High-order homogenization in optimal control by the Bloch wave method

This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient $A^\varepsilon$. The small parameter $\varepsilon>0$ denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error $O(\varepsilon^M)$, where $M\in\mathbb{N}$ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities.


Introduction
Many modern key technologies call for mathematical modeling through partial differential equations (PDEs) with macro-and microstructures. The simplest way to decode a microstructure is to consider periodic coefficients with a small periodicity length ε > 0. In such a situation, the central question concerns the effective or asymptotic behavior of the problem in the homogenization limit ε → 0. The periodic homogenization theory was mainly developed in the late '70s focusing on the effective behavior of elliptic PDEs (see [8,6,26]). Since then, the theory has been generalized to various types of equations and more complex models, including perforated domains [12], high contrast media, and singular geometries [9,23], just to mention a few.
The homogenization theory plays as well a profound role in the optimal control of PDEs with highly oscillatory coefficients. Such problems are particularly encountered in electrical and electronic engineering applications [28,29], which require a substantial extension of the developed techniques by the homogenization theory. We refer to [11,18,19,20,21] for the interplay of homogenization and optimal control based on weak convergence methods. However, those results are only able to characterize the behavior in the limit ε → 0. To the best of the authors' knowledge, neither convergence rates nor higher-order approximations are available in the literature. In the classical homogenization theory, higher-order approximations are achieved by adding highly oscillatory terms to the effective solution, the so-called correctors, which solve microscopic cell problems and characterize the oscillatory behavior of the original solution. Unfortunately, this strategy does not apply to optimal control problems, since the control functions are typically restricted to an admissible set, and adding correctors to the effective control function will destroy its admissibility. To circumvent this difficulty, we propose the use of a spectral method involving the so-called Bloch waves, which allows for a variety of L 2 -conditions in the admissible set. Examples are provided in Section 1.2 below. This paper is aimed at the high-order asymptotic behavior of a linear-quadratic elliptic optimal control problem with ε-periodic coefficients and ε-dependent admissible sets. More precisely, we consider a symmetric and uniformly elliptic coefficient matrix A ∈ L ∞ (R n ; R n×n ) with n ∈ N that is periodic with respect to the unit cube Y := [0, 1] n . Setting A ε (x) := A x ε , we introduce an energy functional E ε : H 1 (R n ) → R and a cost functional J ε : L 2 (R n ) × H 1 (R n ) → R as follows: and J ε (u, y) := µ 1 2 E ε (y − y ε d,1 ) + with fixed constants µ 1 , µ 2 ≥ 0 and κ > 0. Furthermore, y ε d,1 ∈ H 1 (R n ) and y ε d,2 ∈ L 2 (R n ) are given functions denoting, respectively, the desired gradient field and the desired state. In view of (1.2), our focus is set on the following linear-quadratic elliptic optimal control problem: min J ε (u ε , y ε ) over (u ε , y ε ) ∈ U ε ad × H 1 (R n ) s.t. − ∇ · (A ε (x)∇y ε (x)) + y ε (x) = f ε (x) + u ε (x) in weak sense on R n .
(P ε ) In the setting of (P ε ), f ε ∈ L 2 (R n ) is a given function, and ∅ = U ε ad ⊂ L 2 (R n ) is a convex and closed subset representing the admissible control set. By a classical argument [24,25], the optimal control problem (P ε ) admits a unique optimal solution denoted throughout this paper by (u * ε , y * ε ) ∈ U ε ad × H 1 (R n ). Let us emphasize that the admissible control set U ε ad and the quantities f ε , y ε d,1 , y ε d,2 feature highly oscillatory structures depending on ǫ. In particular, the numerical treatment of (P ε ) is extremely costly since the fine scale, represented by the small parameter ε > 0, has to be resolved. The underlying ε-dependence in (P ε ) will be specified in Assumptions 1.8 and 1.10.
More than two decades ago, Kesavan and Saint Jean Paulin [20] analyzed the effective behavior of a similar linear-quadratic elliptic optimal control problem. They disregarded the role of the desired state and specified the desired gradient field to be zero, i.e., µ 2 = 0 and y ε d,1 ≡ 0 in (1.2). However, differently from (P ε ), they allowed for a periodic coefficient B ε in the cost functional that may differ from the coefficient A ε in the state equation. The main contribution of [20] is the weak convergence for the optimal solution as ǫ → 0 towards the solution of an effective optimal control problem with an effective state equation. The effective coefficient A eff in the effective state equation corresponds to the classical effective coefficient from the periodic homogenization theory, while the effective coefficient B eff in the energy functional is a perturbation of the classical H-limit of B ε . In the case A ε = B ε , that is relevant for us, one obtains A eff = B eff . Unfortunately, since the convergence property holds solely in the weak topology, the approximation of the effective optimal control problem could be far from precise. Indeed, from the classical homogenization theory, it is well known that solutions to highly oscillatory problems exhibit highly oscillatory behavior, and the homogenization limit provides an appropriate average of these oscillations. In order to obtain higher-order approximations, one has to capture the fast oscillations by correcting the homogenization limit, wherefore results of this type are referred to as corrector results. The ultimate goal of this paper is therefore to set up a corrector result of the following form: Given M ∈ N, we seek for a proper approximation (u * M , y * M ) of the optimal solution (u * ε , y * ε ) to (P ε ) such that The key tool of our approach is the Bloch wave expansion, which is introduced in the upcoming section.

Bloch expansion
For the sake of completeness, we briefly motivate and collect well-known facts from the classical Bloch theory in the context of homogenization, which can be found, for instance, in [13,14,16,27]. The Bloch expansion is a generalization of the classical Fourier expansion. Every function u ∈ L 2 (R n ; C) can be written as whereû(ξ) := R n u(x)e −2πix·ξ dx denotes the classical Fourier-transform of u. Decomposing every ξ ∈ R n into ξ = k + η with k ∈ Z n and η ∈ Z := [−1/2, 1/2) n , one obtains where for every η ∈ Z the function F (·, η) is Y -periodic with Y := [0, 1] n . It can thus be expanded in the Y -periodic eigenfunctions to the following eigenvalue problem: By the classical spectral theory, one finds a sequence of eigenvalues (µ m (η)) m∈N 0 and corresponding Y -periodic eigenfunctions The subscript ♯ tags spaces of periodic functions. Furthermore we define Note that the quasi-periodic Bloch waves solve the eigenvalue problem This leads to the following classical result.
Proposition 1.2 (Bloch expansion). Every u ∈ L 2 (R n ; C) admits a unique expansion with the Bloch coefficientŝ is ε-periodic, one needs to introduce rescaled Bloch waves: The scaling is designed in such a way that A simple calculation provides the following rescaled version of the classical Bloch expansion.

Proposition 1.3 (Rescaled Bloch expansion).
Every u ∈ L 2 (R n ; C) admits a unique expansion In what follows, we will frequently exploit that due to the eigenvalue property of ω ε m the operator −∇ · (A ε ∇) acts on the rescaled Bloch expansion as a multiplier. More precisely, for u as in (1.10) one has It is well known that in the homogenization process only the lowest eigenvalue, the ground state λ 0 (η), is relevant. By classical perturbation theory it can be shown that λ 0 (η) is simple and analytic in a neighbourhood of η = 0. Moreover, one immediately finds that λ 0 (0) = 0 and that λ 0 is even, wherefore all odd derivatives of λ 0 vanish. Summing up, one obtains the following Taylor expansion: Lemma 1.4 (Taylor expansion of the first Bloch eigenvalue). For every k ∈ N there exists a 2k-th order tensor It is a well-known fact in the Bloch theory that the second order tensor A * 2 ∈ R n×n coincides with the effective coefficient A eff from the classical homogenization theory. As shown, for instance in [4] and [17], the tensors A * 2k can be computed from periodic cell problems. The above expansion plays a major role in our analysis.

Assumptions
In this section, we introduce the assumptions for the given data involved in (P ε ), namely, the admissible control set U ε ad ⊂ L 2 (R n ) and the functions f ε , y ε d,1 , and y ε d,2 . Roughly speaking, we have to demand that the data are well prepared to the periodic microstructure.
We recall that solutions to homogenization problems are characterized by some effective profile, the homogenization limit, and fast lower order oscillations, which adapt the effective profile to the periodic microstructure. These oscillatory corrections can be described by periodic cell solutions, the so-called correctors, which are necessary to construct high-order approximations as in (1.3). In fact, there is an analogue to this adaption process in the Bloch wave approach. As shown in [13,14,16], in a classical homogenization process only the lowest Bloch eigenvalue λ ε 0 and the corresponding eigenfunction ω ε 0 play a role, while higher Bloch modes are negligible. Moreover, it can be shown that the rescaled Bloch eigenfunction ω ε 0 is in fact an adaption of the Fourier basis function e 2πiη·x to the microstructure (see [4]). In view of this, the following definition is reasonable. Definition 1.5 (Adaption). Let f ∈ L 2 (R n ; C) with the Fourier-transformf . Then, the adaption A ε (f ) of f is defined as follows: Remark 1.6. In [14] A ε is referred to as the Bloch approximation. It is shown that A ε (f ) converges almost everywhere to f and, assuming that f is sufficiently smooth withf decaying sufficiently fast at infinity, a higher-order asymptotic expansion of where χ (1) , χ (2) are Y -periodic corrector functions. In particular, the adaption A ε transforms a smooth, non-oscillatory function f to an ε-oscillatory function according to the periodic microstructure.
In our approach we will use the fact that the adaption operator A ε preserves the L 2 -norm when applied to functions that have compact support in Fourier space. Lemma 1.7. Let f ∈ L 2 (R n ; C) have compact support in Fourier space, i.e., supp(f ) ⊂ K with K ⊂ R n compact. Let ε > 0 be such that K ⊂ Z/ε. Then A ε (f ) L 2 (R n ;C) = f L 2 (R n ;C) .
Proof. Let ε > 0 be such that K ⊂ Z/ε. Then where in the second equality we exploited thatf is supported on K ⊂ Z/ε and in the last equality we used Parseval's identity (1.12) and the definition of A ε .
Let us now state the main assumptions for the given data and the admissible set U ε ad involved in (P ε ): Assumption 1.8. There exist f, y d,1 , y d,2 ∈ L 2 (R n ) and a compact set K ⊂ R n such that (1.16) Moreover, the compact set K is assumed to be symmetric in the sense that (1.17) Remark 1.9.
(i) The condition that the Fourier transformsf ,ŷ d,1 ,ŷ d,2 have compact support implies readily that f, y d,1 , y d,2 ∈ H k (R n ) for all k ∈ N. Therefore, along with (1.17), it follows that their adaptions f ε , y ε d,1 , y ε d,2 are real-valued and belong to H k (R n ) for all k ∈ N, cf. [13,14]. Also, notice that Assumption 1.8 demands that f ε , y ε d,1 and y ε d,2 are highly oscillatory according to the micro-structure. Actually this assumption is quite natural as solutions to ε-periodic coefficient equations such as the state equation in (P ε ) typically exhibit a highly oscillatory behavior.
(ii) Relation (1.15) suggests that the difference A ε (f )−f is typically of order ε. For unprepared data, i.e., y ε d,1 = y d,1 , y ε d,2 = y d,2 , and f ε = f , the effective model from Definition 1.13 below is therefore expected to approximate the original problem only up to errors of order ε. In fact, well-prepared data are necessary for many corrector results in the homogenization theory. See, for instance, [5,7,10] in the context of periodic wave equations.
We now state the assumption on the admissible set U ε ad .
Assumption 1.10. Let K ⊂ R n be as in Assumption 1.8. For the admissible set U ε ad ⊂ L 2 (R n ), we assume the following: (i) U ε ad is convex and closed.
(iii) For every u ∈ U ε ad with Bloch expansion it follows that the projection is uniformly bounded as shown in Lemma 1.12. The third condition (iii) demands that for every ε > 0 the projection (U ε ad ) K of U ε ad in Bloch space to the lowest mode m = 0 and η ∈ K is a subset of U ε ad .
For the rest of the article, ε > 0 is always assumed to be sufficiently small such that K ⊂ Z/ε. Furthermore S ǫ : L 2 (R n ) → H 1 (R n ) denotes the control-to-state operator associated with (P ε ). Lemma 1.12. Let Assumptions 1.8 and 1.10 hold. Then, (u * ε ) ε>0 ⊂ L 2 (R n ) is uniformly bounded, i.e., there exists a constant C > 0, independent of ε, such that Proof. According to Assumption 1.10, there exists an element u 0 ∈ L 2 (R n ) such that A ε (u 0 ) ∈ U ε ad . For this reason, Our aim is to find a uniform bound for the right hand side of (1.19). Letû 0 be the Fourier transform of u 0 . By construction of A ε (Definition 1.5), Proposition 1.3 yields that the lowest-order Bloch-coefficient (m = 0) of A ε (u 0 ) corresponds to the Fourier transformû 0 , while all higher-order Bloch modes with m ≥ 1 vanish. In the following, let C > 0 denote a ε-independent generic constant that can vary from line to line. By Lemma 2.1 below, the right-hand side in (1.19) can be written as where in the first inequality we used that λ ε 0 ≥ 0. In the second inequality we used Parseval's identity (1.12) and exploited the fact that K |ŷ d,1 (η)| 2 (1 + λ ε 0 (η)) dη is controlled by the H 1 -norm of y d,1 (see [14] for details). Example 1. Let Assumption 1.8 hold. Then the following admissible control sets satisfy Assumption 1.10.
is convex and closed. The condition (ii) of Assumption 1.10 is satisfied with u 0 = 0. The third condition is a direct consequence of (1.18) and Parseval's identity (1.12).
(iii) Bounds on the L 2 -norm of the state: For L > 0 let As the control-to-state operator S ǫ : L 2 (R n ) → H 1 (R n ) is affine linear and continuous, the set U ε ad ⊂ L 2 (R n ) is convex and closed. The condition (ii) of Assumption 1.10 is satisfied with u 0 = −f since according to Assumption 1.8 (1.20) The third condition of Assumption 1.10 follows from the formula (2.1) and (iv) Bounds on the energy of the state: For L > 0 let Since by (1.1) the energy functional E ε : is again convex and closed. Thanks to (1.20), the choice u 0 = −f satisfies again the condition (ii) of Assumption 1.10. In view of (2.2) withŷ d,1 ≡ 0, the third condition (U ε ad ) K ⊂ U ε ad is again satisfied by Parseval's identity (1.12).
Note that pointwise conditions on u or y are not covered by our approach. However, since solutions to homogenization problems typically exhibit a highly oscillatory behavior, pointwise conditions are not appropriate in the framework of highorder homogenization.

Outline and main results
In the following we will expand the optimal solution (u * ε , y * ε ) of (P ε ) in rescaled Bloch waves according to Proposition 1.3. Our analysis consists of four steps: In the first step, see Proposition 2.2, we find that the optimal solution (u * ε , y * ε ) is in fact an adaption: In the second step, we simplify the (Fourier) expansions ofũ * ε andỹ * ε using the analyticity of the lowest Bloch eigenvalue. Based on this first approximation, in our third step, we derive an M-th order effective optimal control problem of the following form. Definition 1.13 (High-order effective optimal control problem). Let Assumptions 1.8 and 1.10 hold. Then, for every M ∈ N, we define the high-order effective cost functional J * M : L 2 (R n ) × H M (R n ) → R as follows: where the tensors A * 2k are as in Lemma 1.4 and , we consider the effective state equation and propose the following high-order effective optimal control problem: where the effective admissible control set U * ad is given by the inverse image of U ε ad under the adaption operator A ε projected onto K, In the fourth and final step, we show that (P * M ) admits for all sufficiently small ε > 0 a unique optimal solution (u * M , y * M ) ∈ U * ad × H 2M (R n ), which relies on a reformulation of the problem in Fourier space (see Proposition 2.6). As a main result, we prove in Proposition 3.3 and Corollary 3.2 that the optimal solution for some ε-independent constant C > 0. We recall that the optimal solution to (P ε ) Let us comment on the high-order effective optimal control problem (P * M ). This optimization problem still depends on ε. It can nevertheless be regarded as an effective model in the sense of the homogenization theory since the coefficients A * 2k in the effective cost functional J * M and the effective equation (1.23) are ε-independent. Note that, for M = 1, (P * M ) coincides with the classical homogenization limit obtained in [20]. For M > 1 it can be understood as a higher-order approximation. The effective admissible set U * ad may depend on ε. In Section 2.2 we determine U * ad corresponding to the admissible sets from Example 1. is well-posed for ε > 0 lying below a certain threshold ε M . Such a threshold can only be found if the right hand side of (1.23) has compact support in Fourier space. This is why the effective admissible set (1.24) is restricted to functions with compact support K in Fourier space. In Section 4 we provide an alternative M-th order effective optimal control problem which is well-posed independently of ε and which has the same approximation quality as the one from Defintion 1.13. In that situation the effective admissible set may be defined as (A ε ) −1 (U ε ad ). However, the effective optimal control still lies in the projected set U * ad from (1.24). The rest of of the paper is organized as follows. In Section 2.1 we derive and justify the effective model (P * M ), and in Section 2.2 we show that it admits a unique optimal solution for all sufficiently small ε > 0. Section 3 provides the central error estimates and proves the approximation property of (P * M ). In this context a highorder approximation for the adjoint state is obtained. In Section 4 an alternative effective problem is proposed, which is well-posed independently of ε and related to (P * M ) through an algebraic manipulation.

The effective optimal control problem
In this section, we justify the M-th order effective optimal control problem (P * M ). The key steps in the derivation are (1) rewriting the original problem in Bloch space and (2) expanding the lowest Bloch eigenvalue in a Taylor series (see Section 2.1). In Section 2.2 we prove that for all sufficiently small ε > 0, (P * M ) admits a unique optimal solution.

Derivation of the effective model
As the first step, using Proposition 1.3, we expand the optimal solution (u * ε , y * ε ) to (P ε ) in the rescaled Bloch waves as follows: We then specify the Bloch expansion of the optimal state y * ε (x) and rewrite the cost functional J ε in terms of the Bloch transform.
Lemma 2.1. Let Assumptions 1.8 and 1.10 hold. Then, for every control u ε ∈ U ε ad with Bloch coefficientsû ε,m , the corresponding state y ε := S ǫ u ǫ satisfies the following expansion:
In the next step, we show that for the optimal control u * ε all Bloch modes with m ≥ 1 or η ∈ (Z/ε) \ K vanish.

Proposition 2.2. Under the assumptions of Lemma 2.1, it holds that
Proof. Our aim is to show thatû * ε,m (η) = 0 for m ≥ 1 or η ∈ (Z/ε) \ K. Indeed, from the representation formula for J ε (Lemma 2.1), one directly concludes that for u ∈ U ε ad the projection u K in the sense of (1.18) satisfies with equality iff u = u K . Since by Assumption 1.10 the projection u K of every u ∈ U ε ad lies again in U ε ad , the (unique) optimal control satisfies u * ε = (u * ε ) K , which is the claimed result (2.8). The statement (2.9) is then a direct consequence of (2.8) and (2.1). Proposition 2.2 states that the optimal control of (P ε ) is an adaption u * ε = A ε (ũ * ε ), with the Bloch coefficientû * ε,0 supported in K. In order to construct a first approximation for y * ε , we exploit the formula (2.9) and the fact that the lowest Bloch eigenvalue λ 0 is analytic in a neighbourhood of η = 0 (cf. Lemma 1.4). In particular, for M ∈ N arbitrary, we can expand where the error is of order ε 2M uniformly in η ∈ K. With we propose the following approximation forỹ * ε : (2.14) Indeed, the above functionỹ * M,app provides a good approximation forỹ * ε in the following sense. Proposition 2.3. Let ε > 0 be such that 1 + P ε M (η) > 0 for all η ∈ K. Letỹ * ε be as in (2.11), and letỹ * M,app be defined through (2.14). Then there exists a constant C > 0 such that Proof. Using Parseval's identity for Fourier expansions one finds where for the second inequality we exploited that the error between P ε M and λ ε 0 is of order ε 2M uniformly in η ∈ K.
By a direct calculation, the approximationỹ * M,app is a solution to the following (2M)-th order constant coefficient equation whereũ * ε was defined in (2.10) satisfying u * ε = A ε (ũ * ε ). It is therefore reasonable to regard (1.23) as a candidate for an effective PDE-constraint. The effective energy (1.21) can be justified as follows. Let u ∈ U * ad be a function with compact support K in Fourier space, i.e., We readily know that the Bloch coefficientsû ε,m of the adaption A ε (u) ∈ U ε ad satisfŷ u ε,m = 0 for all m ≥ 1 andû ε,0 =û. Therefore, as a consequence of Lemma 2.1, we obtain (2. 16) In order to find a candidate for the effective cost functional, we proceed analogously to the derivation of the effective PDE-constraint by formally replacing all (1+λ ε 0 (η))terms in (2.16) by 1 + P ε M (η). This leads to the following candidate for the effective energyĴ * (2.17) By a direct calculation in Fourier space, the above functional is exactly the controlreduced objective functional associated with (1.21) taking into account the effective state equation (1.23).

Well-posedness of (P * M )
The goal of this section is to prove the existence of a unique optimal solution to (P * M ), provided that ε > 0 is sufficiently small. The main difficulty lies in the well-posedness of the effective equation where the highest-order operator A * 2M D 2M might have no or even the wrong sign. Indeed, while A * 2 is positive definite, for A * 4 it has been shown in [15] that this tensor is negative semidefinite. In particular, for M = 2, (1.23) reads as . This fourth-order equation is ill-posed for general right hand sides. However, in our setting, f and every controlũ ∈ U * ad have compact support K in Fourier space, wherefore for ε > 0 sufficiently small the effective equation can be uniquely solved. The well-posedness is based on the following algebraic observation and the fact that the first effective tensor A * 2 is positive definite, i.e., A * 2 · η ⊗2 ≥ λ|η| 2 for some λ > 0.
Lemma 2.4. Let K ∈ R n be a given compact set and let M ∈ N be fixed. Let λ > 0 be the ellipticity constant of A * 2 as above. Then there exists ε M > 0 such that for every 0 < ε < ε M and every N ∈ N with N ≤ M P ε N (η) ≥ λ 2 |η| 2 for all η ∈ K, (2.18) where P ε N is the polynomial defined in (2.13). Proof. Since A * 2 · η ⊗2 ≥ λ|η| 2 , it is sufficient to prove that there exists some ε M > 0 such that for every 0 < ε < ε M and every N ≤ M where A * 2k denotes an appropriate norm of the tensor A * 2k . Since η ∈ K varies in a bounded set, we can choose ε M > 0 such that for every 0 Then ε M satisfies the requirements of the lemma.
We emphasize that the threshold ε M depends on the compact set K, the order of approximation M and the effective coefficients A * 2 , . . . , A * 2M . With the above auxiliary lemma at hand we can now conclude the well-posedness of the effective Equation (1.23).
Proposition 2.5 (Well-posedness of (1.23)). Suppose that Assumption 1.8 holds. Let M ∈ N and let ε M be as in Lemma 2.4. Then, for every 0 < ε < ε M and every control u ∈ L 2 (R n ) with Fourier transformû satisfying supp(û) ⊂ K, it holds that: i) There exists a unique weak solution y ∈ H k (R n ) for all k ∈ N to the effective state equation (1.23).
ii) The solution y satisfies the a priori estimate Proof. To show existence, let us consider As a function with compact support in Fourier space, y belongs to H k (R n ) for all k ∈ N. By a direct calculation in Fourier space one finds that (2.21) is a solution to the effective equation (1.23). It remains to prove the a priori estimate (2.20), from which uniqueness directly follows. Therefore note that the Fourier transformŷ of every solution satisfies (1 + P ε M )ŷ =f +û. Testing the above relation withŷ, taking into accountf (η) =û(η) =ŷ(η) = 0 for η / ∈ K, and exploiting Lemma 2.4 provides that whereȳ denotes the complex conjugate ofŷ. By Parseval's identity one immediately concludes the desired estimate (2.20).
We have found that the PDE-constraint (1.23) is well-posed. To prove that the corresponding effective optimal control problem (P * M ) has a unique optimal solution, we reformulate (P * M ) in terms of the Fourier transform. In the same manner the original problem (P ε ) can be rewritten in terms of the Bloch transform, which is particularly important for the error analysis in Section 3.
Proposition 2.6 (Reformulation of the optimal control problems). Let Assumptions 1.8 and 1.10 hold, and let U * ad be the effective admissible set from (1.24). Then

(2.22)
Furthermore: (i) The optimal control problem (P ε ) is equivalent to  Proof. By virtue of (1.24), (1.18), and (U ε ad ) K ⊂ U ε ad , we have A ε (U * ad ) = (U ε ad ) K . Furthermore, since for functions with compact support in Fourier space the adaption A ε replaces the Fourier basis function e 2πiη·x in the expansion by the Bloch function ω ε 0 (x; η), it follows that u B = A ε (u F ). In particular the two sets in (2.22) are equal.
To prove (i), we note that by Proposition 2.2 the optimal control u * ε of (P ε ) lies in the projection (U ε ad ) K . This fact along with A ε (U * ad ) = (U ε ad ) K leads to The second claim (ii) follows in an analogous way.
Remark 2.7. Due to its quadratic form, the effective energyĴ * M : L 2 (K; C) → R is convex and continuous.
Finally we prove that the admissible setÛ ad defined in (2.22) is nonempty, convex, and closed. Together with Remark 2.7 this yields the well-posedness of the effective optimal control problem (P * M ) provided ε is sufficiently small, i.e., ε < ε M and K ⊂ Z/ε. Proof. The setÛ ad is nonempty since for every u ∈ U ε ad = ∅ the projection u K satisfies u K ∈ (U ε ad ) K with the Bloch coefficient belonging toÛ ad according to (2.22). In view of (2.22), the convexity ofÛ ad ⊂ L 2 (K; C) holds if U * ad ⊂ L 2 (R n ) is convex. By the linearity of the adaption A ε and the definition of U * ad (1.24), the convexity of U ε ad ⊂ L 2 (R n ) implies immediately the convexity of U * ad ⊂ L 2 (R n ). Due to Parseval's identity (1.12), strong convergence of functions in L 2 (R n ) is equivalent to strong L 2 -convergence of the corresponding Bloch transforms. Thus, it follows from (1.18), (U ε ad ) K ⊂ U ε ad , and the closedness of U ε ad ⊂ L 2 (R n ) that the projected set (U ε ad ) K is closed in L 2 (R n ), and so by (2.22) the closedness of U ad ⊂ L 2 (K; C) follows. Corollary 2.9. Let Assumptions 1.8 and 1.10 hold. Furthermore, let M ∈ N and let ε M be as in Lemma 2.4. Then, for every 0 < ε < ε M , the effective optimal control problem (P * M ) admits a unique optimal solution (u * M , y * M ).
To conclude this section we specify the effective admissible setsÛ ad and U * ad corresponding to the sets from Example 1. At this point, we recall that u ∈ L 2 (R d ; C) is real valued if and only ifû(−η) =û(η). An analogous statement holds for the Bloch transforms.
The specification of the admissible sets for U ε ad from Example 1 in (iii) and (iv) is more subtle and does not allow for a simple form at first sight.
(iii) For U ε ad := u ∈ L 2 (R n ) S ǫ u L 2 (R n ) ≤ L one obtains, using Lemma 2.1, (iv) For U ε ad := u ∈ L 2 (R n ) E ε (S ǫ u) ≤ L one obtains, using again Lemma 2.1, The characterization of the effective setÛ ad for the above two examples uses the εdependent Bloch eigenvalue λ ε 0 . To derive a more practicable condition one can again replace λ ε 0 by its M-th order Taylor expansion P ε M . The resulting approximative admissible setÛ M ad is close toÛ ad . Due to stability of the optimal control problem the minimization can be performed on this approximative set without loosing the approximation quality. Recalling that solutions to the effective PDE-constaint (1.23) have the Fourier transformŷ =f +û 1+P ε M , it follows that where in the last step we have used that |P ε M (η)−λ ε 0 (η)| ≤ Cε 2M uniformly in η ∈ K. For the the second term on the right hand side of (3.6) we argue analogously, f +û * ε,0 L 2 (K;C) + ŷ d,2 L 2 (K;C) .
We next show that the solution y * M of (3.2) is the desired approximation ofỹ * ε .
for an ε-independent constant C.
Proof. In Proposition 2.3 we have already shown that the following functioñ satisfies ỹ * ε −ỹ * M,app L 2 (R n ) ≤ Cε 2M . By triangle inequality it is sufficient to prove Indeed,ỹ * M,app solves the effective equation with right hand side f +ũ * ε , while y * M solves the same equation with right hand side f + u * M . By Corollary 3.2 one has Exploiting the linearity of the effective equation and the a priori estimate (2.20) from Proposition 2.5, one directly concludes which was the claim.
We conclude this section by discussing high-order approximations of the adjoint state corresponding to the optimal control problem (P ε ). By standard arguments (cf. [25]), the adjoint state p * ε is characterized through the following adjoint equation: It can therefore be written as a sum which is well-posed independently of the choice of ε. In the recent publication [3] the Boussinesq trick has been generalized to arbitrary order M > 2 using an induction argument.
Proposition 4.1 (Well posedness of the PDE constraint). Let u ∈ L 2 (R n ) be a given control with Fourier transformû and supp(û) ⊂ K.
i) There exists a unique weak solution y ∈ H 1 (R n ) to Equation (4.3).
ii) The solution y satisfies the a priori estimate with a constant C depending only on K. Here λ > 0 is the ellipticity constant of A * 2 , i.e. such that A * 2 · η ⊗2 ≥ λ|η| 2 for every η ∈ R n .
It is interesting to note how the prefactor where the error is of order ε 4 uniformly in η ∈ K. In particular exploiting the fact that R ε 2 and λ ε 0 are nonnegative, we conclude that 1 + Q ε 2 (η) 1 + R ε 2 (η) uniformly in η ∈ K. We emphasize once more that the advantage of working with is that 1 + R ε 2 ≥ 1 > 0 independently of ε, while this is not the case for 1 + P ε 2 . In view of (4.6), it is reasonable to introduce the following effective cost functional in Fourier form to obtain a second order approximation of the original cost functionalĴ ε from (2.16).
in the setÛ ad .
Let us comment on the above optimization problem. The problem is well-posed independently of ε with a unique minimizerû * 2,Î ∈Û ad . Analogously to Theorem 3.1 we prove in Theorem 4.4 below that û * ε,0 −û * 2,Î L 2 (K;C) ≤ Cε 4 . Exactly as in Proposition 3.3 we then conclude, using the a priori estimate from Proposition 4.1, that y * 2,Î (x) := K 1 + Q ε 2 (η) 1 + R ε 2 (η) (f (η) +û * 2,Î (η))e 2πiη·x dη (4.8) satisfies the error estimate y * ε − y * 2,Î L 2 (R n ) ≤ Cε 4 (4.9) for some ε-independent constant C. The optimization problem from Definition 4.2 is thus an alternative to approximate the original problem up to errors of order ε 4 . Its central advantage lies in the fact that it is well-posed independently of ε. i) As shown in [3] the Boussinesq trick can be generalized to arbitrary order M ∈ N. This fact can be used to derive a higher-order effective optimal control problem that is well-posed independently of ε.
ii) In [7] a different regualarization approach of the effective equation (1.23) is proposed which relies on adding a high-order spatial derivative. [3] provides a brief comparision of both regularization methods.
We now prove the approximation property of the alternative effective optimization problem from Definition 4.2.
We conclude this section by rewriting the minimization problem from Definition 4.2 as an optimal control problem with PDE constraint (4.3). Unfortunately, the factor 1+R ε 2 (η) 1+Q ε 2 (η) in the first part of the cost functionalÎ * 2 does not allow for a simple representation of the cost functional in the corresponding optimal control problem.

Outlook
One of the main applications of Bloch waves in the context of homogenization is the long time behavior of waves in ε-periodic media. Actually, while for fixed times t ∈ [0, T ] the constant coefficient effective wave equation provides a good approximation of the original oscillatory model, for long times t ∈ [0, T /ε 2 ] dispersive effects set in, which are not captured by the standard effective equation (5.1). In the articles [1,2,3,5,22] long time effective disersive models are derived by means of two-scale expansion and in [16,17,27] by means of Bloch wave analysis. We believe that the approach of this paper can also be applied to optimal control problems involving oscillatory wave equations on large time intervals. Another generalization regards the control problem of [20], where the oscillatory matrices in the PDE-constraint and in the energy may differ, cf. the discussion in the introduction. In a future project we hope to extend the Bloch approach to this general framework.