Optimal control problem for Stokes system: Asymptotic analysis via unfolding method in a perforated domain

This article's subject matter is the study of the asymptotic analysis of the optimal control problem (OCP) constrained by the stationary Stokes equations in a periodically perforated domain. We subject the interior region of it with distributive controls. The Stokes operator considered involves the oscillating coefficients for the state equations. We characterize the optimal control and, upon employing the method of periodic unfolding, establish the convergence of the solutions of the considered OCP to the solutions of the limit OCP governed by stationary Stokes equations over a non-perforated domain. The convergence of the cost functional is also established.


Introduction
In this article, we consider the optimal control problem (OCP) governed by generalized stationary Stokes equations in a periodically perforated domain O * ε (see Section 2, on the domain description).The size of holes in the perforated domain is of the same order as that of the period, and the holes are allowed to intersect the boundary of the domain.The control is applied in the interior region of the domain, and we wish to study the asymptotic analysis (homogenization) of an interior OCP subject to the constrained stationary Stokes equations with oscillating coefficients.
One can find several works in the literature regarding the homogenization of Stokes equations over a perforated domain.Using the multiple-scale expansion method, the authors in [16] studied the homogenization of Stokes equations in a porous medium with the Dirichlet boundary condition on the boundary of the holes.They obtained the Darcy's law as the limit law in the homogenized medium.In [9], the authors considered the Stokes system in a periodically perforated domain with non-homogeneous slip boundary conditions depending upon some parameter γ.Upon employing the Tartar's method of oscillating test functions they obtained under homogenization, the limit laws, viz., Darcy's law ( for γ < 1), Brinkmann's law (for γ = 1), and Stokes's type law (for γ > 1).In [25], the author studied a similar problem using the method of periodic unfolding in perforated domains by [10].Further, the type of behavior as seen in [9] was already observed in [12] by the authors while studying the homogeneous Fourier boundary conditions for the two-dimensional Stokes equation.Likewise, in [1,2], the author examined the Stokes equation in a perforated domain with holes of size much smaller than the small positive parameter ε, wherein they considered the boundary conditions on the holes to be of the Dirichlet type in [1] and the slip type in [2].The domain geometry, more specifically, the size of the holes, determines the kind of limit law in these works.Also, the author in [6] employed the Γ− convergence techniques to get comparable results.
A few works concern the homogenization of the OCPs governed by the elliptic systems over the periodically perforated domains with different kinds of boundary conditions on the boundary of holes (of the size of the same order as that of the period).In this regard, with the use of different techiniques, viz., H 0 − convergence in [18], two-sclae convergence in [23], and unfolding methods in [7,21], the homogenized OCPs were thus obtained over the nonperforated domains.Further, in context to the Stokes system, the authors in [22] studied the homogenization of the OCPs subject to the Stokes equations with Dirichlet boundary conditions on the boundary of holes, where the size of the holes is of the same order as that of the period.Here, the authors could obtain the homogenized system, pertaining only to the case when the set of admissible controls was unconstrained.For more literature concerning the homogenization of optimal control problems in perforated domains, the reader is reffered to [13-15, 19, 24] and the references therein.
The present article introduces an interior OCP subject to the generalized stationary Stokes equations in a periodically perforated domain O * ε .On the boundary of holes that do not intersect the outer boundary, the homogeneous Neumann boundary condition is prescribed, while on the rest part of the boundary, the homogeneous Dirichlet boundary condition is prescribed.The underlying objective of this article is to study the homogenization of this OCP.More specifically, we consider the minimization of the L 2 −cost functional (3.1), which is subject to the constrained generalized stationary Stokes equations (3.2).
The Stokes equations are generalized in the sense that we consider a second-order elliptic linear differential operator in divergence form with oscillating coefficients, i.e., − div (A ε ∇), first studied for the fixed domain in [4, Chapter 1], instead of the classical Laplacian operator.
Here, the action of the scalar operator − div (A ε ∇) is defined in a "diagonal" manner on any vector u = (u 1 , . . ., u n ), with components u 1 , . . ., u n in the H 1 Sobolev space.That is, for 1 ≤ i ≤ n, we have (− div (A ε ∇u)) i = − div (A ε ∇u i ).The main difficulty observed during the homogenization was identifying the limit pressure terms appearing in the state and the adjoint systems, which we overcame by introducing suitable corrector functions that solved some cell problems.We thus obtained the limit OCP associated with the stationary Stokes equation in a non-perforated domain.
The layout of this article is as follows: In the next section, we introduce the periodically perforated domain O * ε along with the notations that will be useful in the sequel.Section 3 is devoted to a detailed description of the considered OCP and the derivation of the optimality condition, followed by the characterization of the optimal control.In Section 4, we derive a priori estimates of the solutions to the considered OCP and its corresponding adjoint problem.In Section 5, we recall the definition of the method of periodic unfolding in perforated domains (see, [8,11]) and a few of its properties.Section 6, refers to the limit (homogenized) OCP.Finally, we derive the main convergence results in Section 7.

Domain description
Let {b 1 , ..., b n } be a basis of R n (n ≥ 2), and W be the associated reference cell defined as Let us denote O, W, and W * = W \Y by an open bounded subset of R n , a compact subset of W , and the perforated reference cell, respectively.It is assumed that the boundary of Y is Lipschitz continuous and has a finite number of connected components.Also, let ε > 0 be a sequence that converges to zero and set We take into account the perforated domain O * ε (see Figure 1) given by O * ε = O\Y ε , where The associated perforated domains are defined as which means that Γ ε 1 denotes the boundary of set of holes contained in O ε .In Figure 1, O  * ε and Λ * ε respectively represent the dark perforated part and the remaining part of the perforated domain O * ε .While, Γ ε 1 and Γ ε 0 respectively represent the boundary of holes contained in O * ε and the boundary of holes contained in Λ * ε along with the outer boundary ∂O.In the following, we introduce a few notations that we shall use throughout this article.
• η ε denotes the outward normal unit vector to Γ ε 1 .• η denotes the outward normal unit vector to ∂O.
• M t denotes the transpose of any matrix M .
• ψ is the zero extension of any function ψ outside O * ε to the whole of O.
• |F | is the Lebesgue measure of the measurable set F .
, the proportion of the perforated reference cell W * in the reference cell W .
• M W * (φ) is the mean value of φ on the perforated reference cell W * .
• {D → R}, the set of all real valued functions defined on domain D.
• D(Ω), is the space of infinitely many times differentiable functions with compact support in Ω, for any open set Ω ∈ R n .

Problem description and Optimality condition
Let us consider the following OCP associated with Stokes system: where the desired state u d = (u d 1 , . . ., u dn ) is defined on the space (L 2 (O)) n , θ ε is a control function defined on the space (L 2 (O * ε )) n and τ > 0 is a given regularization parameter.Here, the matrix A ε (x) = A( x ε ), where A(x) = (a ij (x)) 1≤i,j≤n defined on the space (L ∞ (O)) n×n is assumed to obey the uniform ellipticity condition: there exist real constants m 1 , m 2 > 0 such that m 1 ||λ|| 2 ≤ n i,j=1 a ij (x)λ i λ j ≤ m 2 ||λ|| 2 for all λ ∈ R n , which is endowed with an Eucledian norm denoted by || • ||.Also, we understand the action of scalar boundary operator and, for all Here, ( : ) and (•) represent the summation of the component-wise multiplication of the matrix entries and the usual scalar product of vectors, respectively.The existence of a unique weak solution (u 2) follows analogous to [5,Theorem IV.7.1].Also, for each ε > 0, there exists a unique solution to the problem (3.1) that can be proved along the same lines as in [20, Chapter 2, Theorem 1.2].We call the optimal solution to (3.1) by the triplet (u ε , p ε , θ ε ), with u ε , p ε , and θ ε as optimal state, pressure, and control, respectively.
Optimality Condition: The optimality condition is given by J ), where the pair (v ε , q ε ) is the solution to the following adjoint problem: We call v ε and q ε , the adjoint state and pressure, respectively.The existence of unique weak solution (v ε , q ε ) to (3.5) can now be proved in a way similar to that of system (3.2).
The following theorem characterizes the optimal control, the proof of which follows analogous to standard procedure laid in [20, Chapter 2, Theorem 1.4].

A priori estimates
This section concerns the derivation of estimates for the optimal solution to the problem (3.1) and the associated solution to the adjoint problem (3.5).These estimates are uniform and independent of the parameter ε.Towards attaining this aim, we first evoke the following two lemmas: Lemma 4.2 (Lemma 5.1, [12]).For each ε > 0 and Theorem 4.3.For each ε > 0, let u ε , p ε , θ ε be the optimal solution of the problem (3.1) and (v ε , q ε ) solves the corresponding adjoint problem (3.5).Then, one has Proof.Let u ε (0) denotes the solution to (3.2) corresponding to θ ε = 0.In view of Lemma 4.1, one can show that u ε (0 Using this and the optimality of solution (u ε , p ε , θ ε ) to problem (3.1), we have which gives estimate (4.2).Now, let us take u ε as a test function in (3.3).Considering (4.2) and the uniform ellipticity condition of matrix A ε , one obtains upon applying the Cauchy-Schwarz inequality along with the Lemma 4.1, the following: from which estimate (4.3) follows.
Owing to Lemma 4.2, for given In view of (4.1), (4.2) and (4.3), and the uniform ellipticity condition of the matrix A ε , one obtains from (4.7) upon employing the Cauchy-Schwarz inequality and Lemma 4.1, the following: ) n×n , which gives the estimate (4.5).Likewise, one can easily obtain the estimates (4.4) and (4.6) following the above discussion.Finally, from (3.6), we obtain that 5 The method of periodic unfolding for perforated domains We evokes the definition of the periodic unfolding operator and few of its properties as stated in [8,11].Given x ∈ R n , we denote the greatest integer and the fractional parts of x respectively by [x] W and {x} W .That is, [x] W = n j=1 k j b j be the unique integer combination of periods and {x} W = x − [x] W .In particular, we have for ε > 0, Definition 5.1.The unfolding operator T * ε : {O * ε → R} → {O × W * → R} is defined as Also, for any domain D ⊇ O * ε and vector u In the following there are the properties of the unfolding operator: Then, there exists 6 Limit optimal control problem This section presents the limit (homogenized) system corresponding to the problem (3.1), which we considered in the beginning.
Let us consider the function space which is a Hilbert space for the norm We now consider the limit OCP associated with the Stokes system subject to where the tensor B = (b αβ ij ) = (b αβ ij ) 1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, is given by : ∇ y P α i dy as the entries of the constant tensor A 0 , P β j = P β j (y) = (0, . . ., y j , . . ., 0) with y j at the β-th position, and for 1 ≤ j, β ≤ n, the correctors (χ The existence of this unique pair (u, p) ∈ (H 1 0 (O)) n × L 2 (O) can be found in [4, Chapter 1].Further, the problem (6.1) is a standard one and there exists a unique weak solution to it, one can follow the arguments introduced in [20, Chapter 2, Theorem 1.2].We call the triplet (u, p, θ) ) n , the optimal solution to (6.1), with u, p, and θ as the optimal state, pressure, and control, respectively.Now, we introduce the limit adjoint system associated with (6.2): Find a pair (v, q) ∈ (H 1 0 (O)) n × L 2 (O) which solves the system where the tensor B t = (b βα ji ) = (b βα ji ) 1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, is given by : ∇ y P α i dy as the entries of the constant tensor In the following, we state a result similar to Theorem 3.1 that characterizes the optimal control θ in terms of the adjoint state v and the proof of which follows analogous to the standard procedure laid in [20, Chapter 2, Theorem 1.4].
Theorem 6.1.Let u, p, θ be the optimal solution to (6.1) and (v, q) be the corresponding adjoint solution to (6.4), then the optimal control is characterized by Conversely, suppose that a triplet (ǔ, p, θ) , respectively, satisfy the following systems: Then, the triplet ǔ, p, − 1 τ v is the optimal solution to (6.1).

Convergence results
We present here the key findings on the convergence analysis of the optimal solutions to the problem (3.1) and its corresponding adjoint system (3.5) by using the method of periodic unfolding for perforated domains described in Section 5.
Theorem 7.1.For given ε > 0, let the triplets (u ε , p ε , θ ε ) and (u, p, θ), respectively, be the optimal solutions of the problems (3.1) and (6.1).Then where A 0 is a tensor as defined in Section 6, I is the n × n identity matrix, θ is characterized through (6.6) and the pairs (v ε , q ε ) and (v, q) solve respectively the systems (3.5) and (6.4).Moreover, Proof.First, upon using Proposition 5.2 (vi) on the entries of the matrix A ε , we obtain (7.1a) under the passage of limit ε → 0. Similarly, one can prove the convergence for the matrix A t ε under unfolding.Next, in view of Theorem 4.3 and the fact that the triplet (u ε , p ε , θ ε ) is an optimal solution to problem (3.1), one gets uniform estimates for the sequences Thus, by weak compactness, there exists a subsequence not relabelled and a function θ in (L 2 (O × W * )) n , such that Now, using Proposition 5.2 (vii) in (7.3) gives where, θ 0 = M W * ( θ).Employing Proposition 5.2 (i), we have the uniform boundedness of the sequences {T ε (u ε )}, {T ε (∇u ε )}, and Likewise, for the associated adjoint counterparts, viz., v ε , and q ε , one obtains that there exist subsequences not relabelled and functions v with M W * (v) = 0, v 0 , and q in spaces (L 2 (O; , respectively, such that ) The identification of the limit functions û, v, p, q, M W * (p) and M W * (q) is carried out in subsequent steps.
Step 4: Now, we will furnish the proof of the energy convergence for the L 2 −cost functional.Thus, from equations (7.33) and (7.35), we get (7.2).This completes the proof of Theorem 7.1.

Conclusions
We have addressed the limiting behavior of an interior OCP corresponding to Stokes equations in an nD (n ≥ 2) periodically perforated domain O * ε via the technique of periodic unfolding in perforated domains (see, [8,11]).We employed the Neumann boundary condition on the part of the boundary of the perforated domain.Firstly, we characterized the optimal control in terms of the adjoint state.Secondly, we deduced the apriori optimal bounds for control, state, pressure, and their associated adjoint state and pressure functions.Thereafter, the limiting analysis for the considered OCP is carried out upon employing the periodic unfolding method in perforated domains.We observed the convergence between the optimal solution to the problem (3.1) posed on the perforated domain O * ε and the optimal solution to that of the limit problem (6.1) governed by stationary Stokes equation posed on a non-perforated domain O. Finally, we established the convergence of energy corresponding to L 2 −cost functional.
Now, let us denote O ε as the interior of the largest union of ε(ζ + W ) cells such that ε(ζ + W ) ⊂ O, while Λ ε ⊂ O as containing the parts from ε(ζ + W ) cells intersecting the boundary ∂O.More precisely, we write Λ ε = O\ O ε , where

Figure 1 :
Figure 1: The Perforated domain O * ε and the reference cell W.

. 1 )
§ The symbol C represents a generic constant that is positive and independent of ε.