Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 80, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS: ASYMPTOTIC ANALYSIS VIA UNFOLDING METHOD IN A PERFORATED DOMAIN SWATI GARG, BIDHAN CHANDRA SARDAR Communicated by Jesús Ildefonso Dı́az Abstract. This article’s subject matter is the study of the asymptotic analy- sis 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. 1. Introduction In this article, we consider the optimal control problem (OCP) governed by generalized stationary Stokes equations in a periodically perforated domain O∗ε (refer Section 2, for detailed configuration of the domain). 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 [18] 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 [11], 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 [34], the author studied 2020 Mathematics Subject Classification. 35B27, 35B40, 35Q93, 49J20, 76D07. Key words and phrases. Stokes equations; homogenization; optimal control; perforated domain; unfolding operator. ©2023. This work is licensed under a CC BY 4.0 license. Submitted May 23, 2023. Published December 6, 2023. 1 2 S. GARG, B. C. SARDAR EJDE-2023/80 a similar problem using the method of periodic unfolding in perforated domains by [12]. Further, the type of behavior as seen in [11] was already observed in [14] by the authors while studying the homogeneous Fourier boundary conditions for the two-dimensional Stokes equation. Likewise, in [3, 1], 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 [3]. The domain geometry, more specifically, the size of the holes, determines the kind of limit law in these works. Also, the author in [8] employed the Γ− convergence techniques to obtain 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., H0− conver- gence in [23], two-scale convergence in [28], and unfolding methods in [9, 26], the homogenized OCPs were thus obtained over the non-perforated domains. Further, the homogenization of OCPs subject to the boundary value problems concerning the steady Stokes equations mainly comprise the boundary conditions of the type: Dirichlet, Navier slip-friction, Neumann, Mixed, etc. The authors in [27] 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 un- constrained. For more literature concerning the homogenization of optimal control problems in perforated domains, the reader is referred to [15, 32, 17, 16] and the references therein. Also, over another type of oscillating domain, one can refer to the recent work [29] for the case of mixed boundary value problem for the Stokes system, wherein the authors homogenized the stationary Stokes system subject to the mixed boundary condition comprising of the Neumann boundary condition on the highly oscillating boundary and the homogeneous Dirichlet boundary condi- tion on the base part of the domain’s boundary. Very recently, the authors in [20] studied the asymptotic analysis of the Stokes system with mixed boundary con- ditions of similar type on the thin oscillating domain. Furthermore, pertaining to the Navier-Stokes equations, the existence of the solutions to the mixed boundary value problem has been established by the authors in [5] for 2D bounded domain. For more literature related to the Stokes system with mixed boundary conditions, one may refer to [19, 31, 25] and the references therein. This article introduces an interior OCP subject to the generalized stationary Stokes equations in a periodically perforated domain O∗ε . We employ mixed bound- ary data on the boundary of the perforated domain, i.e., 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 arti- cle is to study the homogenization of this OCP. More specifically, we consider the minimization of the L2−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, EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 3 i.e., −div(Aε∇), first studied for the fixed domain in [6, Chapter 1], instead of the classical Laplacian operator, which later on was studied by various authors for different types of ε−dependent varying domains. For instance, we studied in [21] the generalized stationary Stokes equation for the two-dimensional oscillating do- main. Here, the action of the scalar operator −div(Aε∇) is defined in a ”diagonal” manner on any vector u = (u1, . . . , un), with components u1, . . . , un in the H1 Sobolev space. That is, for 1 ≤ i ≤ n, we have (−div(Aε∇u))i = −div(Aε∇ui). 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, [13, 10]) and a few of its properties. Section 6, refers to the limit (homogenized) OCP. Finally, we derive the main convergence results in Section 7 followed by some important remarks. 2. Domain description and notation 2.1. Domain description. Let {b1, . . . , bn} be a basis of Rn (n ≥ 2), and W be the associated reference cell defined as W = { w ∈ Rn : w = n∑ i=1 wibi, (w1, . . . , wn) ∈ (0, 1)n } . Let us denote O, W , and W ∗ = W\Y by an open bounded subset of Rn, 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 T = { ζ ∈ Rn : ζ = n∑ i=1 zibi, (z1, . . . , zn) ∈ Zn } , Zε = { ζ ∈ T : ε(ζ +W ) ⊂ O } . We take into account the perforated domain O∗ε (see Figure 1) given by O∗ε = O\Yε, where Yε = ∪ζ∈T ε(ζ + Y ). Now, let us denote Ôε as the interior of the largest union of ε(ζ +W ) cells such that ε(ζ + W ) ⊂ O, while Λε ⊂ O as containing the parts from ε(ζ + W ) cells intersecting the boundary ∂O. More precisely, we write Λε = O\Ôε, where Ôε = interior{∪ζ∈Zε ε(ζ +W )}. The associated perforated domains are defined as Ô∗ε = Ôε\Yε, Λ̂∗ε = O∗ε\Ô∗ε . Also, we denote the boundary of the perforated domain O∗ε as ∂O∗ε = Γε1 ∪ Γε0, where Γε1 = ∂Ôε ∩ ∂Yε and Γε0 = ∂O∗ε\Γε1, 4 S. GARG, B. C. SARDAR EJDE-2023/80 Figure 1. Perforated domain O∗ε and the reference cell W . which means that Γε1 denotes the boundary of set of holes contained in Ôε. In Figure 1, Ô∗ε and Λ̂∗ε respectively represent the dark perforated part and the remaining part of the perforated domain O∗ε . While, Γε1 and Γε0 respectively represent the boundary of holes contained in Ô∗ε and the boundary of holes contained in Λ̂∗ε along with the outer boundary ∂O. In the following, we introduce a few notations that we shall use throughout this article. 2.2. Notation. • Aε(x) = A(x/ε) a.e. in O, for all ε > 0. • vε = (vε1, . . . , vεn), for any bold symbol vector function vε. • v = (v1, . . . , vn), for any bold symbol vector function v. • τ > 0 is a given regularization parameter. • ηε denotes the outward normal unit vector to Γε1. • η denotes the outward normal unit vector to ∂O. • R : S denotes the element-wise product followed by summation between any matrices R and S. • v ·w denotes the standard dot product between any vector functions v and w. • M t denotes the transpose of any matrix M . • ψ̃ is the zero extension of any function ψ outside O∗ε to the whole of O. • ψ̃ = (ψ̃1, · · · , ψ̃n), for any vector function ψ. • |F | is the Lebesgue measure of the measurable set F . • The symbol C represents a generic constant that is positive and independent of ε. • Θ = |W∗| |W | , the proportion of the perforated reference cell W ∗ in the refer- ence cell W . • MW∗(φ) is the mean value of φ on the perforated reference cell W ∗. • MW∗(φ) = (MW∗(φ1), · · · ,MW∗(φn)), for vector function φ. • {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 ∈ Rn. EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 5 3. Problem description and optimality condition Let us consider the following OCP associated with Stokes system inf θε∈(L2(O∗ ε ))n { Jε(θε) = 1 2 ∫ O∗ ε |uε(θε)− ud|2 + τ 2 ∫ O∗ ε |θε|2 } , (3.1) subject to − div(Aε∇uε) +∇pε = θε in O∗ε , div(uε) = 0 in O∗ε , ηε ·Aε∇uε − pεηε = 0 on Γε1, uε = 0 on Γε0, (3.2) where the target state ud = (ud1 , . . . , udn) is defined on the space (L2(O))n, and θε is a control function defined on the space (L2(O∗ε))n. Here, the matrix Aε(x) = A(xε ), where A(x) = (aij(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 m1, m2 > 0 such that m1‖λ‖2 ≤ ∑n i,j=1 aij(x)λiλj ≤ m2‖λ‖2 for all λ ∈ Rn, which is endowed with an Eucledian norm denoted by ‖ · ‖. Also, we understand the action of scalar boundary operator ηε ·Aε∇ on the vector uε|Γε 1 in a ”diagonal” manner: (ηε · Aε∇uε)i = ηε · Aε∇uεi, for 1 ≤ i ≤ n. We introduce the func- tion space (H1 Γε 0 (O∗ε))n := {φ ∈ (H1(O∗ε))n : φ|Γε 0 = 0}. This is a Banach space endowed with the norm ‖φ‖(H1 Γε 0 (O∗ ε ))n := ‖∇φ‖(L2(O∗ ε ))n×n , ∀φ ∈ (H1 Γε 0 (O∗ε))n. Definition 3.1. We say a pair (uε, pε) ∈ (H1 Γε 0 (O∗ε))n ×L2(O∗ε) is a weak solution to (3.2) if, for all φ ∈ (H1 Γε 0 (O∗ε))n,∫ O∗ ε Aε∇uε : ∇φ dx− ∫ O∗ ε pε div(φ) dx = ∫ O∗ ε θε · φ dx, (3.3) and, for all w ∈ L2(O∗ε), ∫ O∗ ε div(uε) w dx = 0. (3.4) The existence of a unique weak solution (uε(θε), pε) ∈ (H1 Γε 0 (O∗ε))n × L2(O∗ε) of the system (3.2) follows analogous to [7, Theorem 4.7.1]. 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 [24, Chapter 2, Theorem 1.2]. We call the optimal solution to (3.1) by the triplet (uε, pε,θε), with uε, pε, and θε as optimal state, pressure, and control, respectively. Optimality Condition: The optimality condition is given by J ′ε(θ) · (θ−θε) ≥ 0, for all θ ∈ (L2(O∗ε))n (see, [24, Chapter 2, Page 48]). One can obtain the further simplification of this condition as ∫ O∗ ε (vε+τθε) ·(θ−θε) ≥ 0, for all θ ∈ (L2(O∗ε))n (see, [24, Chapter 2]), where the pair (vε, qε) is the solution to the following adjoint problem: −div(Atε∇vε) +∇qε = uε − ud in O∗ε , div(vε) = 0 in O∗ε , ηε ·Atε∇vε − qεηε = 0 on Γε1, vε = 0 on Γε0. (3.5) 6 S. GARG, B. C. SARDAR EJDE-2023/80 We call vε and qε, the adjoint state and pressure, respectively. The existence of unique weak solution (vε, qε) 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 [24, Chapter 2, Theorem 1.4]. Theorem 3.2. Let (uε, pε,θε) be the optimal solution of the problem (3.1) and (vε, qε) solves (3.5), then the optimal control is characterized by θε = −1 τ vε a.e. in O∗ε . (3.6) Conversely, suppose that a triplet (ǔε, p̌ε, θ̌ε) ∈ (H1 Γε 0 (O∗ε))n ×L2(O∗ε)× (L2(O∗ε))n and a pair (v̌ε, q̌ε) ∈ (H1 Γε 0 (O∗ε))n × L2(O∗ε) solves the system −div(Aε∇ǔε) +∇p̌ε = −1 τ v̌ε in O∗ε , −div(Atε∇v̌ε) +∇q̌ε = ǔε − ud in O∗ε , div(ǔε) = 0, div(v̌ε) = 0 in O∗ε , ηε ·Aε∇ǔε − p̌εηε = 0 on Γε1, ηε ·Atε∇v̌ε − q̌εηε = 0 on Γε1, v̌ε = 0, ǔε = 0 on Γε0. Then the triplet (ǔε, p̌ε,− 1 τ v̌ε) is the optimal solution of (3.1). 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 4.1 ([4, Lemma A.4]). There exists a constant C ∈ R+, independent of ε, such that ‖v‖L2(O∗ ε )n ≤ C‖∇v‖(L2(O∗ ε ))n×n , ∀v ∈ (H1 Γε 0 (O∗ε))n. Lemma 4.2 ([14, Lemma 5.1]). For each ε > 0 and qε ∈ L2(O∗ε), there exists gε ∈ (H1 Γε 0 (O∗ε))n and a constant C ∈ R+, independent of ε, such that div(gε) = qε and ‖∇gε‖(L2(O∗ ε ))n×n ≤ C(O) ‖qε‖L2(O∗ ε ). (4.1) Theorem 4.3. For each ε > 0, let (uε, pε,θε) be the optimal solution of the problem (3.1) and (vε, qε) solves the corresponding adjoint problem (3.5). Then, one has θε ∈ (H1 Γε 0 (O∗ε))n and there exists a constant C ∈ R+, independent of ε such that ‖θ̄ε‖(L2(O∗ ε ))n ≤ C, (4.2) ‖ūε‖(H1 Γε 0 (O∗ ε ))n ≤ C, (4.3) ‖v̄ε‖(H1 Γε 0 (O∗ ε ))n ≤ C, (4.4) ‖p̄ε‖L2(O∗ ε ) ≤ C, (4.5) ‖q̄ε‖L2(O∗ ε ) ≤ C. (4.6) EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 7 Proof. Let uε(0) denote the solution to (3.2) corresponding to θε = 0. In view of Lemma 4.1, one can show that ‖uε(0)‖(L2(O∗ ε ))n ≤ 0, i.e., uε(0) = 0 in (L2(O∗ε))n. Using this and the optimality of solution (uε, pε,θε) to problem (3.1), we have ‖uε(θ)− ud‖2(L2(O∗ ε ))n + τ‖θε‖2(L2(O∗ ε ))n ≤ ‖uε(0)− ud‖2(L2(O∗ ε ))n ≤ C, which gives estimate (4.2). Now, let us take uε as a test function in (3.3). Con- sidering (4.2) and the uniform ellipticity condition of matrix Aε, one obtains upon applying the Cauchy-Schwarz inequality along with the Lemma 4.1, the inequality m1‖∇uε‖2(L2(O∗ ε ))n×n ≤ ∫ O∗ ε Aε∇uε : ∇uε dx ≤ C ‖θε‖(L2(O∗ ε ))n‖∇uε‖(L2(O∗ ε ))n×n , from which estimate (4.3) follows. Owing to Lemma 4.2, for given pε ∈ L2(O∗ε), there exists gε ∈ (H1 Γε 0 (O∗ε))n satisfying div(gε) = pε. Corresponding to θε, taking v = gε in (3.3), we obtain ‖pε‖2L2(O∗ ε ) = ∫ O∗ ε Aε∇uε : ∇gε dx− ∫ O∗ ε θε · gε dx. (4.7) In view of (4.1), (4.2) and (4.3), and the uniform ellipticity condition of the matrix Aε, one obtains from (4.7) upon employing the Cauchy-Schwarz inequality and Lemma 4.1, the inequality ‖pε‖2L2(O∗ ε ) ≤ (m2‖∇uε‖(L2(O∗ ε ))n×n + C‖θε‖(L2(O∗ ε ))n)‖∇gε‖(L2(O∗ ε ))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 θε ∈ (H1 Γε 0 (O∗ε))n. � 5. The method of periodic unfolding for perforated domains We evoke the definition of the periodic unfolding operator and few of its prop- erties as stated in [13, 10]. Given x ∈ Rn, 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 kjbj be the unique integer combination of periods and {x}W = x− [x]W . In particular, we have for ε > 0, x = ε ([x ε ] W + {x ε } W ), ∀x ∈ Rn. Definition 5.1. The unfolding operator T ∗ε : {O∗ε → R} → {O ×W ∗ → R} is defined as T ∗ε (u)(x, y) = { u(ε [ x1 ε ] W + εy) a.e. for (x, y) ∈ Ôε ×W ∗, 0 a.e. for (x, y) ∈ Λ̂∗ε ×W ∗. Also, for any domain D ⊇ O∗ε and vector u = (u1, · · · , un) ∈ ({D → R})n, we define its unfolding by T ∗ε (u) := (T ∗ε (u1), . . . , T ∗ε (un)). Proposition 5.2. The unfolding operator has the following properties: (1) T ∗ε is linear and continuous from L2(O∗ε) to L2(O ×W ∗). (2) Let u, v ∈ L2(O∗ε). Then T ∗ε (uv) = T ∗ε (u)T ∗ε (v). (3) Let u ∈ L2(O). Then T ∗ε (u)→ u strongly in L2(O ×W ∗). 8 S. GARG, B. C. SARDAR EJDE-2023/80 (4) Let u ∈ L1(O∗ε). Then∫ Ô∗ ε u(x) dx = ∫ O∗ ε u(x) dx− ∫ Λ̂∗ ε u(x) dx = 1 |W ∗| ∫ O×W∗ T ∗ε (u)(x, y) dx dy. (5) For each ε > 0, let {uε} ∈ L2(O) and uε → u strongly in L2(O). Then T ∗ε (uε)→ u strongly in L2(O ×W ∗). (6) Let v ∈ L2(W ∗) be a W -periodic function and vε(x) = v(xε ). Then T ∗ε (vε)(x, y) = { v(y) a.e. for (x, y) ∈ Ôε ×W ∗, 0 a.e. for (x, y) ∈ Λε ×W ∗. (7) Let fε ∈ L2(O∗ε) be uniformly bounded. Then there exists f ∈ L2(O ×W ∗) such that T ∗ε (fε) ⇀ f weakly in L2(O ×W ∗), and f̃ε ⇀ 1 |W | ∫ W∗ f(·, y) dy weakly in L2(O). Proposition 5.3. Let O ⊂ Rn be bounded with Lipschitz boundary. Let fε ∈ H1(O∗ε) be such that fε = 0 on ∂O ∩ ∂O∗ε and satisfy, ‖∇fε‖(L2(O∗ ε ))n ≤ C. Then there exists f ∈ H1 0 (O) and f̂ ∈ L2(O;H1 per(W ∗)) with MW∗(f̂) = 0, such that up to a subsequence, T ∗ε (∇fε) ⇀ ∇f +∇y f̂ weakly in (L2(O ×W ∗))n, T ∗ε (fε)→ f strongly in L2(O;H1(W ∗)). 6. Limit optimal control problem This section presents the limit (homogenized) system corresponding to problem (3.1), which we considered in the beginning. Let us consider the function space (H1 0 (O))n := {ϕ ∈ (H1(O))n : ϕ|∂O = 0}, which is a Hilbert space for the norm ‖ϕ‖(H1 0 (O))n := ‖∇ϕ‖(L2(O))n×n ∀ϕ ∈ (H1 0 (O))n. We now consider the limit OCP associated with the Stokes system inf θ∈(L2(O))n { J(θ) = Θ 2 ∫ O |u− ud|2 dx+ τΘ 2 ∫ O |θ|2 dx } , (6.1) subject to − n∑ j,α,β=1 ∂ ∂xα ( bαβij ∂uj ∂xβ ) +∇p = θ in O, div(u) = 0 in O, u = 0 on ∂O, (6.2) where the tensor B = (bαβij ) = (bαβij )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, is given by bαβij = aαβij − 1 |W ∗| ∫ W∗ A(y)∇y(P β j − χ β j ) : ∇yχαi dy, EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 9 with aαβij = 1 |W∗| ∫ W∗ A(y)∇y(P β j − χ β j ) : ∇yPαi dy as the entries of the constant tensor A0, P β j = P β j (y) = (0, . . . , yj , . . . , 0) with yj at the β-th position, and for 1 ≤ j, β ≤ n, the correctors (χβj ,Π β j ) ∈ (H1(W ∗))n×L2(W ∗) solves the cell problem −divy ( A(y)∇y(P β j − χ β j ) ) +∇yΠβ j = 0 in W ∗, η ·A(y)∇y(P β j − χ β j )−Πβ j η = 0 on ∂W ∗\∂W, divy(P β j − χ β j ) = 0 in W ∗, (χβj ,Π β j )W ∗-periodic, MW∗(χβj ) = 0. (6.3) The existence of this unique pair (u, p) ∈ (H1 0 (O))n×L2(O) can be found in [6, 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 [24, Chapter 2, Theorem 1.2]. We call the triplet (u, p,θ) ∈ (H1 0 (O))n × L2(O) × (L2(O))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) ∈ (H1 0 (O))n × L2(O) which solves the system − n∑ j,α,β=1 ∂ ∂xα ( bβαji ∂vj ∂xβ ) +∇q = u− ud in O, div(v) = 0 in O, (6.4) where the tensor Bt = (bβαji ) = (bβαji )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, is given by bβαji = aβαji − 1 |W ∗| ∫ W∗ At(y)∇y(P β j −H β j ) : ∇yHα i dy, with aβαji = 1 |W∗| ∫ W∗ A t(y)∇y(P β j −H β j ) : ∇yP α i dy as the entries of the constant tensor At0. Also, for 1 ≤ j, β ≤ n, the correctors (Hβ j , Z β j ) ∈ (H1(W ∗))n×L2(W ∗) solves the cell problem −divy(At(y)∇y(P β j −H β j )) +∇yZβj = 0 in W ∗, η ·At(y)∇y(P β j −H β j )− Zβj η = 0 on ∂W ∗\∂W, divy(P β j −H β j ) = 0 in W ∗, (Hβ j , Z β j )W ∗-periodic, MW∗(Hβ j ) = 0. (6.5) In the following, we state a result similar to Theorem 3.2 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 [24, Chapter 2, Theorem 1.4]. Theorem 6.1. Let (u, p,θ) be the optimal solution to (6.1) and (v, q) be the cor- responding adjoint solution to (6.4), then the optimal control is characterized by θ = −1 τ v a.e. in O. (6.6) 10 S. GARG, B. C. SARDAR EJDE-2023/80 Conversely, suppose that a triplet (ǔ, p̌, θ̌) ∈ (H1 0 (O))n×L2(O)× (L2(O))n and a pair (v̌, q̌) ∈ (H1 0 (O))n × L2(O), respectively, satisfy the following systems: − n∑ j,α,β=1 ∂ ∂xα ( bαβij ∂ǔj ∂xβ ) +∇p̌ = −1 τ v̌ in O, div(ǔ) = 0 in O, and − n∑ j,α,β=1 ∂ ∂xα ( bβαji ∂v̌j ∂xβ ) +∇q̌ = ǔ− ud in O, div(v̌) = 0 in O. Then the triplet (ǔ, p̌,− 1 τ v̌) is the optimal solution to (6.1). 7. 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 (uε, pε,θε) and (u, p,θ), respec- tively, be the optimal solutions of the problems (3.1) and (6.1). Then T ∗ε (Aε)→ A strongly in (L2(O ×W ∗))n×n, (7.1a) θ̃ε ⇀ Θθ weakly in (L2(O))n, (7.1b) ũε ⇀ Θu weakly in (H1 0 (O))n, (7.1c) ṽε ⇀ Θv weakly in (H1 0 (O))n, (7.1d) p̃ε ⇀ Θ n A0∇u : I + Θp weakly in L2(O), (7.1e) q̃ε ⇀ Θ n At0∇v : I + Θq weakly in L2(O), (7.1f) where A0 is a tensor as defined in Section 6, I is the n× n identity matrix, θ is characterized through (6.6) and the pairs (vε, qε) and (v, q) solve respectively the systems (3.5) and (6.4). Moreover, lim ε→0 Jε(θε) = J(θ). (7.2) Proof. First, upon using Proposition 5.2(6) on the entries of the matrix Aε, we obtain (7.1a) under the passage of limit ε → 0. Similarly, one can prove the con- vergence for the matrix Atε under unfolding. Next, in view of Theorem 4.3 and the fact that the triplet (uε, pε,θε) is an optimal solution to problem (3.1), one gets uniform estimates for the sequences {θε}, {uε}, {pε}, {vε}, and {qε} in the spaces (L2(O∗ε))n, (H1 Γε 0 (O∗ε))n, L2(O∗ε), (H1 Γε 0 (O∗ε))n, and L2(O∗ε), respectively. Using the uniform estimate of the sequence {θε} in the space (L2(O∗ε))n and Propo- sition 5.2(1), we have the sequence {T ∗ε (θε)} to be uniformly bounded in the space (L2(O ×W ∗))n. Thus, by weak compactness, there exists a subsequence not rela- belled and a function θ̂ in (L2(O ×W ∗))n, such that T ∗ε (θε) ⇀ θ̂ weakly in (L2(O ×W ∗))n. (7.3) EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 11 Now, using Proposition 5.2(7) in (7.3) gives θ̃ε ⇀ 1 |W | ∫ W∗ θ̂(x, y) dy = Θθ0 weakly in (L2(O))n, (7.4) where, θ0 =MW∗(θ̂). Employing Proposition 5.2(1), we have the uniform bound- edness of the sequences {T ε(uε)}, {T ε(∇uε)}, and {T ε(pε)} in the respective spaces (L2(O;H1(W ∗)))n, (L2(O ×W ∗))n×n, and L2(O ×W ∗). Further, upon employ- ing Proposition 5.3 and Proposition 5.2(7), there exist subsequences not relabelled and functions û with MW∗(û) = 0, u0, and p̂ in spaces (L2(O;H1 per(W ∗)))n, (H1 0 (O))n, and L2(O ×W ∗), respectively, such that T ∗ε (uε)→ u0 strongly in (L2(O;H1(W ∗)))n, (7.5a) T ∗ε (∇uε) ⇀ ∇u0 +∇yû weakly in (L2(O ×W ∗))n×n, (7.5b) ũε ⇀ Θu0 weakly in (H1 0 (O))n, (7.5c) T ∗ε (pε) ⇀ p̂ weakly in L2(O ×W ∗), (7.5d) p̃ε ⇀ ΘMW∗(p̂) weakly in L2(O). (7.5e) Likewise, for the associated adjoint counterparts, viz., vε, and qε , one obtains that there exist subsequences not relabelled and functions v̂ withMW∗(v̂) = 0, v0, and q̂ in spaces (L2(O;H1 per(W ∗)))n, (H1 0 (O))n, and L2(O ×W ∗), respectively, such that T ∗ε (vε)→ v0 strongly in (L2(O;H1(W ∗)))n, (7.6a) T ∗ε (∇vε) ⇀ ∇v0 +∇yv̂ weakly in (L2(O ×W ∗))n×n, (7.6b) ṽε ⇀ Θv0 weakly in (H1 0 (O))n, (7.6c) T ∗ε (qε) ⇀ q̂ weakly in L2(O ×W ∗), (7.6d) q̃ε ⇀ ΘMW∗(q̂) weakly in L2(O). (7.6e) The identification of the limit functions û, v̂, p̂, q̂,MW∗(p̂) andMW∗(q̂) is carried out in subsequent steps. Step 1: (Claim): For all ϕ ∈ (H1 0 (O))n, ψ ∈ (L2(O;H1 per(W ∗)))n, and w ∈ L2(O), we claim that the ordered quadruplet (u0, û, p̂,θ0) belongs to (H1 0 (O))n × (L2(O;H1 per(W ∗)))n×L2(O×W ∗)× (L2(O))n is a unique solution to the following limit system: 1 |W | ∫ O×W∗ A(y)(∇u0 +∇yû(x, y)) : (∇ϕ+∇yψ)dx dy − 1 |W | ∫ O×W∗ p̂(x, y) · (div(ϕ) + divy(ψ))dx dy = Θ ∫ O θ0 ·ϕ dx,∫ O div(u0)w dx = 0, (7.7) 12 S. GARG, B. C. SARDAR EJDE-2023/80 and the ordered triplet (v0, v̂, q̂) ∈ (H1 0 (O))n× (L2(O;H1 per(W ∗)))n×L2(O×W ∗) is a unique solution to the limit adjoint system 1 |W | ∫ O×W∗ At(y)(∇v0 +∇yv̂(x, y)) : (∇ϕ+∇yψ)dx dy − 1 |W | ∫ O×W∗ q̂(x, y)(div(ϕ) + divy(ψ))dx dy = Θ ∫ O (u0 − ud) ·ϕ dx,∫ O div(v0)w dx = 0. (7.8) Proof of the Claim: Towards the proof of (7.7), let us consider a test function ϕ ∈ (D(O))n in (3.3) and use properties (1), (2), and (4) of Proposition 5.2 to obtain 1 |W | ∫ O×W∗ T ∗ε (Aε)T ∗ ε (∇uε) : T ∗ε (∇ϕ) dx dy + ∫ Λ̂∗ ε Aε∇uε : ∇ϕ dx − ∫ Λ̂∗ ε pε div(ϕ) dx− 1 |W | ∫ O×W∗ T ∗ε (pε)T ∗ ε (div(ϕ)) dx dy = 1 |W | ∫ O×W∗ T ∗ε (θε) · T ∗ε (φε) dx dy + ∫ Λ̂∗ ε θε ·ϕ dx. (7.9) Using Proposition 5.2(3), the fact that limε→0 |Λ̂∗ε| = 0, and convergences (7.3), (7.1a), (7.5b), (7.5d), we have under the passage of limit as ε→ 0 in (7.9), 1 |W | ∫ O×W∗ A(y)(∇u0 +∇yû(x, y)) : ∇ϕ dx dy − 1 |W | ∫ O×W∗ p̂(x, y) div(ϕ) dx dy = Θ ∫ O θ0 ·ϕ dx, (7.10) which remains valid for every ϕ ∈ (H1 0 (O))n, by density. Now, consider the function φε(x) = εφ(x)ξ(xε ), where φ ∈ D(O) and ξ ∈ (H1 per(W ∗))n. Employing properties (2), (3), and (6) of Proposition 5.2, one can easily obtain T ∗ε (φε)(x, y)→ 0 strongly in (L2(O ×W ∗))n, (7.11a) T ∗ε (∇φε)(x, y)→ φ(x)∇yξ(y) strongly in (L2(O ×W ∗))n×n. (7.11b) Let us use the test function φε in (3.3) and employ properties (1), (2), and (4) of Proposition 5.2 to obtain 1 |W | ∫ O×W∗ T ∗ε (Aε)T ∗ ε (∇uε) : T ∗ε (∇φε) dx dy + ∫ Λ̂∗ ε Aε∇uε : ∇φε dx − ∫ Λ̂∗ ε pε div(φε) dx− 1 |W | ∫ O×W∗ T ∗ε (pε)T ∗ ε (div(φε)) dx dy = 1 |W | ∫ O×W∗ T ∗ε (θε) · T ∗ε (φε) dx dy + ∫ Λ̂∗ ε θε · φε dx. (7.12) EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 13 In above equality, the absolute value of each integral over Λ̂∗ε is bounded above with a bound of order ε|Λ̂∗ε| or |Λ̂∗ε|. This with the fact that limε→0 |Λ̂∗ε| = 0, and convergences (7.3), (7.1a), (7.5b), (7.5d), and (7.11), gives under the passage of limit ε→ 0, 1 |W | ∫ O×W∗ A(y)(∇u0 +∇yû(x, y)) : ∇yψ dx dy − 1 |W | ∫ O×W∗ p̂(x, y) divy(ψ) dx dy = 0, (7.13) which remains valid for every φ ξ = ψ ∈ (L2(O;H1 per(W ∗)))n, by density. Further, for all w ∈ L2(O), we have ∫ O∗ ε div(uε)w dx = 0. (7.14) Now, upon applying unfolding on (7.14) and using properties (1), (2), and (3) of Proposition 5.2 along with convergence (7.5b), we obtain under the passage of limit ε→ 0 1 |W | ∫ O×W∗ (div(u0) + divy(û))w dxdy = 0, which eventually gives upon using the fact that û is W ∗-periodic, for all w ∈ L2(O),∫ O div(u0)w dx = 0. (7.15) Finally, upon adding (7.10) with (7.13) and considering (7.15), we establish (7.7). Likewise, one can easily establish (7.8). This settles the proof of the claim. Step 2: First, we are going to identify the limit functions û, v̂, p̂, and q̂. Next, using these identifications, we will identify MW∗(p̂) and MW∗(q̂). Identification of û, v̂, p̂, q̂: Taking successively ϕ ≡ 0 and ψ ≡ 0 in (7.7), yields − divy(A(y)∇yû(x, y)) +∇yp̂(x, y) = divy(A(y))∇u0(x) in O ×W ∗, − divx (∫ W∗ A(y)(∇u0(x) +∇yû(x, y))dy ) +∇ (∫ W∗ p̂(x, y)dy ) = |W ∗|θ0 in O, div(u0) = 0 in O, û(x, ·) is W ∗-periodic. (7.16) In the first line of (7.16), we have the y-independence of ∇u0(x) and the linearity of operators, viz., divergence and gradient, which suggests û(x, y) and p̂(x, y) to be of the following form (see, for e.g., [22, Page 15]): û(x, y) = − n∑ j,β=1 χβj (y) ∂u0j ∂xβ + u1(x), p̂(x, y) = n∑ j,β=1 Πβ j (y) ∂u0j ∂xβ + p0(x). (7.17) where the ordered pair (u1, p0) ∈ (H1(O))n×L2(O), and for 1 ≤ j, β ≤ n, the pair (χβj ,Π β j ) satisfy the cell problem (6.3). Likewise we obtain for the corresponding 14 S. GARG, B. C. SARDAR EJDE-2023/80 adjoint weak formulation (7.8), −divy(A(y)∇yv̂(x, y)) +∇y q̂(x, y) = divy(A(y))∇v0(x) in O ×W ∗, − divx (∫ W∗ A(y)(∇v0(x) +∇yv̂(x, y))dy ) +∇ (∫ W∗ q̂(x, y)dy ) = |W ∗| (u0 − ud) in O, div(v0) = 0 in O, v̂(x, ·) is W ∗-periodic, (7.18) and v̂(x, y) = − n∑ j,β=1 Hβ j (y) ∂v0j ∂xβ + v1(x), q̂(x, y) = n∑ j,β=1 Zβj (y) ∂v0j ∂xβ + q0(x), (7.19) where the ordered pair (v1, q0) ∈ (H1(O))n×L2(O), and for 1 ≤ j, β ≤ n, the pair (Hβ j , Z β j ) satisfy the cell problem (6.5). Identification of MW∗(p̂) and MW∗(q̂): Choosing y = (y1, . . . , yn) as a test function in the weak formulation of (6.3), we obtain n∑ i,l,k,α=1 ∫ W∗ alk ∂ ∂yk (P β j − χ β j ) · ∂P α i ∂yl ∂yi ∂yα dy = n ∫ W∗ Πβ j dy. (7.20) In view of (7.5e), (7.17), and (7.20), we observe that MW∗(p̂) = 1 n|W ∗| n∑ i,j,l,k,α,β=1 ∫ W∗ alk ∂ ∂yk ( P β j − χ β j ) · ∂P α i ∂yl ∂yi ∂yα ∂u0j ∂xβ dy + p0, which upon using the definition of aαβij , gives MW∗(p̂) = 1 n n∑ i,j,α,β=1 aαβij ∂u0j ∂xβ ∂yi ∂yα + p0. (7.21) Also, we re-write equation (7.21) to obtain the identification of MW∗(p̂) as MW∗(p̂) = 1 n A0∇u0 : I + p0. (7.22) Likewise, one can obtain the identification of MW∗(q̂) as MW∗(q̂) = 1 n At0∇v0 : I + q0. (7.23) Thus, from (7.5e) and (7.22); (7.6e) and (7.23), we have the following weak conver- gences: p̃ε ⇀ Θ n A0∇u0 : I + Θp0 weakly in L2(O), (7.24a) q̃ε ⇀ Θ n At0∇v0 : I + Θq0 weakly in L2(O). (7.24b) Step 3: (Claim): The pairs (u0, p0) and (v0, q0) solve the systems (6.2) and (6.4), respectively. EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 15 Proof of the Claim: We now prove that the pair (u0, p0) solves the system (6.2). The proof that the pair (v0, q0) solves the system (6.4) follows analogously. Substituting the values of û(x, y) and p̂(x, y) from expression (7.17) into equation (7.10), we obtain 1 |W | n∑ l,k=1 ∫ O×W∗ alk (∂u0 ∂xk − n∑ j,β=1 ∂χβj ∂yk ∂u0j ∂xβ ) ∂ϕ ∂xl dx dy − 1 |W | n∑ j,β=1 ∫ O×W∗ Πβ j ∂u0j ∂xβ div(ϕ) dx dy −Θ ∫ O p0 div(ϕ) dx = Θ ∫ O θ0 ·ϕ dx. (7.25) Considering P β j = (0, . . . , yj , . . . , 0) with yj at the β−th position, we can express the terms ∂u0 ∂xk , ∂ϕ∂xl , and div(ϕ) as ∂u0 ∂xk = n∑ j,β=1 ∂P β j ∂yk ∂u0j ∂xβ , ∂ϕ ∂xl = n∑ i,α=1 ∂P α i ∂yl ∂ϕi ∂xα , div(ϕ) = n∑ i,α=1 divy(P α i ) ∂ϕi ∂xα . Substituting these expressions in (7.25), we obtain n∑ i,j,α,β=1 ∫ O ( 1 |W ∗| n∑ l,k=1 ∫ W∗ alk ∂ ∂yk (P β j − χ β j ) ∂P α i ∂yl dy )∂u0j ∂xβ ∂ϕi ∂xα dx − n∑ i,j,α,β=1 ∫ O ( 1 |W ∗| ∫ W∗ Πβ j divy(P α i ) dy )∂u0j ∂xβ ∂ϕi ∂xα dx− ∫ O p0 div(ϕ) dx = ∫ O θ0 ·ϕ dx. (7.26) Now, choosing the test function χαi in the weak formulation of (6.3), upon using the fact that divy(χαi ) = divy(P α i ) = δiα, where δ denotes the Kronecker delta function, we obtain∫ W∗ A(y)∇y(P β j − χ β j ) : ∇yχαi dy = ∫ W∗ Πβ j δiα dy. (7.27) Further, substituting (7.27) in (7.26), we obtain n∑ i,j,α,β=1 ∫ O ( 1 |W ∗| n∑ l,k=1 ∫ W∗ alk ∂ ∂yk (P β j − χ β j ) ∂ ∂yl (P α i − χαi ) dy )∂u0j ∂xβ ∂ϕi ∂xα dx − ∫ O p0 div(ϕ) dx = ∫ O θ0 ·ϕ dx. Also, we can write this equality as n∑ i,j,α,β=1 ∫ O bαβij ∂u0j ∂xβ ∂ϕi ∂xα dx− ∫ O p0 div(ϕ) dx = ∫ O θ0 ·ϕ dx, (7.28) 16 S. GARG, B. C. SARDAR EJDE-2023/80 which holds true for all ϕ ∈ (H1 0 (O))n. Also, from equation (7.15), we have∫ O div(u0)w dx = 0, for every w ∈ L2(O). This together with equation (7.28) implies that, for θ = θ0, the pair (u0, p0) ∈ (H1 0 (O))n × L2(O) satisfies the varia- tional formulation of the system (6.2). Therefore, we obtain the optimality system for the minimization problem (6.1). Also, in view of Theorem 6.1, we conclude that the triplet (u0, p0,θ0) is indeed an optimal solution to the problem (6.1). Finally, upon considering the optimal solution’s uniqueness, we establish that the subsequent pair of triplets are equal: (u, p,θ) = (u0, p0,θ0). (7.29) Hence, upon comparing (7.5c), (7.6c), (7.24a), (7.24b), and (7.4) with (7.29), we obtain convergences (7.1c), (7.1d), (7.1e), (7.1f), and (7.1b), respectively. Step 4: Now, we will furnish the proof of the energy convergence for the L2-cost functional. Choosing the test function (uε − ud) in the weak formulation of system (3.5), we obtain under unfolding upon passing ε→ 0, lim ε→0 ∫ O∗ ε |uε − ud|2 dx = 1 |W | lim ε→0 ∫ O×W∗ T ∗ε (Atε)T ∗ ε (∇vε) : T ∗ε (∇(uε − ud)) dx dy + 1 |W | lim ε→0 ∫ O×W∗ T ∗ε (qε) T ∗ ε (div(ud)) dx dy, which gives in view of (7.29), Proposition 5.2(3) and convergences (7.6a), (7.5b), and (7.6d) lim ε→0 ∫ O∗ ε |uε − ud|2 dx = 1 |W | ∫ O×W∗ At(y)(∇v +∇yv̂(x, y)) : ∇y(u− ud) dx dy + 1 |W | ∫ O×W∗ q̂(x, y) div(ud) dx dy. (7.30) Also, using (7.19) in (7.30) along with (7.29), we have upon simplification that lim ε→0 ∫ O∗ ε |uε − ud|2 dx = Θ ( n∑ i,j,α,β=1 ∫ O bβαji ∂vi ∂xα ∂(u− ud)j ∂xβ dx− ∫ O q div(u− ud) dx ) . (7.31) Now, using the test function (u− ud) in the weak formulation of system (6.4), we obtain the following upon comparing with the right-hand side of equation (7.31), lim ε→0 ∫ O∗ ε |uε − ud|2 dx = Θ ∫ O |u− ud|2 dx. (7.32) EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 17 Furthermore, in view of (3.6), (7.6a), and (7.29), we obtain under unfolding upon the passage of limit ε→ 0, lim ε→0 τ 2 ∫ O∗ ε |θε|2 dx = lim ε→0 1 2|W | ∫ O×W∗ |T ∗ε (θε)|2 dx dy = lim ε→0 1 2τ |W | ∫ O×W∗ |T ∗ε (vε)|2 dx dy = 1 2τ |W | ∫ O×W∗ |v|2 dx dy. (7.33) Also, since v is independent of y and comparing the right-hand side of (7.33) with (6.6), we obtain lim ε→0 τ 2 ∫ O∗ ε |θε|2 dx = Θτ 2 ∫ O |θ|2 dx. (7.34) Thus, from equations (7.32) and (7.34), we obtain (7.2). This completes the proof of Theorem 7.1. � Remark 7.2. We observe that in our case, where the size of holes is of the same order as that of the period, i.e., when the size of the holes is large (limε→0 σε = 0), with Neumann data on the part of the boundary ∂O∗ε , the homogenized problem is an interior OCP governed by stationary Stokes System. Where, we define (upon following the convention of Allaire [2]) σε as the ratio between the actual size of the holes and the critical size, with bε denoting the size (say, diameter) of holes: σε = { ε| log( bεε )|1/2 for n = 2,( εn bn−2 ε )1/2 for n ≥ 3. (7.35) Regarding the OCP governed by Stokes equations with homogeneous Dirichlet boundary condition on the boundary of the perforated domain, the authors in [32] studied the cases when the size of the holes is critical and smaller. Concerning the case of smaller size holes, i.e., when limε→0 σε = +∞, they obtained under homogenization the OCP governed by Stokes law, while in the case of critical size holes, i.e., when limε→0 σε = r, where r > 0 is finite, they obtained under homog- enization the OCP governed by Brinkman-type law, leading to the appearance of ‘strange term’ in the limit state equation (see, [32, Theorem 2.2, Page 164-165]). The situation concerning the case of larger size holes, i.e., when limε→0 σε = 0, was left open by the authors in [32] which is then settled by the authors in [27], wherein they employed two-scale convergence method to obtain under homogenization the OCP governed by Darcy’s law (see, [27, Theorem 2.8., Page 7]). One can notice that in our setting, due to Neumann data on the part of the boundary ∂O∗ε , we obtained under homogenization the OCP governed by Stokes system, unlike Darcy law which the authors obtained in [27] due to Dirichlet data on the boundary of the perforated domain. The homogenization of the OCP (3.1) with Neumann data for the cases where the size of holes is smaller (limε→0 σε = +∞) and critical (limε→0 σε = r, where r > 0 is finite), are left open to be explored. It would be interesting to see the type of laws the limit OCP would obey in each case mentioned above. Remark 7.3. For the time-dependent (evolution) Stokes and incompressible Navier- Stokes type equations over periodically perforated domains with holes of large size, 18 S. GARG, B. C. SARDAR EJDE-2023/80 one can find, for example, in the works of [30, 33] that the homogenous/non- homogeneous Dirichlet boundary data is prescribed on the boundary of perforated domain and under homogenization the Darcy-type Law is obtained. However, in the case of Neumann Data on the boundary of perforated domain, it remains a question of one’s interest that owing to the vector value spaces involved of the form (L2(0, T ;V ))n and (L2(0, T ;V ′))n, where, V := {u ∈ (H1(O∗ε))n : div(u) = 0 in O∗ε} and V ′ being the topological dual of V , how one devises an approach to deal with the difficulties that may arise in establishing the homogenization results. 8. 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 tech- nique of periodic unfolding in perforated domains (see, [13, 10]). We employed the Neumann boundary condition on the part of the boundary of the perforated domain. Firstly, we characterized the unique minimizer of the problem (3.1) in terms of the adjoint state. Secondly, we deduced the a priori optimal bounds for control, state, pressure, and their corresponding 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 station- ary Stokes equation posed on a non-perforated domain O. Finally, we established the convergence of energy corresponding to L2−cost functional. Acknowledgments. S. Garg wants to thank the Ministry of Education, Govern- ment of India, for Prime Minister’s Research Fellowship (PMRF-2900953). B. C. Sardar is thankful for the partial support from the Science & Engineering Research Board (SERB) (SRG/2019/000997), Government of India. References [1] G. Allaire; Homogénéisation des équations de Stokes dans un domaine perforé de petits trous répartis périodiquement, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 11, 741–746. [2] G. Allaire; Homogenization of the navier-stokes equations in open sets perforated with tiny holes. ii. noncritical sizes of the holes for a volume distribution and a surface distribution of holes, Arch. Rational Mech. Anal. 113 (1990), no. 3, 261–298. [3] G. Allaire; Homogenization of the Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 44 (1991), no. 6, 605–641. [4] G. Allaire, F. Murat; Homogenization of the Neumann problem with nonisolated holes, As- ymptotic Anal. 7 (1993), no. 2, 81–95, With an appendix written jointly with A. K. Nan- dakumar. [5] M. Beneš, P. Kučera; Solutions to the Navier-Stokes equations with mixed boundary condi- tions in two-dimensional bounded domains, Math. Nachr. 289 (2016), no. 2-3, 194–212. [6] A. Bensoussan, J.-L. Lions, G. Papanicolaou; Asymptotic analysis for periodic struc- tures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. [7] F. Boyer, P. Fabrie; Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013. [8] A. Brillard; Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman’s law via epi-convergence methods, Ann. Fac. Sci. Toulouse Math. (5) 8 (1986/87), no. 2, 225–252. EJDE-2023/80 OPTIMAL CONTROL PROBLEM FOR STOKES SYSTEMS 19 [9] B. Cabarrubias; Homogenization of optimal control problems in perforated domains via pe- riodic unfolding method, Appl. Anal. 95 (2016), no. 11, 2517–2534. [10] D. Cioranescu, A. Damlamian, P. Donato, G. Griso, R. Zaki; The periodic unfolding method in domains with holes, SIAM J. Math. Anal. 44 (2012), no. 2, 718–760. [11] D. Cioranescu, P. Donato, H. I. Ene; Homogenization of the Stokes problem with non- homogeneous slip boundary conditions, Math. Methods Appl. Sci. 19 (1996), no. 11, 857–881. [12] D. Cioranescu, P. Donato, R. Zaki; Periodic unfolding and Robin problems in perforated domains, C. R. Math. Acad. Sci. Paris 342 (2006), no. 7, 469–474. [13] D. Cioranescu, P. Donato, R. Zaki; The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006), no. 4, 467–496. [14] C. Conca; On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl. (9) 64 (1985), no. 1, 31–75. [15] C. Conca, P. Donato, E. C. Jose, I. Mishra; Asymptotic analysis of optimal controls of a semilinear problem in a perforated domain, J. Ramanujan Math. Soc. 31 (2016), no. 3, 265– 305. [16] J. I. Diaz, A. V. Podolskiy, T. A, Shaposhnikova; On the convergence of controls and cost functionals in some optimal control heterogeneous problems when the homogenization process gives rise to some strange terms, J. Math. Anal. Appl. 506 (2022), no. 1, Paper No. 125559, 13. [17] J. I. Diaz, A. V. Podolskiy, T. A, Shaposhnikova; On the homogenization of an optimal control problem in a domain perforated by holes of critical size and arbitrary shape, Dokl. Math. 105 (2022), no. 1, 6–13. [18] Horia I. Ene, Enrique Sánchez-Palencia; Équations et phénomènes de surface pour l’écoulement dans un modèle de milieu poreux, J. Mécanique, 14 (1975), 73–108. MR 371242 [19] J. Fabricius; Stokes flow with kinematic and dynamic boundary conditions, Quart. Appl. Math. 77 (2019), no. 3, 525–544. [20] J. Fabricius, S. Manjate, P. Wall; On pressure-driven Hele-Shaw flow of power-law fluids, Appl. Anal. 101 (2022), no. 14, 5107–5137. [21] S. Garg, B. C. Sardar; Asymptotic analysis of an interior optimal control problem governed by stokes equations, Math. Meth. Appl. Sci. 46 (2023), no. 1, 745–764. [22] S. Gu; Homogenization of Stokes Systems with Periodic Coefficients, ProQuest LLC, Ann Arbor, MI, 2016, Thesis (Ph.D.)–University of Kentucky. [23] S. Kesavan, J. Saint Jean Paulin; Optimal control on perforated domains, Journal of Mathe- matical Analysis and Applications 229 (1999), no. 2, 563–586. [24] J.-L. Lions; Optimal control of systems governed by partial differential equations, Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York- Berlin, 1971. [25] V. Maz’ya, J. Rossmann; Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Ration. Mech. Anal. 194 (2009), no. 2, 669–712. MR 2563641 [26] I. Mishra; Homogenization of boundary optimal control problem, Electron. J. Differential Equations 12 (2022), 1– 23. [27] T. Muthukumar, A. K. Nandakumaran; Darcy-type law associated to an optimal control problem, Electron. J. Differential Equations (2008), No. 16, 12. [28] T. Muthukumar, A. K. Nandakumaran; Homogenization of low-cost control problems on perforated domains, J. Math. Anal. Appl. 351 (2009), no. 1, 29–42. [29] T. Muthukumar, B. C. Sardar; Homogenization of Stokes with Neumann condition on oscil- lating boundary, Asymptot. Anal. 112 (2019), no. 3-4, 125–150. [30] A. K. Nandakumar; Steady and evolution Stokes equations in a porous media [medium] with nonhomogeneous boundary data: a homogenization process, Differential Integral Equations 5 (1992), no. 1, 73–93. [31] P. A. Nguyen, J.-P. Raymond; Boundary stabilization of the Navier-Stokes equations in the case of mixed boundary conditions, SIAM J. Control Optim. 53 (2015), no. 5, 3006–3039. [32] J. Saint Jean Paulin, H. Zoubairi; Optimal control and “strange term” for a Stokes problem in perforated domains, Port. Math. (N.S.) 59 (2002), no. 2, 161–178. [33] L. Signing; Periodic homogenization of the non-stationary Navier-Stokes type equations, Afr. Mat. 28 (2017), no. 3-4, 515–548. 20 S. GARG, B. C. SARDAR EJDE-2023/80 [34] R. Zaki; Homogenization of a Stokes problem in a porous medium by the periodic unfolding method, Asymptot. Anal. 79 (2012), no. 3-4, 229–250. Swati Garg Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar-140001, Punjab, India Email address: swati.19maz0006@iitrpr.ac.in, swatigargmks@gmail.com Bidhan Chandra Sardar Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar-140001, Punjab, India Email address: bcsardar@iitrpr.ac.in 1. Introduction 2. Domain description and notation 2.1. Domain description 2.2. Notation 3. Problem description and optimality condition 4. A priori estimates 5. The method of periodic unfolding for perforated domains 6. Limit optimal control problem 7. Convergence results 8. Conclusions Acknowledgments References