ω × Y ; L(V)) and are symmetric and uniformly coercive, i.e., there exist constants c > and b > such that In addition, B ∈ L ∞ (Ω × Y ; L(V; R d ×d sym )) is a given linear mapping.
Next, we de ne the function spaces for the various variables in ( . ): Definition . (Function spaces). Let s ∈ [ , ∞). For the quantities in ( . ), we de ne the following spaces: • space for the macro displacement u: • space for the micro displacement : For the latter, we denote by C ∞ per (Y ; R d ) the space of C ∞ (R d ; R d ) functions which are Y -periodic, identi ed with their restriction on Y , and de ne W ,s per (Y ; R d ) to be the closure of C ∞ per (Y ; R d ) with respect to the W ,s (Y ; R d ) norm. Further, W ,s ⊥ (Y ; R d ) is the closed subspace of W ,s (Y ; R d ) consisting of functions of mean , and We set the norm on V s to be with which V s becomes a Banach space and for the case s = a Hilbert space with the obvious scalar product.
Assumption . (Maximal monotone operator). The maximal monotone operator A from the evolution law in ( . d) is a set-valued map in the Hilbert space H = Z , i.e., A: Z → Z . It is assumed to satisfy our standing assumptions from Section . Moreover, we assume that there is a sequence of operators {A n } from Z to Z satisfying Assumption . . In Section . below, we will investigate the maximal monotone operator arising in the case of linear kinematic hardening with von Mises yield condition and show how to construct the approximating sequence of smooth operators for this particular case. With the above de nitions at hand, we are now in the position to de ne our precise notion of solutions to ( . ): Definition . (Weak solutions). Let ∈ H ( ,T ; (U ) * ) and z ∈ Z . Then we say that a tuple is a solution of ( . ), if, for almost all t ∈ ( ,T ), there holdŝ where π : S → L (Ω; R d ×d sym ) is the average mapping from ( . ) and In the following, we will frequently consider π − r in di erent domains and ranges, for simplicity denoted by the same symbol.
. In the following, we reduce the system ( . ) to an equation in the internal variable z only and it will turn out that this equation has exactly the form of our general equation ( . ). To this end we proceed analog to [ , Chapter ]. For this purpose, let us de ne the following operators: Definition . . Let s ∈ [ , ∞). Then we de ne For its adjoint, we write With a slight abuse of notation, we denote these operators for di erent values of s always by the same symbol. Lemma . . Let Assumption . be ful lled. Then there is an indexs > such that, for every s ∈ [s ,s] and every (f, g) ∈ (U s ) * × (V s ) * , there exists a unique solution (u, ) ∈ U s × V s of and there is a constant C s > , independent of f and g, such that H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . .
Proof. The claim is equivalent to − Div (x, y ) C∇ s (x, y ) being a topological isomorphism between U s × V s and its dual space (U s ) * × (V s ) * for s ∈ [s ,s]. We start with the case s = . For this, the left hand side of ( . ) gives rise to a bilinear form b on the Hilbert space U Clearly, b is bounded. Due to Poincaré's inequality for functions with zero mean value, which implies it is also coercive so that the claim and isomorphism property for − Div (x, y ) C∇ s (x, y ) for s = follows from the Lax-Milgram lemma.
We next extrapolate this isomorphism property to U s × V s for s around using the fundamental stability theorem by Šneȋberg [ ]. (See also the more accessible and extensive [ , Appendix A].) More precisely, we show that the spaces U s × V s and their duals form complex interpolation scales in s. Then the stability theorem shows that the set of scale parameters s such that − Div (x, y ) C∇ s (x, y ) is a topological isomorphism between U s × V s and its dual space is open. Since the set includes , as seen above, this then implies the claim.
To establish the interpolation scale, it is enough to consider the primal case, since the dual interpolation scale is inherited from the primal one by duality properties of the complex interpolation functor [ , Theorem . . ]. So, we show that for all s , s ∈ ( , ∞) and θ ∈ ( , ). It is moreover su cient to consider each component in the interpolation separately. For the U s = W ,s D (Ω; R d ) spaces, the interpolation scale property is well known by now in the setting of Assumption . and even much more general ones; we refer to [ ]. The result for V s is proven by reducing the problem to the W ,s per,⊥ (Y ; R d ) spaces and showing that these are complemented subspaces of W ,s (Y ; R d ) and thus inherit the latter's interpolation properties. This is done in the appendix, Theorem . , and nishes the proof.
Remark . . In general, one cannot expects to be signi cantly larger than , due to both the irregular coe cient tensors and the mixed boundary conditions, see e.g. [ , , ]. This issue will become crucial in the discussion of second-order necessary optimality conditions in . below. Now we are in the position to reduce ( . ) to an equation in the variable z only. For this purpose, we need the following Definition . (Q and R for the case of homogenized plasticity). Let s ∈ [s ,s] be given. By Lemma . , the solution operator associated with ( . ), denoted by is well de ned, linear and bounded. The components of G are abbreviated by Based on this solution operator, we moreover de ne H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . .

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Now, we have everything at hand to de ne the mappings R and Q from our general equation ( . ) for the special case of homogenized plasticity: Again, with a slight abuse of notation, we denote all of the above operators for di erent values of s ∈ [s ,s] always by the same symbol.
The reason for de ning the operators Q and R in the way we did in De nition . is the following: Owing to Lemma . , given z ∈ Z , one can solve ( . a)-( . c) for u, , and Σ so that the tuple (u, , Σ) ∈ U ×V ×S is uniquely determined by z. Even more, using the operators from De nition . , we see that the solution of ( . a)-( . c) for given z is Inserting the last equation in ( . d) and employing the de nition of Q and R in ( . ) and ( . ) then yields i.e., exactly an evolution equation of the general form in ( . ). This shows that the system ( . ) of homogenized elastoplasticity can equivalently be rewritten as an abstract operator evolution equation of the form ( . ).
For the di erentiability properties needed in sections and , a norm gap is required such that it is no longer su cient to consider just the Hilbert space H = Z . In accordance with the de nitions of R and Q, we therefore de ne the spaces in the abstract setting in our concrete application problem as follows: Definition . (Spaces in case of homogenized plasticity). The spaces Y, Z, H , and W from Section are set to The integrability indices s , s , and s depend crucially on the di erentiability properties of the regularized version of A and will be speci ed for a concrete realization of A in Section . below. Moreover, we choose Furthermore, the control space is given by Due to s ≥ , X c is re exive and embeds compactly in X by Sobolev embedding and trace theorems. Therefore, all our standing assumptions on the spaces in Section are ful lled. Of course, elements in X c are identi ed with those in X by In order to apply our general theory to the present setting, we need the following assumption on the regularity of the linear equation ( . ). As we will see in subsections . and . below, this assumption may become fairly restrictive, if one aims to establish second-order su cient optimality conditions, since, in this case, s and the conjugate index s may be rather large. where s and s are the integrability indices from ( . ). Proposition . . Under Assumption . and with the spaces de ned in De nition . , the operators R and Q from ( . ) and ( . ), respectively, satisfy the standing assumptions from Section , that is, R is linear and bounded from (U s ) * to Z s and Q is a linear and bounded operator from Z s to Z s for all s ∈ [s , s ] and, considered as an operator in Z , coercive and self-adjoint.
Proof. The required mapping properties of Q and R directly follow from their construction in De nition . in combination with Lemma . and Assumption . , respectively. It remains to show that Q is coercive and self-adjoint. Since B is symmetric and coercive according to Assumption . , it is su cient to prove that the operator B CB − T : Z → Z is symmetric and positive. To prove the symmetry, rst observe that B CB is symmetric by the symmetry of C. The symmetry of C moreover implies that G : (U ) * × (V ) * → U × V , i.e., the solution operator of ( . ), is self-adjoint. Therefore, the construction of T in De nition . implies for all z , z ∈ Z that so that T is also symmetric. To show the positivity of B CB − T , let z ∈ Z be arbitrary. To shorten the notation, we abbreviate (u z , z ) G(− Div (x, y ) (CBz)). Then, by testing the equation for (u z , z ), i.e., ( . ) with (f, g) = − Div (x, y ) (CBz), with (−u z , − z ), we arrive at Since, by construction, T z = B C∇ s (x, y ) (u z , z ), the coercivity of C therefore implies As z was arbitrary, this proves the positivity.
We point out that the whole analysis in sections and is carried out in the Hilbert space H = Z . Therefore, for the mere existence and approximation results from these two sections, the critical regularity condition in Assumption . is automatically ful lled by setting s = s = s = (so that Y = Z = W = H = Z ). Note that, in this case, the Lax-Milgram lemma guarantees the assertion of Assumption . without any further regularity assumptions, see the proof of Lemma . . The additional crucial regularity assumption only comes into play, when rst-and second-order optimality conditions are investigated, see Remark . . In Section . below, we will elaborate in detail, where the critical Assumption . is needed to ensure the required di erentiability properties of the regularized control-to-state map for the example of a speci c yield condition.
We collect our ndings so far in the following Theorem . (Homogenized plasticity as abstract evolution VI). Under the Assumptions . and . , the system of homogenized elastoplasticty in its weak form in ( . ) is equivalent to an abstract operator di erential equation of the form In addition, Q and R satisfy the standing assumptions from Section (provided that the function spaces are chosen according to De nition . ). Therefore, the existence and approximation results of Sections and hold for ( . ), in particular: • For every ∈ U(z , D(A)), there is a unique solution (u, , z, Σ) of the weak system of homogenized plasticity in ( . ), cf. Theorem . .
• Optimal control problems governed by the weak system of homogenized elastoplasticity admit globally optimal solutions, provided that the standing assumptions on the objective are ful lled, cf. Theorem . .
• The approximation results of Theorem . and Corollary . apply in case of homogenized elastoplasticity.
Remark . . Existence and uniqueness of solutions to ( . ) was already established in [ ].
Remark . . The example for objective functionals mentioned above, i.e., with u D ∈ L (Ω; R d ) and σ D ∈ L (Ω; R d ×d sym ) and α, β ≥ , γ > , satis es the standing assumptions on the objective functional, as we will see in the following. In this case, the functional Ψ : (z, ) → R in the general setting consists of the two integrals at end point T . Let us consider the rst one containing the displacement u. Since the latter is given by the rst component of the solution operator of ( . ), which maps H ( ,T ; Z ) × H ( ,T ; X c ) to H ( ,T ; U ) → C([ ,T ]; L (Ω; R d )), this integral is well de ned. (Note that the operators in ( . ) just act pointwise in time and the time regularity of z and carries over to u and .) Clearly, this solution operator is linear and bounded. In case of the second integral involving Σ, one can argue completely analogously based on the solution operator of ( . ). Thus, Ψ is convex and continuous, hence weakly lower semicontinuous, and in addition, bounded from below by zero. Moreover, the purely control part of the objective is given by Φ(f , ) = γ (f , ) H ( ,T ;X c ) and therefore clearly weakly lower semicontinuous and coercive as required. Thus all standing assumptions are ful lled as claimed. Of course, various other objective functionals are possible as well, such as tracking type objectives over the whole space-time-cylinder, but to keep the discussion concise, we just mention the example above.
. In the following section, we establish necessary and su cient optimality conditions for the optimal control of regularized homogenized elastoplasticity. To be more precise, we consider a single element of the sequence of regularizations of the maximal monotone operator A from Assumption . , which we again denote by A s , and apply the general theory from Section and Section . In view of the norm gap needed for the di erentiability of A s , we will consider A s in di erent domains and ranges (denoted by the same symbol) and assume that A s maps Y = Z s to itself. Accordingly, we treat the regularized version of the state equation (with A s instead of A) in the same manner, i.e., with integrability index s instead of , see ( . ) below. To keep the discussion concise, we moreover assume in all what follows that s ≥ is such that p = r = satisfy the conditions in ( . ). For d = dim(Ω) = , this implies s < and, in case without boundary control, i.e., ≡ , s < is su cient. This will become important in the discussion of second-order su cient conditions, as we will see below. Motivated by ( . ), we H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . .

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consider an optimal control problem of the form Since, in many applications, displacement and stress on the macro level are of special interest, especially at end time, we focus on objectives with this particular structure with continuously Fréchet di erentiable mappings In order to apply our general theory, we not only have to reduce the state system to an equation of the form ( . ), but also have to reduce the objective. For this purpose, let us denote the solution operators of ( . ) and ( . ) by u : ( , z) → u and S : ( , z) → Σ. To shorten the notation, we will consider u and S with di erent domains and ranges, e.g. u : (U s ) * × Z s → U s and u : L ( ,T ; (U s ) * ) × L ( ,T ; Z s ) → L ( ,T ; U s ) with s ∈ [s ,s] and analogously for S. Note again that the time regularity of z and directly carries over to the time regularity of u and Σ. Given these operators, we de ne so that the objective in ( . ) becomes i.e., exactly an objective of the form in ( . ) and ( . ), respectively. Since u and S are linear and bounded and F and F are assumed to be continuously Fréchet di erentiable, the chain rule implies the di erentiability of Ψ and Ψ so that Assumption . (i) is met, if we set s = so that W = Z . To apply the results of Section in order to establish an optimality system for ( . ), we additionally need that A s satis es Assumption . (ii), which is ensured by the following Assumption . . We set s = s = , i.e., W = Z = Z , and assume that A s ful lls Assumption . (ii) with Y = Z s , s ≥ , i.e., in particular that A s is Fréchet di erentiable from Z s to Z .
In light of Lemma . , Assumption . does not impose any restriction for practical realizations of A s , as we will see in Section . below. Given this assumption, Theorem . and Example . imply for a locally optimal solution ( ) = (f , ) with associated optimal internal variable z: .

( . c)
Then, owing to the precise structure of R, Q, Ψ , and Ψ , this leads us to the following H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . .
Theorem . (KKT-system for optimal control of homogenized plasticity). Let Assumption . be satis ed and assume that A s ful lls Assumption . . Suppose moreover that the regularity condition in Assumption . is satis ed, i.e.,s ≥ s . Then, if (f , ) ∈ H ( ,T ; L (Ω)) × H ( ,T ; L (Γ N )) is locally optimal for ( . ) with associated state (u, , z, Σ) ∈ H ( ,T ; U s )×H ( ,T ; V s )×H ( ,T ; Z s )×H ( ,T ; S s ), then there exists an adjoint state such that the following optimality system is satis ed: State equation: Remark . . A passage to the limit w.r.t. the regularization in order to obtain an optimality system for the original optimal control problem involving the maximal monotone operator A would of course be of particular interest. The results of [ ] however indicate that the optimality conditions obtained in this way are in general rather weak. In [ ], an optimal control problem governed by quasi-static elastoplasticity (without homogenization) is considered, which provides substantial similarities to ( . ). This system could also be treated by means of a reduction to the internal variable similar to our procedure for ( . ). In [ ] however, a time discretization followed by a regularization was employed for the derivation of rst-order optimality conditions. The reason for the comparatively weak optimality conditions obtained for the original (non-smooth) problem is the poor regularity of the dual variables in the limit, in particular the adjoint state. We expect a similar behavior in case of ( . ), when the regularization is driven to zero. This however is subject to future research.
Next, we turn to second-order su cient optimality conditions. Now, Ψ , Ψ , and A s have to ful ll Assumption . . For this purpose, we require the following Assumption . . In contrast to Assumption . , this assumption is very restrictive. If we assume that A s arises as a Nemyzki operator from a nonlinear function A s : V → V (for simplicity denoted by the same symbol), then the last condition in Assumption . , i.e., with W = Z , may only hold-even in case A s (z) ∈ L ∞ (Ω × Y ; L(V, V))-provided that s ≥ s = .
In order to have that A s is Fréchet di erentiable from Z s to Z s , we therefore need s > such that Assumption . indeed becomes very restrictive, see Remark . and Remark . below. Moreover, as described at the beginning of this subsection, if one sets r = , i.e., considers to boundary loads in H ( ,T ; L (Γ N ; R d )), then, in view of ( . ), s < has to hold (at least in three spatial dimensions) so that our second-order analysis cannot be applied in case of boundary controls (at least if r = and d = ). Therefore, we restrict to volume forces, i.e., controls in H ( ,T ; L (Ω; R d )) in what follows. Then, given all these assumptions, we can apply Theorem . and Corollary . to ( . ) to obtain the following Theorem . (Second-order su icient conditions for optimal control of homogenized plasticity). Let Assumptions . and . hold and suppose thats ≥ s so that Assumption . is ful lled. Assume moreover that f ∈ H ( ,T ; L (Ω; R d )) together with its associated state (u, , z, Σ) and an adjoint state (w, q, φ, ϒ, w T , q T , ϒ T ) satis es the optimality system ( . a)-( . i) (without ( . j) because boundary controls are omitted) and, in addition, that there exists δ > such that is the solution of the linearized state system associated with h, i.e., − Div (x, y ) η Σ = (h, ), Then f is a strict local minimizer of ( . ) and satis es the quadratic growth condition ( . ).
Remark . . As indicated above, Assumption . in combination with Assumption . is very restrictive. One can however weaken these assumptions, if the objective provides certain properties. Let us for instance consider an objective of the form with a twice Fréchet di erentiable functional In this case, it is su cient to choose s such that u : (z, ) → u maps W × X c with W = Z s to W ,p (Ω; R d ) → L (Ω; R d ), i.e., p ≥ / for d = . Moreover, as seen above, in order to have that the Nemyzki operator A s ful lls ( . ), we need s > s . Thus, we are tempted to choose s as small as possible. However, the crucial, limiting condition is the regularity assumption in Assumption . involving the conjugate exponent, i.e.,s ≥ max{s , s }, and this leads to the following equilibration H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . .
), page of of s and s in the case d = : s > , s = / (such that s = ). Then, in view of ( . ), u maps Z s to W ,s (Ω; R d ), which is continuously embedded in L (Ω; R d ) for d ≤ as desired. In the next subsection, we will present an example for a Nemyzki operator A s ful lling all assumptions for s arbitrarily close to . But nonetheless,s > in Assumption . is still a rather restrictive assumption and will not be satis ed in general (depending on the regularity of C and the boundary of Ω). This shows that the second-order analysis for problems of type (P) (resp. its regularized counterparts, to be precise) is in general a delicate issue, mainly due to the quasi-linear structure of the state equation. .
In the following, we will discuss a concrete realization of the maximal monotone operator A and its regularization, respectively, in order to demonstrate how the Assumptions . , . , and . can be satis ed in practice and how restrictive they are, in particular Assumption . . We consider the case of linear kinematic hardening with von Mises yield condition, cf. [ ] for a detailed description of this model. In this case, the nite dimensional space for the internal variable is given by V = R d ×d sym and B : R d ×d sym → R d ×d sym is the identity so that Z = S . Moreover, A is the convex subdi erential of the indicator functional I K of following set of admissible stresses where τ D τ − d tr(τ )I is the deviator of τ ∈ R d ×d sym and σ denotes the initial uni-axial yield stress, a given material parameter. The domain of A = ∂I K is trivially K, which is closed and convex. Furthermore, it is easily seen that A (τ ) = for all τ ∈ D(A) = K so that all of our standing assumptions are ful lled in this case. Note moreover that A satis es Assumption . so that the second approximation result on the convergence of the optimal states in Corollary . applies in this case. For the Yosidaapproximation of ∂I K , one obtains where π K denotes the projection on K in Z , i.e., π K (σ ) arg min τ ∈K τ −σ Z . Herein, with a slight abuse of notation, we denote the Nemyzki operator in L ∞ (Ω × Y ) associated with the pointwise maximum, i.e., R r → max{ , r } ∈ R, by the same symbol. In addition, we set max{ , − σ /r } , if r = .
The precise form of A λ in ( . ) immediately suggests the following regularization of the Yosida approximation: where max ϵ is a regularized version of the max-function, depending on a regularization parameter ϵ > . To be more precise, max ϵ : L ∞ (Ω × Y ) → L ∞ (Ω × Y ) is the Nemyzki operator associated with a real valued function (again denoted by the same symbol) with the following properties: . For every ϵ > , there holds max ϵ ∈ C (R), . max ϵ (r ) = max{ , r } for |r | ≥ and every < ϵ ≤ / , Example . . An example for a function satisfying the above conditions is One easily veri es that max ε is twice continuously di erentiable and that | max ε (r ) = max{ , r }| ≤ ϵ.
H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . . Lemma . . Let {λ n } n ∈N ⊂ R + and {ϵ n } n ∈N ⊂ R + be null sequences satisfying and de ne A n A λ n ,ϵ n : Z → Z . Then, the sequence {A n } n ∈N satis es Assumption . . Thus, Assumption . is ful lled in this case so that the approximation results from Theorem . apply.
Proof. Based on our assumptions on max ϵ , we nd for every τ ∈ Z and all n ∈ N such that ϵ n ≤ / The coupling of ϵ n and λ n in ( . ) then implies that ( . ) is ful lled.
Remark . . We point out that we neither claim that the coupling of λ and ϵ in ( . ) is optimal nor that our regularization approach is the most e cient one for this speci c ow rule.
Let us now x n ∈ N and set λ s λ n , max s max ϵ n , and A s A n . As before, we will denote the Nemyzki operators generated by max s and its derivatives by the same symbol. The following result can be proven as in [ , Prop. . ] by using [ , Theorem ]: Lemma . . Let s > r ≥ be arbitrary. Then A s is continuously Fréchet di erentiable from Z s to Z r and its directional derivative at τ ∈ Z s in direction h ∈ Z r is given by As a consequence of this result, we obtain the following Corollary . . Assumption . is ful lled by setting s s, wheres > is the exponent, whose existence is guaranteed by Lemma . . Thus, in case of linear kinematic hardening with von Mises yield condition and the regularization introduced above, the optimality condition in Theorem . are indeed necessary for local optimality without any further assumptions (except our standing Assumption . ).
Proof. We have to verify Assumption . (ii) for Y = Zs and Z = H = Z . The Fréchet di erentiability from Zs to Z follows from Lemma . . Moreover, the (global) Lipschitz continuity of A s in Zs can be deduced from the smoothness of max s and the condition max s (r ) = max{ , r } for all |r | ≥ / . The latter condition also guarantees that A s (y)h Z ≤ C h Z for all y ∈ Zs and all h ∈ Z . This completes the proof. Furthermore, following the lines of [ ] and [ , Theorem ], one proves the following Lemma . . For every s > and ≥ r < s/ , A s is twice Fréchet di erentiable and its second derivative at τ ∈ Z s in directions h , h ∈ Z r is given by Corollary . . The conditions on A s in Assumption . are satis ed, if the indexs from Lemma . ful lls s > . Proof. If we set s =s > ,s ∈] , s [, and s = , then Lemma . implies the di erentiability conditions in Assumption . (iii) with Y = Z s , Z = Z s , and W = Z s . The Lipschitz continuity of A s from Z s to L(Z s ) as well as the estimate A s (y)w Z ≤ C w Z follow from the condition max s (r ) = max{ , r } for all |r | ≥ / . This condition also ensures that A s (y)[z , z ] Z ≤ C z Z z Z , which in turn implies the last condition in Assumption . (iii) thanks to s > .
Remark . . As indicated in Remark . , the assumptions > is very restrictive. However, if W = Z , then any Nemyzki operator is only twice Fréchet di erentiable from Y to W, if Y = Z s with s > , see e.g. [ ] and the references therein. In light of this observation, the above regularization is rather well-behaved. As explained in Remark . , one can reduce the value of s , if only the L -norm of the displacement appears in the objective. However, one still needss > in this case, which is not guaranteed by Lemma . in general. But again, one can show that anys > is su cient for our concrete example, no matter how close it is to . This concrete realization of A s thus allows for the weakest possible regularity assumptions.
Before we are in the position to show that S is twice Fréchet di erentiable, we need the following result on the Lipschitz continuity of the rst derivative, which is also needed in the proof of Lemma . . Proof of Proposition . . Let , , h ∈ H ( ,T ; X) be arbitrary and abbreviate Using the rst Lipschitz-assumption in Assumption . (ii), we deduce for almost all t ∈ [ ,T ] that Gronwall's inequality and the de nition of y and y then yield where we used Lemma . and the estimate in Theorem . . Now, we are ready to prove that the solution operator is twice Fréchet-differentiable. The proof is based on an estimate of the remainder term and thus similar to the one of Theorem . .

Proof.
The proof is similar to the one of Theorem . . Let , h , h ∈ H ( ,T ; X) be arbitrary and de ne z We rst address the existence of solutions to ( . ). We argue similarly to Lemma . and set w : From the estimate in Assumption . (iii) it follows that H. Meinlschmidt, C. Meyer, S. Walther Optimal control of an abstract evolution variational . . . and, since the limit of Bochner measurable functions is Bochner measurable, we obtain w ∈ L ( ,T ; W).
Since A s (y) is assumed to be bounded in W by . (ii), we can now follow the proof of Lemma . (with W instead of Z) to deduce the existence of a unique solution ξ ∈ H ( ,T ; W) of ( .
so that Gronwall's inequality and the estimate in Theorem . give This shows also ( . ) (after having proved that ξ = S s ( ) [h , h ]). It only remains to prove the remainder term property. To this end, we de ne Then, the equations for η , , η , and ξ lead to where r (y; ζ ) A s (y +ζ )−A s (y)−A s (y)ζ ∈ L ( ,T ; L(Z; W)) denotes the corresponding remainder term. The estimate in Assumption . (iii) thus implies for almost all t ∈ [ ,T ] such that Gronwall's inequality yields Hence, thanks to the Fréchet di erentiability of A s : Y → L(Z; W), we have for almost all t ∈ [ ,T ] . Furthermore, using the Lipschitz continuity of A s : Y → L(Z; Z), the estimate for A s in Assumption . (iii) and again ( . ), we deduce for almost all t ∈ [ ,T ]. The convergence in ( . ) now follows from Lebesgue's dominated convergence theorem.

V s
We prove that the spaces V s = L s (Ω;W ,s per,⊥ (Y ; R d )) de ned in Section form a complex interpolation scale in s. This result is a cornerstone in the proof of Lemma . . Lemma . . Let θ ∈ ( , ) and s , s ∈ ( , ∞) and set s = −θ s + θ s . Then Proof. The proof relies on the complemented subspace interpolation theorem [ , Theorem . . ] which essentially says that one can transfer interpolation properties to complemented subspaces provided there exists a common projection onto these subspaces on all involved spaces.
In this spirit, we rst consider a larger regular domain Y # ⊃ Y which includes a nite open covering of Y , and, for all < r < ∞, identifyW ,r per (Y ; R d ) isomorphically with the closed subspaceW ,r per,Y (Y # ; R d ) of W ,r (Y # ; R d ) consisting of periodic extensions of W ,r per (Y ; R d )-functions. This is possible since the periodic extension of a W ,r per (Y ; R d ) function will preserve the Sobolev regularity [ , Proposition . ]. We next argue that there exists a projection P per mapping W ,r (Y # ; R d ) onto W ,r per,Y (Y # ; R d ). (We will not give a detailed proof of this since the details are somewhat tedious and lengthy.) To this end, we rst wrap u ∈ W ,r (Y # ; R d ) around the torus T d in a smooth manner using a xed smooth partition of unity on T d derived from the open covering of Y , and then pull it back. One checks that this indeed yields a function P per u which is periodic on Y . Moreover, P per is in fact a continuous linear operator on W ,r (Y # ; R d ), which in addition acts as the identity on the periodic extensions of C ∞ per (Y ; R d ). This implies that P per is indeed the searched-for projection of W ,r (Y # ; R d ) onto W ,r per,Y (Y # ; R d ). Here, the interpolation of W ,r (Y # ; R d ) is classical since we have assumed Y # to be regular. Overall, this gives the claim.
Proof. We again argue via the complemented subspace interpolation theorem. For every < r < ∞, the operator as desired.

Proof.
We argue by contradiction. Assume that there exists ε > and a strictly monotonically increasing function n : N → N, such that for all k ∈ N there exists x k ∈ M with ε ≤ d N (G n(k) (x k ), G(x k )) for all k ∈ N. Since M is compact, we can extract a subsequence x k j of x k such that x k j → x in M, thus d N (G n(k j ) (x k j ), G(x k j )) ≤ d N (G n(k j ) (x k j ), G n(k j ) (x)) + d N (G n(k j ) (x), G(x k j )) ≤ Ld M (x k j , x) + d N (G n(k j ) (x), G(x k j )) → , which gives the contradiction.
Lemma . . Let M be a compact metric space and N a metric space. Furthermore, let {G n } n ∈N ⊂ C(M; N ), G ∈ C(M; N ) with G n → G in C(M; N ). De ne U n G n (M) and U G(M). Then the set U ∪ ∞ n= U n is compact.
Proof. Let {y k } k ∈N ⊂ U . Since a nite union of compact sets is compact, we can assume that there exists a subsequence {y k j } j ∈N and a strictly monotonically increasing function n : N → N, such that y k j ∈ U n(j) . Then there exists a sequence {x j } j ∈N ⊂ M, with y k j = G n(j) (x j ). Because M is compact we can select a subsequence, again denoted by x j , and a limit x ∈ M, such that x j → x, hence, d N (y k j , G(x)) ≤ d N (y k j , G(x j )) + d N (G(x j ), G(x)) → , thus the proof is complete.