Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 12, pp. 1–23.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

HOMOGENIZATION OF BOUNDARY OPTIMAL

CONTROL PROBLEM

INDIRA MISHRA

Communicated by Jesus Ildefonso Diaz

Abstract. In this article, we study the asymptotic behavior of solutions to

some optimal control problems, governed by an elliptic boundary value prob-
lem with Robin boundary conditions in a periodically perforated domain. The

coefficients of the differential operator in the state equation and in the cost-

functional are rapidly oscillating. We also study the boundary homogenization
of some optimal control problems.

1. Introduction

In this article, we study the convergence of solutions of an optimal control prob-
lem governed by a second order elliptic boundary value problem in a periodically
perforated domain. The sizes of the holes are same as the period, the holes can
intersect the boundary of the domain. The coefficients of the state equation and
the cost-functional are rapidly oscillating. The cost-functional involves a Dirichlet
type integral of the state function. We prescribe a linear Robin condition on the
boundary of holes and the homogeneous Dirichlet condition on the external bound-
ary. The Robin conditions appear in several physical situations such as chemical
reactive flows [12] or climatization [20]. We use periodic unfolding method for the
homogenization.

Periodic unfolding method was introduced for the perforated domain in [8]. In
general, one does not need any extension operator in this method, which makes
things simpler while dealing with problems involving non-homogeneous boundary
conditions.

A version of our problem was studied by Kesavan et al. [16] in a perforated
domain. Then Muthukumar et al. [19] studied this problem in a periodically per-
forated domain using two-scale convergence. Recently Diaz et al. [13] considered
an optimal control problem in a perforated domain for the case of critically small
holes, which after homogenization, gives rise to strange terms in the limit equations.
Cabarrubias [5] studied similar problems using unfolding method, where they prove
the energy convergence with L2-cost functional only. However in this paper, we es-
tablish the energy convergence associated with Dirichlet-cost functional.

2020 Mathematics Subject Classification. 35B27, 35J20, 35J56, 35Q93, 49J20.
Key words and phrases. Homogenization; optimal control; periodic unfolding method;

elliptic problems; boundary control.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted June 21, 2021. Published February 17, 2022.

1



2 I. MISHRA EJDE-2022/12

In homogenization of state and adjoint equation with of Robin’s boundary condi-
tion in a periodically perforated domain, we face two main difficulties. The first one
arises in obtaining the a-priori estimates for the state and adjoint variables. This
is because of presence of some surface terms due to Robin boundary conditions,
thanks to the boundary unfolding operators (cf. Section 2), we overcame this.

The second difficulty lies in the homogenization of the adjoint equation for which
we used the special cell problems introduced by Allaire [1] and further used in [19].
The presence of the Robin conditions on the boundary of holes, and the cost-
functional (1.3) involving gradients and the oscillating coefficients Bε together led
us to play repeatedly with these cell problems, which contributed the nontrivial
terms in the homogenized equation. Our paper generalizes the existing results for
the elliptic case in a periodically perforated domain with a more general condition
(Robin conditions) on the boundary of holes. Our second main result comprises of
studying the boundary homogenization of some optimal control problems.

The organization of this article is as follows: In Section 1, we introduce the
problem and the method used. In Section 1.1, we give the preliminaries and briefly
describe the setting of the problem and the optimality conditions. Section 2 is the
brief review of the periodic unfolding method for the perforated domain. In Section
3, we introduce some cell problems, state and prove our first main result Theorem
3.2. We also establish Theorem 3.3, the ellipticity of the bilinear form defined
by perturbed matrix B# (see (3.6)). In Section 3.1, we observe the existence of
unique optimal controls over certain convex sets. In Section 4, we study our second
main result where we study the boundary homogenization of some optimal control
problems, in a sense that control is acted upon on a part of the external boundary.

1.1. Notation and problem setting. Suppose Ω is an open bounded set of RN
(N ≥ 2) with a Lipschitz continuous boundary ∂Ω such that |∂Ω| = 0. Let Y , T
and Y ∗ be as follows: Y = (0, 1)N is a reference cell, or more generally a set having
the paving property with respect to a basis (b1, . . . , bN ) defining the periods,

Y =
{
y ∈ RN : y =

N∑
i=1

yibi, (y1, y2, . . . , yN ) ∈ (0, 1)N
}
.

T ⊂ Y is an open set with Lipschitz boundary with finite number of connected
components such that ∂T does not contain the summits of Y . Perforated reference
cell Y ∗ = Y \T is a connected open set. Let {ε} be a positive sequence that
converges to zero and we set

G = {ξ ∈ RN : ξ =

N∑
i=1

kibi (k1, . . . , kN ) ∈ ZN}, Ξε = {ξ ∈ G : ε(ξ + Y ) ⊂ Ω}.

The perforated domain Ω∗ε is defined by

Ω∗ε = Ω\Tε, where Tε = ∪ξ∈Gε(ξ + T ).

Following the notation introduced in [9] for the periodic unfolding method in per-
forated domains, we set

Ω̂ε = interior
(
∪ξ∈Ξε

ε(ξ + Y )
)
, Λε = Ω\Ω̂ε.

By construction, Ω̂ε is the interior of the largest union of ε(ξ + Y ) cells such that
ε(ξ + Y ) is included in Ω and Λε is the subset of Ω containing the parts from the



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 3

Figure 1. Perforated domain Ω∗ε and the reference set Y .

ε(ξ+Y ) cells intersecting the boundary ∂Ω. We define the corresponding perforated
sets as

Ω̂∗ε = Ω̂ε\Tε, Λ∗ε = Ω∗ε\Ω̂∗ε.
We decompose the boundary of the perforated domain Ω∗ε as (see Figure 1)

∂Ω∗ε = Γε0 ∪ Γε1, where Γε1 = ∂Ω̂∗ε ∩ ∂Tε and Γε0 = ∂Ω∗ε\Γε1,

so that Γε1 is the boundary of set of holes included in Ω̂ε.
Boundary of holes inside Ω∗ε is Γε1, the remaining part including holes inside dark

boundary is Λ∗ε and the boundary of holes contained in this part together with the
external boundary ∂Ω is Γε0. Hence Ω∗ε is a periodically perforated domain where
the size of the holes are of same order as the period.

We shall use the following notation throughout the paper.

• |E| denotes the Lebesgue measure of the measurable set E.
• Y ∗ = Y \ T .
• Θ = |Y ∗|/|Y | the proportion of the material.
• MY (v) is the mean value of v over the measurable set Y .
• χE is the characteristic function of the set E.
• ũ is the extension by zero on E of a function u defined on Eε(= E ∩ Ωε).
• (nε) = (niε)

N
i=1 the unit external normal vector with respect to Ωε.

• Aε(x) = A
(
x
ε

)
a.e. in Ω, for any ε > 0.

• ‖u‖0,E and ‖u‖1,E , represents respectively L2 and H1-norms defined over
the set E.
• C∞per(Y ) is a subset of C∞(RN ), and it consists of Y -periodic functions.

• H1
per(Y

∗) is the closure of C∞per(Y ) with respect to H1-norm.

• H1
per(Y

∗)/R is the space of equivalence classes defined by: ‘u ' v′ ⇔ u− v
is constant, for all u, v ∈ H1

per(Y
∗).

• L2(Ω;H1
per(Y

∗)) is the space of functions f taking x ∈ Ω to f(x, ·) ∈
H1

per(Y
∗) and ‖f(x, ·)‖H1

per(Y
∗) ∈ L2(Ω).



4 I. MISHRA EJDE-2022/12

and the constants at different places are denoted by C, which are independent of
ε. Note that

χΩ∗ε
⇀ Θ = |Y ∗|/|Y | weak∗ in L∞(Ω).

Now we discuss the setting of the problem. Let Uεad ⊂ L2(Ω∗ε) be a closed, convex
subset. For given constants 0 < αm ≤ αM , we denote by M(αm, αM ,Ω) the set of
all N ×N matrices A = A(x) such that

A ∈ L∞(Ω)N×N ,

(A(x)ξ, ξ) ≥ αm|ξ|2 and |A(x)ξ| ≤ αM |ξ|, ∀ξ ∈ RN and a.e. x ∈ Ω.
(1.1)

Let Aε ∈ M(αm, αM ,Ω), Bε ∈ M(βm, βM ,Ω), where 0 < βm ≤ βM are given
constants.

Assume that Bε is symmetric for every ε > 0. We consider the optimal control
problem governed by the boundary value problem

−div(Aε∇uε) = f + θε in Ω∗ε,

Aε∇uε · nε + hεuε = εgε on Γε1,

uε = 0 on Γε0,

(1.2)

where f ∈ L2(Ω), θε ∈ L2(Ω∗ε), nε is unit outward normal to Γε1, h is a real, positive
number, gε(x) = g(x/ε), where g is Y -periodic function in L2(∂T ). For N0 > 0,
the following cost functional is associated with (1.2),

Jε(θε) =
1

2

∫
Ω∗ε

Bε∇uε∇uε dx+
N0

2

∫
Ω∗ε

θ2
ε dx. (1.3)

Then the optimal control problem given by (1.2) and (1.3) admits a unique
solution θ∗ε ∈ Uεad minimizing the cost-functional (1.3) over Uεad. We wish to study

the limiting behavior of θ∗ε as ε→ 0. Further if θ̃∗ε ⇀ θ∗0 weakly in L2(Ω), we would
like to characterize θ∗0 as the optimal control of a similar problem in a fixed domain
Ω. Using periodic unfolding, we study the homogenization of state (1.2) and its
adjoint equations in periodically perforated domain, when the holes are of same
size as period.

We introduce the space

Vε = {v ∈ H1(Ω∗ε) : v = 0 on Γε0}.

This is a Banach space equipped with

‖v‖Vε
:= ‖∇v‖[L2(Ω∗ε)]N , ∀u ∈ Vε. (1.4)

The weak formulation of (1.2) is given as follows: Find uε ∈ Vε solution of∫
Ω∗ε

Aε∇uε∇ϕdx+ hε

∫
Γε

1

yεϕdσ(x) =

∫
Ω∗ε

(f + θ)ϕdx+ ε

∫
Γε

1

gεϕdσ(x), (1.5)

for all ϕ ∈ Vε.
The assumption that ∂T is Lipschitz continuous, is necessary, in order to write

surface integrals on the boundary of holes, appearing in the variational formulation
of the problem. We obtain the following result as an application of Lax Milgram
theorem.

Theorem 1.1. There exists a unique solution of (1.5) when the assumptions on
the data stated after (1.2) holds.



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 5

1.2. Optimality conditions. The optimality conditions [18, Theorem 1.3] are
given by

∫
Ω∗ε

∂Jε
∂θ (θ − θ∗ε) ≥ 0. Now, in view of [21, Chapter 2] and [15, page 140],

these conditions can be rewritten as∫
Ω∗ε

(p∗ε +Mθ∗ε)(θ − θ∗ε) ≥ 0,

where p∗ε = pε(θ
∗
ε) is the solution of the adjoint equation of (1.2), for θ = θ∗ε ,

−div(tAε∇pε) = −div(Bε∇uε) in Ω∗ε,

(tAε∇pε) · nε − pεhε = (Bε∇uε) · nε on Γε1,

pε = 0 on Γε0.

(1.6)

We consider the following optimal control problem for the cost functional (1.3):{
inf Jε(uε, θ) : (uε, θ) ∈ Vε × L2(Ω∗ε), (uε, θ) satisfies (1.2)

}
. (1.7)

We have the following results, which can be proved along the same lines as in [18]
and [21, Chapter 2].

Theorem 1.2. For each ε > 0, the optimal control problem (1.7) admits a unique
solution.

Let u∗ε = uε(θ
∗
ε) be the optimal state. Then the characterization of θ∗ε is given

below.

Theorem 1.3. For (u∗ε, θ
∗
ε), the optimal solution of (1.7), let p∗ε = pε(θ

∗
ε) be the

optimal adjoint state. Then the optimal control is given by

θ∗ε = − 1

M
p∗ε. (1.8)

2. The periodic unfolding method for perforated domain

In this section, we briefly recall the definitions and properties of the unfolding
operator T ∗ε and the boundary unfolding operator T bε . For more details on this
topic, we refer to [8].

Let [x]Y =
∑N
j=1 kjbj be the unique integer combination of periods such that for

any x ∈ RN , x− [x]Y is in Y . Set {x}Y = x− [x]Y . In particular, for any ε > 0,

x = ε
([x
ε

]
Y

+
{x
ε

}
Y

)
for all x ∈ RN .

Definition 2.1. For any Lebesgue-measurable function φ on Ω∗ε, the unfolding
operator T ∗ε is defined as

T ∗ε (φ)(x, y) =

{
φ
(
ε
[
x
ε

]
Y

+ εy
)
, a.e. for (x, y) ∈ Ω̂ε × Y ∗,

0, a.e. for (x, y) ∈ Λε × Y ∗.

In what follows, if φ is a function on a domain containing Ω∗ε, we will use the
notation T ∗ε (φ) instead of T ∗ε (φ|Ω∗ε ).

Proposition 2.2. Let p ∈ (1,+∞). Then

(1) T ∗ε is linear and continuous from Lp(Ω∗ε) to Lp(Ω× Y ∗).
(2) T ∗ε (φψ) = T ∗ε (φ)T ∗ε (ψ) for every φ, ψ ∈ Lp(Ω∗ε).
(3) For w ∈ Lp(Ω), T ∗ε (w)→ w strongly in Lp(Ω× Y ∗).



6 I. MISHRA EJDE-2022/12

(4) For all φ ∈ L1(Ω∗ε) we have∫
Ω̂∗ε

φ(x) dx =

∫
Ω∗ε

φ(x) dx−
∫

Λ∗ε

φ(x) dx =
1

|Y |

∫
Ω×Y ∗

T ∗ε (φ)(x, y) dx dy.

Moreover, if {φε} is a bounded sequence in Lr(Ω∗ε) for some r > 1, then

lim
ε→0

∫
Ω∗ε

φε(x) dx = lim
ε→0

1

|Y |

∫
Ω×Y ∗

T ∗ε (φε)(x, y) dx dy.

(5) Let φε ∈ Lp(Ω) be such that φε → φ strongly in Lp(Ω). Then T ∗ε (φε)→ φ
strongly in Lp(Ω× Y ∗).

(6) Let ϕ ∈ Lp(Y ∗) be a Y -periodic function and set ϕε(x) = ϕ
(
x
ε

)
. Then

T ∗ε (ϕε)(x, y) = ϕ(y) a.e. in Ω̂ε × Y ∗.

Proposition 2.3. Suppose Ω is a bounded open subset of RN with Lipschitz bound-
ary ∂Ω. Let wε ∈W 1,p

0 (Ω∗ε; ∂Ω ∩ ∂Ω∗ε) satisfy

‖∇wε‖Lp(Ωε) ≤ C,

where C is a positive constant independent of ε. Then there exist w0 ∈ W 1,p
0 (Ω)

and ŵ ∈ Lp(Ω;W 1,p
per(Y

∗)) with MY ∗(ŵ) = 0, such that up to a subsequence,

T ∗ε (wε)→ w0 strongly in Lp(Ω;W 1,p(Y ∗)),

T ∗ε (∇wε) ⇀ ∇xw0 +∇yŵ weakly in (Lp(Ω× Y ∗))N ,
w̃ε ⇀ Θw0 weakly in Lp(Ω),

(2.1)

as ε tends to zero, where Θ = |Y ∗|/|Y |.

We now recall the definition and properties of the boundary unfolding operator
T bε .

Definition 2.4. For any function ϕ, Lebesgue-measurable on ∂Ω̂∗ε∩∂Tε, the bound-
ary unfolding operator is defined as

T bε (ϕ)(x, y) =

{
ϕ
(
ε[xε ]Y + εy

)
, a.e. for (x, y) ∈ Ω̂ε × ∂T,

0, a.e. for (x, y) ∈ Λε × ∂T.

Proposition 2.5 ([8]). Let p ∈ (1,+∞). Then

(1) T bε is a linear operator from Lp(∂Tε) to Lp(Ω× ∂T ).
(2) T bε (φψ) = T bε (φ)T bε (ψ) for every φ, ψ ∈ Lp(∂Tε).
(3) Let φ ∈ Lp(∂T ) be a Y -periodic function. Set φε(x) = φ(xε ). Then

T bε (φε)(x, y) = φ(y) a.e. in Ω∗ε × ∂T.
(4) For all φ ∈ L1(∂Tε), the integration formula is given by∫

Γε
1

φ(x) dσ(x) =
1

ε|Y |

∫
Ω×∂T

T bε (φ)(x, y) dx dσ(y).

(5) Let φ ∈ Lp(∂T ∗ε ). Then T bε (φ)→ φ strongly in Lp(Ω× ∂T ).

Proposition 2.6 (see [8, Proposition 5.7]). We have the following convergence
results:

(1) Let ϕ ∈ L2(Ω). Then as ε→ 0, one has the convergence∫
RN×∂T

T bε (ϕ)(x, y) dx dσ(y)→
∫
RN×∂T

ϕ̃ dx dσ(y). (2.2)



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 7

(2) Let ϕ ∈ L2(Ω). Then T bε (ϕ)→ ϕ̃ strongly in L2(RN × ∂T ).
(3) Let ϕε ∈ L2(∂Tε) for every ε, such that T bε (ϕε) ⇀ ϕ̂ weakly in L2(RN×∂T ).

Then

ε

∫
∂Tε

ϕεψ dσ(x)→ 1

|Y |

∫
RN×∂T

ϕ̂(x, y)ψ(x) dx dσ(y),

for all ψ ∈ H1(Ω).
(4) Let ϕε ∈ H1(Ωε) for every ε and ϕ̂ ∈ H1(Ω) such that T ∗ε (ϕε) ⇀ ϕ̂ weakly

in L2
loc(Ω;H1(Y ∗)). Then

T bε (ϕε) ⇀ ϕ̂ weakly in L2
loc(Ω;H

1
2 (∂T )).

Let g be a Y -periodic function in L2(∂T ) and let gε(x) = g(x/ε) for all x in the
set RN\ ∪ξ∈ZN ε(ξ + T ). Then we have the following two results (cf. [8, Cor 5.4,
Prop. 5.6]), which will be needed for the homogenization of some problems.

Proposition 2.7. For every Φ ∈ Vε and gε as above, the following inequality holds∣∣ ∫
Γε

1

g
(x
ε

)
Φ(x) dσ(x)

∣∣ ≤ C

ε
(|M∂T (g)|+ ε) ‖∇Φ‖[L2(Ω∗ε)]N . (2.3)

3. Main results

To state and prove our first main result, we need the following hypothesis:

(H1) Let Aε ∈M(αm, αM ,Ω), Bε ∈M(βm, βM ,Ω).
(H2) Let A = A(x, y) ∈ M(αm, αM ,Ω × Y ∗), B ∈ M(βm, βM ,Ω × Y ∗), such

that

T ∗ε (Aε)→ A(x, y) a.e. in Ω× Y ∗,
T ∗ε (Bε)→ B(x, y) a.e. in Ω× Y ∗.

(H3) A, B are Y -periodic with respect to the second variable y.

Now we define some cell problems, needed to state our first main results, and for
the identification of the limits. Such cell problems were introduced by Allaire [1],
and further used by Muthukumar et al. [19].

For 1 ≤ i ≤ N , let µi ∈ H1
per(Y

∗)/R be the solution of the cell problem

−div(A(x, y)[∇yµi(x, y) + ei]) = 0 in Y ∗,

A(x, y)[∇yµi(x, y) + ei] · ν = 0 on ∂Y ∗\∂Y,
y 7→ µi(x, y) is Y -periodic.

(3.1)

Let ωi ∈ H1
per(Y

∗)/R be the solution of the adjoint cell problem

−div(tA(x, y)[∇yωi(x, y) + ei]) = 0 in Y ∗,
tA(x, y)[∇yωi(x, y) + ei] · ν = 0 on ∂Y ∗\∂Y,

y 7→ ωi(x, y) is Y -periodic.

(3.2)

Let ψi ∈ H1
per(Y

∗)/R, be the solution of the cell problem

−div
(
tA(x, y)∇yψi(x, y)−B(x, y)(∇yµi + ei)

)
= 0 in Y ∗,(

tA(x, y)∇yψi(x, y)−B(x, y)(∇yµi + ei)
)
· ν = 0, on ∂Y ∗\∂Y,

y 7→ µi(x, y) is Y -periodic.

(3.3)



8 I. MISHRA EJDE-2022/12

Here {e1, e2, . . . , eN} is the standard basis of RN . The homogenized matrix A0 is
defined as

(A0)ij =

∫
Y ∗
A(x, y)[∇yµi(x, y) + ei] · [∇yµj + ej ] dy. (3.4)

The homogenized transposed matrix tA0 is

(tA0)ij =

∫
Y ∗

tA(x, y)[∇yωi(x, y) + ei] · [∇yωj(x, y) + ej ] dy, (3.5)

and the perturbed matrix B# is

(B#)ei =

∫
Y ∗
{B(x, y)[∇yµi(x, y) + ei]− tA(x, y)∇yψi(x, y)} dy. (3.6)

To see the forms of the above homogenized matrices, we refer to [7, Propositions
6.8, 6.9]. We now state a Poincaré inequality result for the perforated domain Ω∗ε
(when the holes can even meet the boundary).

Lemma 3.1 (see [2, Lemma A.4]). There exists a positive constant C, independent
of ε, such that

‖u‖0,Ω∗ε ≤ C‖∇u‖[L2(Ω∗ε)]N . (3.7)

The above lemma gives an equivalent norm on Vε, as ‖u‖Vε = ‖∇u‖0,Ωε . Now,
we state our first main result of the paper.

Theorem 3.2. Assume (H1)–(H3) hold. Let uε ∈ Vε and pε ∈ Vε be the solutions
of the following systems, respectively

−div(Aε∇uε) = f + θε in Ω∗ε,

Aε∇uε · nε + hεuε = εgε on Γε1,

uε = 0 on Γε0,

(3.8)

and
−div

(
tAε∇pε −Bε∇uε

)
= 0 in Ω∗ε,

(tAε∇pε −Bε∇uε) · nε = pεhε on Γε1,

pε = 0 on Γε0,

(3.9)

where f ∈ L2(Ω) is fixed, θε ∈ L2(Ωε), h > 0 is a real number, gε(x) = g(x/ε),
where g is Y -periodic function in L2(∂T ). Then there exist functions u0, p0 ∈
H1

0 (Ω) and θ0 ∈ L2(Ω) such that for a subsequence, we have

ũε ⇀ Θu0 weakly in L2(Ω),

p̃ε ⇀ Θp0 weakly in L2(Ω),

θ̃ε ⇀ Θθ0 weakly in L2(Ω).

(3.10)

The functions u0 and p0 satisfy the following systems respectively,

−div(A0∇u0) + h
|∂T |
|Y |

u0 = Θ(f + θ0) +
|∂T |
|Y |
M∂T (g) in Ω,

u0 = 0 on ∂Ω,

(3.11)

and

−div(tA0∇p0 −B#∇u0) = h
|∂T |
|Y |

p0 in Ω,

p0 = 0 on ∂Ω,

(3.12)



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 9

where Θ = |Y ∗|/|Y |, the matrices A0, tA0 and B# are defined by (3.4), (3.5), and
(3.6) respectively.

Proof. We prove this theorem in three main steps. In the first step, we establish
the convergence of solutions (3.10). In the second step, we homogenize the state
equation, and in the third step, we homogenize the adjoint equation using the cell
problems introduced in the beginning of this section.

Step 1: Convergence of solutions of (3.8) and (3.9). Taking ϕ ∈ D(Ω) as a test
function in (3.8), we have∫

Ω∗ε

Aε∇uε∇ϕdx+hε

∫
∂Tε

uεϕdσ(x) =

∫
Ω∗ε

(f+θε)ϕdx+ε

∫
∂Tε

gεϕdσ(x). (3.13)

We first establish “a priori estimates” for uε. Considering uε as a test-function in
(3.13), we have

αm‖∇uε‖2[L2(Ω∗ε)]N + hε‖uε‖2L2(∂Tε)

≤ (‖f‖L2(Ω) + ‖θε‖L2(Ω∗ε))‖uε‖0,Ω∗ε + ε
∣∣ ∫
∂Tε

gεuε dσ
∣∣. (3.14)

Using the Poincaré inequality (3.7) and Proposition 2.7 in (3.14), we obtain

αm‖∇uε‖2[L2(Ω∗ε)]N ≤ C(1 + ε+ |M∂T (g)|)‖∇uε‖[L2(Ω∗ε)]N .

Hence we have

‖uε‖H1(Ω∗ε) ≤ C̃. (3.15)

Thus we deduce that there exists U0 ∈ H1(Ω) such that

ũε ⇀ U0 weakly in L2(Ω).

By [8, Theorem 3.2 (2, 3, 4)], there exist u0 ∈ H1
0 (Ω) and û ∈ L2(Ω, H1

per(Y
∗))

such that

(i) T ∗ε (uε) ⇀ u0 weakly in L2
loc(Ω;H1(Y ∗)),

(ii) T ∗ε (∇x(uε)) ⇀ ∇xu0 +∇yû weakly in L2
loc(Omega× Y ∗).

To identify U0, let ϕ ∈ D(Ω) and consider∫
Ω

ũεϕdx =

∫
Ω∗ε

uεϕdx =
1

|Y |

∫
Ω×Y ∗

T ∗ε (uε)T ∗ε (ϕ) dx dy.

The former convergences yield∫
Ω∗ε

uεϕdx→
1

|Y |

∫
Ω×Y ∗

u0(x)ϕ(x) dx dy =
|Y ∗|
|Y |

∫
Ω

u0ϕdx = Θ

∫
Ω

u0ϕdx.

But we know that
∫

Ω
ũεϕdx ⇀

∫
Ω
U0ϕdx, as ε → 0. Consequently, we have

U0 = Θu0 and we obtain the first convergence of (3.10). We note that u0 is a
function of x only. Letting pε as a test function in the adjoint eq. (3.9), we have∫

Ω∗ε

tAε∇pε∇pε dx−
∫

Ω∗ε

Bε∇uε∇pε dx = hε

∫
∂Tε

pε dσ(x).

By ellipticity of Aε, boundedness of Bε, and Cauchy-Schwarz inequality, we have

βm‖∇pε‖2[L2(Ω∗ε)]N

≤ βM‖∇uε‖[L2(Ω∗ε)]N ‖∇pε‖[L2(Ω∗ε)]N + hε
∣∣ ∫
∂Tε

1.pε dσ(x)
∣∣. (3.16)



10 I. MISHRA EJDE-2022/12

It follows by (3.15), Proposition 2.7 (with gε=1) and Poincaré inequality (3.7) in
the last term, that

βm‖∇pε‖2[L2(Ω∗ε)]N ≤ (βMC + hε)‖∇pε‖[L2(Ω∗ε)]N .

This implies that
‖pε‖H1(Ω∗ε) ≤ C. (3.17)

A similar argument, as to get (3.10)(i), would give us the existence of p0 ∈ H1
0 (Ω)

such that (3.10)(ii) holds. Now from (1.8) and (3.17), we have

‖θ̃ε‖L2(Ω∗ε) ≤ C,

where is a positive constant. Thus there exists θ0 ∈ L2(Ω) such that the convergence
(3.10)(iii) holds.

Step2: Homogenization of the state equation (3.8). We pass to the limit in (3.13),
as ε→ 0. Using the Proposition 2.2, Proposition 2.5(4) and by unfolding in (3.13),
we obtain∫

Ω̃×Y ∗
T ∗ε (Aε)T ∗ε (∇uε)T ∗ε (∇ϕ) dx dy + h

∫
Ω̃×∂T

T bε (uε)T bε (ϕ) dx dσ(y)

=

∫
Ω̃×Y ∗

T ∗ε (f)T ∗ε (ϕ) dx dy +

∫
Ω̃×Y ∗

T ∗ε (θε)T ∗ε (ϕ) dx dy

+

∫
Ω̃×∂T

T bε (gε)T bε (ϕ) dx dσ(y).

(3.18)

From Proposition 2.2(3) and Proposition 2.6(2, 4), we obtain by passing to the
limit that∫

Ω×Y ∗
A(x, y)(∇xu0 +∇yû)∇ϕ(x) dx dy + h

∫
Ω×∂T

u0ϕdx dσ(y)

=

∫
Ω×Y ∗

(f + θ0)ϕdx dy +

∫
Ω×∂T

gϕ dx dσ(y).

Since u0, f , θ0, and ϕ are functions of x only, we have∫
Ω×Y ∗

A(x, y)(∇xu0 +∇yû)∇ϕ(x) dx dy + h|∂T |
∫

Ω×∂T
u0ϕdx dσ(y)

= |Y ∗|
∫

Ω

(f + θ0)ϕdx+

∫
Ω

ϕdx

∫
∂T

g dσ(y).

(3.19)

By denseness, this result is valid for all ϕ ∈ H1
0 (Ω). We define a new test-function

vε = εϕ(x)ξ
(x
ε

)
, where ϕ ∈ D(Ω), ξ ∈ H1

per(Y
∗).

We observe that T ∗ε (vε) = εT ∗ε (ϕ)ξ and ∇vε = ε∇xϕξ
(
.
ε

)
+ ϕ∇yξ

(
.
ε

)
. Hence

T ∗ε (vε) ⇀ 0 weakly in L2(Ω;H1(Y ∗)),

T ∗ε (∇vε) ⇀ ϕ∇yξ weakly in L2(Ω× Y ∗).
Taking vε as a test-function in (3.8) and by unfolding operator in the resulting
variational-formulation, we obtain∫

Ω̃×Y ∗
T ∗ε (Aε)T ∗ε (∇uε)T ∗ε (∇vε) dx dy + h

∫
Ω̃×∂T

T bε (uε)T bε (vε) dx dσ(y)

=

∫
Ω̃×Y ∗

T ∗ε (f)T ∗ε (vε) dx dy +

∫
Ω̃×Y ∗

T ∗ε (θε)T ∗ε (vε) dx dy



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 11

+

∫
Ω̃×∂T

T bε (gε)T bε (vε) dx dσ(y).

When passing to the limit, we obtain∫
Ω×Y ∗

A(x, y)(∇xu0 +∇yû)ϕ(x)∇yξ(y) dx dy = 0.

It is known from [4], that the space C∞c (Ω)
⊗
C∞per(Y ) consisting of all finite sums

of functions of the form g(x)h(y) (g ∈ C∞c (Ω), h ∈ C∞per(Y )) is dense in Lp(Ω× Y ),

for 1 ≤ p <∞. It follows by denseness that for all ζ ∈ L2(Ω;H1
per(Y

∗)), we have∫
Ω×Y ∗

A(x, y)(∇xu0 +∇yû)∇yζ(x, y) dx dy = 0. (3.20)

Finally adding (3.19) (for ϕ ∈ H1
0 (Ω)) and (3.20), we obtain the following varia-

tional formulation of (3.8): For all ϕ ∈ H1
0 (Ω) and all ζ ∈ L2(Ω;H1

per(Y
∗)),∫

Ω×Y ∗
tA(x, y)(∇xu0 +∇yû)(∇xϕ+∇yζ(x, y)) dx dy + h|∂T |

∫
Ω

u0ϕdx

= |Y ∗|
∫

Ω

(f + θ0)ϕdx+

∫
Ω

ϕdx

∫
∂T

g dσ(y),

(3.21)

where (u0, û) is the unique solution of (3.21). The proof that u0 is the solution of
(3.11) follows along the same lines as in the proof given in [7, Chapter 9].

Taking successively ϕ = 0 and ζ = 0 in (3.21) yields

û(x, y) =

N∑
i=1

µi(x, y)
∂u0

∂xi
(x), (3.22)

where µi satisfies the cell problem (3.1). Substituting the value of û obtained in
(3.22) in the variational formulation (3.21) and integrating by parts with respect
to x, after putting ζ = 0, we obtain the expression for A0. By a standard argument
[7], it is easily seen that the matrix A0 is elliptic. Then the uniqueness of u0 as a
solution of state equation (3.11), follows from the Lax-Milgram theorem.

Step3: Homogenization of the adjoint equation. Taking ϕ ∈ D(Ω) as a test-
function in (3.9), we have∫

Ω∗ε

tAε∇pε∇ϕdx−
∫

Ω∗ε

Bε∇uε∇ϕdx−
∫
∂Ω∗ε

pεhεϕ dx = 0.

By unfolding and using Propositions 2.5 and 5.2, we obtain∫
Ω̃∗ε×Y ∗

T ∗ε (tAε)T ∗ε (∇pε)T ∗ε (∇ϕ) dx dy −
∫

Ω̃∗ε×Y ∗
T ∗ε (Bε)T ∗ε (∇uε)T ∗ε (∇ϕ) dx dy

−
∫

Ω̃∗ε×∂T
T bε (pε)hT bε (ϕ) dx dσ(y) = 0.

We pass the limit as ε→ 0 and obtain∫
Ω×Y ∗

tA(x, y)(∇xp0 +∇yp̂)∇ϕdx dy

−
∫

Ω×Y ∗
B(x, y)(∇xu0 +∇yû)∇ϕdx dy

= h|∂T |
∫

Ω

p0ϕdx.

(3.23)



12 I. MISHRA EJDE-2022/12

Next we take vε = εϕξ
(
x
ε

)
as a test-function in (3.9), where ϕ ∈ D(Ω) and ξ ∈

H1
per(Y

∗),∫
Ω̃∗ε×Y ∗

T ∗ε (tAε)T ∗ε (∇pε)T ∗ε (∇vε) dx dy −
∫

Ω̃∗ε×Y ∗
T ∗ε (Bε)T ∗ε (∇uε)T ∗ε (∇vε) dx dy

− h
∫

Ω̃∗ε×∂T
T bε (pε)T bε (vε) dx dσ(y) = 0,

which by passing to the limit gives∫
Ω×Y ∗

tA(x, y)(∇xp0 +∇yp̂)ϕ(x)∇yξ(x, y) dx dy

=

∫
Ω×Y ∗

B(x, y)(∇xu0 +∇yû)ϕ(x)∇yξ(x, y) dx dy.

Using denseness arguments as before, for all ζ ∈ L2(Ω;H1
per(Y

∗), we have∫
Ω×Y ∗

tA(x, y)(∇xp0 +∇yp̂)∇yζ(x, y) dx dy

=

∫
Ω×Y ∗

B(x, y)(∇xu0 +∇yû)∇yζ(x, y) dx dy.

(3.24)

Adding (3.23) and (3.24), we obtain the variational formulation of the adjoint
problem: For all ϕ ∈ H1

0 (Ω) and all ζ ∈ L2(Ω;H1
per(Y

∗)),∫
Ω×Y ∗

tA(x, y)(∇xp0 +∇yp̂)(∇xϕ+∇yζ(x, y)) dx dy − h|∂T |
∫

Ω

p0ϕdx

=

∫
Ω×Y ∗

B(x, y)(∇xu0 +∇yû)(∇xϕ+∇yζ(x, y) dx dy,

(3.25)

where (p0, p̂) is the unique solution of (3.25).
Now, taking u0 as a test-function in (3.3), we have∫

Ω×Y ∗
B(x, y)(∇yµi + ei)∇u0 =

∫
Ω×Y ∗

tA(x, y)∇yψi(x, y)∇u0. (3.26)

Letting ϕ = 0 in the variational-formulation (3.25), and putting û =
∑N
i=1 µi

∂u0

∂xi
,

we have ∫
Ω×Y ∗

tA(x, y)(∇xp0 +∇yp̂)∇yζ(x, y) dx dy

=

∫
Ω×Y ∗

B(x, y)(ei +∇yµi)∇xu0∇yζ(x, y) dx dy.

=

∫
Ω×Y ∗

tA(x, y)∇yψi(x, y)∇xu0∇yζ(x, y) dx dy ,

which follows by (3.26). This implies that∫
Ω×Y ∗

tA(x, y)∇yp̂ ∇yζ(x, y) dx dy

=

∫
Ω×Y ∗

tA(x, y)∇yψi(x, y)∇xu0∇yζ(x, y) dx dy

−
∫

Ω×Y ∗
tA(x, y)∇xp0∇yζ(x, y) dx dy.

(3.27)



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 13

Taking p0 as a test-function in (3.2), we obtain∫
Ω×Y ∗

tA(x, y)(∇yωi + ei)∇p0 = 0. (3.28)

Adding and subtracting
∫

Ω×Y ∗
tA(x, y)(∇yωi+ei)∇xp0∇yζ(x, y) dx dy from (3.27),

we have ∫
Ω×Y ∗

tA(x, y)∇yp̂ ∇yζ(x, y) dx dy

=

∫
Ω×Y ∗

tA(x, y)∇yψi(x, y)∇xu0∇yζ(x, y) dx dy

−
∫

Ω×Y ∗
tA(x, y)∇y(ωi + ei)∇xp0∇yζ(x, y) dxdy

+

∫
Ω×Y ∗

tA(x, y)∇yωi∇xp0∇yζ(x, y) dx dy .

Integrating by parts with respect to y and using (3.28), we have

−divy(tA(x, y)∇yp̂) = −divy(tA(x, y)∇yψi(x, y))∇xu0

− divy(tA(x, y)∇yωi(x, y))∇xp0.
(3.29)

This equality shows that

p̂ =

N∑
i=1

ωi
∂p0

∂xi
+

N∑
i=1

ψi
∂u0

∂xi
.

Further taking ζ = 0 in variational-formulation (3.25) and integrating by parts with
respect to x, we obtain the desired homogenized adjoint equation:

−div(tA0∇p0 −B#∇u0) = h
|∂T |
|Y |

p0 in Ω,

p0 = 0 on ∂Ω.

�

Below we prove that the bilinear form defined by B# (the perturbed matrix of
B0),is H1

0 (Ω)-elliptic. A key step involves in establishing that

lim
ε→0

∫
Ω∗ε

Bε∇wε∇wε dx =

∫
Ω

B#∇w∇w dx, (3.30)

where wε satisfies (3.33) below. The Robin condition on the boundary of holes in
(3.8) and hence the limit equation with nontrivial terms, leads us to play with the
various integration by parts, for passing to the limit in (3.30). An analogous result
with Neumann condition on the boundary of holes can be seen [16, Theorem 3.3].

To prove the ellipticity of the bilinear form, we shall need [8, Proposition 6.6]
involving extension operator and the unfolding method. We have the existence of
sequence {P ε} of linear extension operators [10], such that for any ε > 0,

P ε ∈ L(Vε, H
1
0 (Ω)),

P εv = v in Ω∗ε, ∀v ∈ Vε
‖P εv‖L2(Ω) ≤ C‖v‖L2(Ω∗ε), ∀v ∈ Vε

‖∇P εv‖[L2(Ω)]N ≤ C‖∇v‖[L2(Ω∗ε)]N ∀v ∈ Vε.

(3.31)



14 I. MISHRA EJDE-2022/12

Although in [10], the set of holes do not intersect the external boundary Γε0, it is
easy to check (see [6, page 6]) that if we extend a function in Vε as in [10] inside

the holes contained in Ω̂∗ε and extend it by zero elsewhere, we obtain an extension
operator that still verify (3.31).

Theorem 3.3. For every w ∈ H1
0 (Ω), we have∫

Ω

B#∇w∇w dx ≥ βmC−2‖w‖2H1
0 (Ω),

where Bε ∈M(βm, βM ,Ω) and C > 0 is a constant appearing in (3.31).

Proof. Let

F = −div(A0∇w)− |∂T |
|Y |
M∂T (g) + h

|∂T |
|Y |

w. (3.32)

Then F ∈ H−1(Ω). Let wε ∈ Vε, be the solution of the problem:

−div(Aε∇wε) = (P ε)∗F in Ω∗ε,

Aε∇wε · nε + hεuε = εgε on Γε1,

uε = 0 on Γε0.

(3.33)

Using ellipticity of Aε and bounds of (3.32), we obtain ‖∇wε‖[L2(Ω∗ε)]N ≤ C. It
follows by (3.31)(iv) and [8, Proposition 6.6] (also see [10]) that

P εwε ⇀ w weakly in H1
0 (Ω).

Let us define qε ∈ Vε as the solution of the problem

−div
(
tAε∇qε −Bε∇wε

)
= 0 in Ω∗ε,

(tAε∇qε −Bε∇wε) · nε = qεhε on Γε1,

qε = 0 on Γε0.

(3.34)

Its homogenized equation is

−div(tA0∇q0 −B#∇w) = h
|∂T |
|Y |

q0 in Ω,

q0 = 0 on ∂Ω.

(3.35)

Multiplying (3.34) by wε and integrating by parts, we obtain∫
Ω∗ε

tAε∇qε∇wε =

∫
Ω∗ε

Bε∇wε∇wε dx+

∫
Γε

1

εwεhqε. (3.36)

Taking qε ∈ Vε as a test-function in (3.33), we have∫
Ω∗ε

Aε∇wε∇qε −
∫

Γε
1

Aε∇wε · nε∇qε = 〈F, P εqε〉H−1(Ω),H1
0 (Ω). (3.37)

Substituting F from (3.32) and using (3.36) in the above equation, we obtain∫
Ω∗ε

Bε∇wε∇wε dx = 〈− div(A0∇w), P εqε〉 − ε
∫

Γε
1

(2hwε − gε)qε

+
|∂T |
|Y |
〈hw −M∂T (g), P εqε〉H−1(Ω),H1

0 (Ω) .

(3.38)

By unfolding, the second term of the right-hand side of (3.38), it can be written as

lim
ε→0

∫
Ω̃×∂T

(2hT bε (wε)− T bε (gε))T bε (qε) dxdσ(y) =
|∂T |
|Y |

(
2hw −M∂T (g)

)
. (3.39)



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 15

Now passing to the limit as ε→ 0 in (3.38), and using (3.39) we obtain

lim
ε→0

∫
Ω∗ε

Bε∇wε∇wε dx =

∫
Ω

A0∇w∇q0 − h
|∂T |
|Y |

q0. (3.40)

Multiplying (3.35) by w and integrating by parts, we obtain∫
Ω

tA0∇q0∇w =

∫
Ω

B#∇w∇w + h
|∂T |
|Y |

q0w. (3.41)

Substituting this value of
∫

Ω
tA0∇q0∇w in (3.40), we obtain (3.30).

We use the ellipticity of Bε and (3.31) to obtain∫
Ω∗ε

Bε∇wε∇wε dx ≥ βm‖wε‖2Vε
≥ βmC−2‖P εwε‖2H1

0 (Ω). (3.42)

Passing to the limit in (3.42) as ε → 0, in view of (3.30) and [8, Proposition 6.6],
we obtain∫

Ω

B#∇w∇w dx ≥ lim inf
ε→0

βmC
−2‖P εwε‖2H1

0 (Ω) ≥ βmC
−2‖w‖2H1

0 (Ω), (3.43)

which shows the ellipticity of B#. �

Remark 3.4. A byproduct of the above proof is

−div(Aε∇wε) = (P ε)∗fε in Ω∗ε,

Aε∇wε · nε + hεuε = εgε on Γε1,

uε = 0 on Γε0,

where fε → f strongly in H−1(Ω). Then the proof of Theorem 3.3 can be adapted
to prove that if P εwε ⇀ w0 weakly in H1

0 (Ω), then we have

lim
ε→0

∫
Ω∗ε

Bε∇wε∇wε dx =

∫
Ω

B#∇w∇w dx. (3.44)

3.1. Convergence of optimal controls. Now we consider some optimal control
problems, where we take the convex set Uεad ⊂ L2(Ω∗ε), to be of obstacle type. Here
we will use the notation χε for χΩ∗ε

.
We consider Uεad as one of the following sets:

Uεad = L2(Ω∗ε), (3.45)

Uεad = {θ ∈ L2(Ω∗ε)|θ ≥ ψ a.e. in Ω∗ε}

= {θ ∈ L2(Ω∗ε)|θ̃ ≥ χεψ a.e. in Ω},
(3.46)

Uεad = {θ ∈ L2(Ω∗ε)|χεψ1 ≤ θ̃ ≤ χεψ2 a.e. in Ω}, (3.47)

where ψ, ψ1 and ψ2 are given functions in L2(Ω). The first case considered above
is unconstrained, the second one is unilateral and the third one corresponds to
bilateral constraint. We define the limiting set Uad ⊂ L2(Ω) corresponding to
(3.45), (3.46) and (3.47) as follows:

Uad = L2(Ω), (3.48)

Uad = {θ ∈ L2(Ω)| θ ≥ Θψ a.e. in Ω}, (3.49)

Uad = {θ ∈ L2(Ω∗ε)| Θψ1 ≤ θ̃ ≤ Θψ2 a.e. in Ω}. (3.50)



16 I. MISHRA EJDE-2022/12

Now, in view of classical linear control theory of Lions [18], there exists a unique
minimal-norm control θ∗ε ∈ Uεad, minimizing the cost-functional (1.3) over any of
the closed convex sets (3.45)-(3.47). So that we have

N0

2

∫
Ω∗ε

θ∗ε
2 dx ≤ Jε(θ∗ε) ≤ Jε(Φε), (3.51)

where

Φε =


χε in case of (3.45),

χεψ in case of (3.46),

χεψ2 in case of (3.47).

In all the cases defined in (3.45)-(3.47), Jε(Φε) is uniformly bounded, since ‖u‖Vε

of the solution uε of (3.8) is bounded independent of ε and Bε ∈ M(βm, βM ,Ω).

Thus {θ̃∗ε} ⊂ L2(Ω) will be bounded sequence, hence there exists a subsequence
converging weakly to θ∗0 in L2(Ω). By (3.7), {ũ∗ε} and {p̃∗ε} (u∗ε, p

∗
ε are solutions of

(3.8) and of (3.9) with θε = θ∗ε) will be bounded independent of ε. Therefore by
extracting a further subsequence (if necessary) we have ũ∗ε ⇀ Θu∗0 weakly in L2(Ω)
and p̃∗ε ⇀ Θp∗0 weakly in L2(Ω).

We define below the limiting optimal control problem. For that let θ0 ∈ Uad,
and let u0 ∈ H1

0 (Ω) be the solution of

−div(A0∇u0) + h
|∂T |
|Y |

u0 = Θ(f + θ0) +
|∂T |
|Y |
M∂T (g) in Ω,

u0 = 0 on ∂Ω,

and the cost-functional by

J0(θ) =
1

2

∫
Ω

B#∇u0∇u0 dx+
N0

2

∫
Ω

θ2
0

Θ
dx,

where A0 and B# are same as defined in (3.4) and (3.6). With this framework, we
state the results on convergence of optimal controls. A proof of which follows by
Remark 3.4, (also see Kesavan et al. [16, Theorem 4.1]).

Theorem 3.5. The limit θ∗0 satisfies the optimality conditions

J0(θ∗0) = min
θ∈Uad

J0(θ),

lim
ε→0

Jε(θ
∗
ε) = J0(θ∗0).

4. Boundary homogenization of optimal control problems

In this section, we study the boundary homogenization of some boundary control
problems. Let Ωε = Ω\Tε, where {Tε} is a sequence of admissible holes in the sense
of H0-convergence and Tε do not intersect the external boundary ∂Ω. We assume
that ∂Ω is partitioned into two parts Γ0,ε and Γ1,ε (corresponding to an ε-periodic
structure for example) such that each connected component of Γ0,ε and/ or Γ1,ε

has diameter going to zero with ε or a directional thickness going to zero with ε for
some given linear subspace direction. These kinds of problems (considered below)
arise in oil exploitation.



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 17

Let Aε satisfy (1.1), gε ∈ L2(∂Ω), and Uεad ⊂ L2(Γ1,ε) be a closed, convex subset
representing the admissible controls. For θ ∈ Uεad the state equation is

− div(Aε∇uε) + uε = 0 in Ωε,

Aε∇uε · nε = gε + θ on Γ1,ε,

uε = cε (unknown constant) on Γ0,ε∫
Γ0,ε

Aε∇uε · nε ds =

∫
Γ0,ε

gε ds

Aε∇uε · nε = 0 on ∂Tε.

(4.1)

We refer [14] for the physical significance of the boundary conditions on Γ0,ε. For
N0 > 0, we associate the cost functional

Jε(θ) =
1

2

∫
Ωε

∇uε · ∇uε dx+
N0

2

∫
Γ1,ε

θ2 ds (4.2)

For the existence and uniqueness of solutions of (4.1), (4.2), we refer to [21, Section
2.7, 2.2], [17, Section 5, p. 265], and [18]. To characterize the optimal control θ∗ε ,
minimizing (4.2), we introduce the adjoint state which is the solution of

−div(tAε∇pε) + pε = −∆uε in Ωε,

tAε∇pε · nε = ∇uε · nε on Γ1,ε,

pε = dε (unknown constant) on Γ0,ε,∫
Γ0,ε

tAε∇pε · nε ds =

∫
Γ0,ε

∇uε · nε ds on Γ1,ε

tAε∇pε · nε = ∇uε · nε on ∂Tε.

(4.3)

Let u∗ε ∈ H1(Ωε), p
∗
ε ∈ H1(Ωε) be the state and the adjoint state for θ = θ∗ε . Then

θ∗ε satisfies the extremality condition∫
Γ1,ε

(p∗ε +N0θ
∗
ε)(θ − θ∗ε) ds ≥ 0 ∀ θ ∈ Uεad (4.4)

We need the following hypotheses:

(H4) The characteristic function χΓ0,ε of Γ0,ε satisfies

∃χ ∈ L∞(∂Ω), 0 < χ(x) for a.e. x ∈ ∂Ω and (1− χ)−1 ∈ L∞(∂Ω).

and χΓ0,ε ⇀ χ weak∗ L∞(∂Ω).
(4.5)

(H5) gε is bounded in L2(∂Ω) and there exists g0 ∈ R,∫
∂Ω

gε ds→ g0.

(H6) Aε H0-converges to A0 in the sense of Briane et al. [3].
(H7) Every weak∗ limit point in L∞(Ω) of χΩε

is positive a.e. in Ω, that is
χΩε

⇀ χ0 weak∗ L∞(Ω), then χ−1
0 ∈ L∞(Ω).

(H58 For each ε > 0, there exists an extension operator P ε ∈ L(Vε, H
1(Ω)) such

that for all u ∈ Vε,
P εu|Ωε

= u, ‖P εu‖1,Ω ≤ C‖u‖1,Ωε
,

where

Vε = {u ∈ H1(Ωε); u = M (constant) on Γ0,ε}.



18 I. MISHRA EJDE-2022/12

Here and in the sequel, C denotes the several positive constants independent of ε.
Then in view of assumptions (H4)–(H8), the state and adjoint equations (4.1) and
(4.3) can be homogenized to

−div(A0∇u0) + χ0u0 = 0 in Ω,

−div(tA0∇p0 −B#
I ∇u0) + χ0p0 = 0 in Ω,

u0 = constant on ∂Ω,∫
∂Ω

A0∇u0 · nds,= g0 +

∫
∂Ω

(1− χ)θ ds,∫
∂Ω

(tA0∇p0 −B#
I ∇u0) · nds = 0.

(4.6)

The homogenization is known to us by Prof C. Conca [11], (cf. see Section 5).

4.1. Optimal controls. For each ε > 0, let the optimal control be denoted by θ∗ε
which is characterized by (4.4). Let σ ∈ L2(∂Ω) be a given function, let Uεad and
Uad be the following sets of admissible controls

Uεad =
{
θε ∈ L2(Γ1,ε) : θε(x) ≥ σ(x)|Γ1,ε a.e. x ∈ ∂Ω

}
,

Uad =
{
θε ∈ L2(∂Ω) : θ(x) ≥ (1− χ)σ(x) for a.e. x ∈ Γ1,ε

}
.

Let θ ∈ Uad and let us define

θε = χΓ1,ε

θ

1− χ
∈ Uεad. (4.7)

Theorem 4.1. Let Jε and θε be given by (4.2), and (4.7). Then we can pass to
the limit in the cost functional

lim
ε→0

Jε(θε) = J0(θ),

where

J0(θ) =
1

2

∫
Ω

B#
I ∇u0∇u0 dx+

N0

2

∫
∂Ω

θ2

1− χ
ds . (4.8)

Then θ∗0 ∈ Uad is the unique minimizer of the above cost-functional over Uad, that
is

J0(θ∗0) ≤ J0(θ) ∀θ ∈ Uad. (4.9)

To prove the above theorem, we shall need the following key lemma.

Lemma 4.2. Let

θε = χΓ1,ε

θ

1− χ
⇀ θ weakly in L2(∂Ω).

Then

lim inf
ε→0

∫
∂Ω

∣∣χΓ1,ε

θ

1− χ
∣∣2ds ≥ ∫

∂Ω

θ2

1− χ
ds.

Proof. By (H4),
χΓ1,ε

(1−χ)θ0 ∈ L2(Ω). We define the convex functional ϕ : L2(Ω)→ R
as

ϕ(θ) =

∫
∂Ω

|θ|2 ds.

Using convexity arguments and some simple calculations, we obtain

ϕ(θ̃ε)− ϕ
( χΓ1,ε

1− χ
θ0

)
≥ 2

∫
∂Ω

χΓ1,ε

1− χ
θ0

(
θ̃ε −

χΓ1,ε

1− χ
θ0

)
ds. (4.10)



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 19

Using that χΓ1,ε θ̃ε = θ̃ε and χrΓ1,ε = χΓ1,ε for all r > 0, the right-hand side of (4.10)
becomes

2

∫
∂Ω

θ0

1− χ

(
θ̃ε −

χΓ1,ε

1− χ
θ0

)
ds,

which tends to 0 as ε→ 0, by the weak convergence of θ̃ε ⇀ θ0 L
2(∂Ω)-weakly and

the convergence χΓ1,ε
L∞−−−−⇀

weak∗
(1− χ). Further we have

ϕ
( χΓ1,ε

1− χ
θ0

)
=

∫
∂Ω

χ2
Γ1,ε

(1− χ)2
|θ0|2ds

=

∫
∂Ω

1 + χ2
Γ0,ε − 2χΓ0,ε

(1− χ)2
|θ0|2ds

=

∫
∂Ω

1− χΓ0,ε

(1− χ)2
|θ0|2ds

→
∫
∂Ω

|θ0|2

(1− χ)
ds.

Then the result follows from the above observations. �

Proof of Theorem 4.1. Using the state equation associated with θε, and [16, Re-
mark 3.3], we have

Jε

(
χΓ1,ε

θ

1− χ

)
=

1

2

∫
Ωε

∇uε · ∇uε +
N0

2

∫
Γ1,ε

χΓ1,ε

θ2

(1− χ)2
ds −→ J0(θ) (4.11)

where J0(θ) is defined by (4.8).
On the other hand, using [16, Remark 3.3] (when Bε = I for all ε > 0), we have

lim
ε→0

∫
Ωε

|∇u∗ε|2 dx =

∫
Ω

B#
I ∇u

∗
0∇u∗0 dx.

Now we pass to the limit in the inequality

Jε(θ
∗
ε) ≤ Jε

(
χΓ1,ε

θ

1− χ

)
,

in view of (4.11), we have

J0(θ) ≥ 1

2

∫
Ω

B#
I ∇u

∗
0 · ∇u∗0 + lim sup

ε→0

N0

2

∫
∂Ω

χΓ1,ε(θ∗ε)2 ds. (4.12)

Thus taking θ = θ∗0 , we have

θ∗ε = χΓ1,ε

θ∗0
1− χ

.

Now putting θ∗ε in (4.12) we have

lim sup
ε→0

∫
∂Ω

χΓ1,ε

(
χΓ1,ε

θ∗0
(1− χ)

)2

ds ≤
∫
∂Ω

(θ∗0)2

1− χ
ds (4.13)

which implies

lim sup
ε→0

∫
∂Ω

χΓ1,ε

(
χΓ1,ε

θ∗0
(1− χ)

)2

ds ≤
∫
∂Ω

(θ∗0)2

1− χ
ds. (4.14)

Using χΓ1,ε = 1− χΓ0,ε , we have

(1− χΓ0,ε)2 = 1 + (χΓ0,ε)2 − 2χΓ0,ε = 1− χΓ0,ε



20 I. MISHRA EJDE-2022/12

Similarly,

(1− χΓ0,ε)3 = (1− χΓ0,ε)(1− χΓ0,ε)2 = (1− χΓ0,ε)(1− χΓ0,ε)

= (1− χΓ0,ε) = χΓ1,ε .

With the above observations, (4.14) implies

lim sup
ε→0

∫
∂Ω

χΓ1,ε

( θ∗0
1− χ

)2

ds ≤
∫
∂Ω

(θ∗0)2

1− χ
ds. (4.15)

On the other hand, by a convexity argument, see Lemma 4.2, also see [16, Propo-
sition 2.2], we have

lim inf
ε→0

∫
∂Ω

χΓ1,ε

( θ∗0
1− χ

)2

ds ≥
∫
∂Ω

(θ∗0)2

1− χ
ds. (4.16)

From (4.12) and the last two inequalities (4.15) and (4.16), we have

J0(θ) ≥ J0(θ∗0) ∀θ ∈ Uad. (4.17)

This completes the proof . �

5. Appendix by C. Conca

The variational formulation of (4.1) is as follows: Find uε ∈ Vε such that for all
ϕ ∈ Vε,∫

Ωε

Aε∇uε∇ϕdx+

∫
Ωε

uεϕdx =

∫
Γ1,ε

(gε + θ)ϕds+ ϕ|Γ0,ε

∫
Γ0,ε

gε ds (5.1)

Taking ϕ = uε as a test function in (5.1),∫
Ωε

Aε∇uε∇uε dx+

∫
Ωε

uε · uε =

∫
∂Ω

gεuε + θ|Γ1,ε

∫
Γ1,ε

uε ds. (5.2)

Since uε is constant on Γ0,ε, and using (1.1), (H8)–(H10), we have

αm‖uε‖21,Ωε
≤ ‖uε‖0,∂Ω

(
‖gε‖0,Ωε

+ ‖θ‖0,Γ1,ε

)
≤ C‖uε‖1,Ωε

(
‖gε‖0,Ωε

+ ‖θ‖0,Γ1,ε

)
‖uε‖1,Ωε

≤ C
(
g0 + ‖θ‖0,Γ1,ε

)
;

thus we can extract a subsequence (still denoted by ε) {uε} such that

P εuε ⇀ u0 weakly in H1(Ω),

˜(Aε∇uε) ⇀ (A0∇u0) weakly in [L2(Ω)]N
(5.3)

where u0 satisfies

− div(A0∇u0) + χ0u0 = 0 in Ω. (5.4)

We set

V = {u ∈ H1(Ω) : u ∈ H1(Ω), u = M (constant) on ∂Ω}.
Clearly, V ⊂ Vε for all ε > 0. Then for all ϕ ∈ V , we have∫

Ωε

˜(Aε∇uε) · ∇ϕdx+

∫
Ω

χΩε
P εuεϕdx

= ϕ|∂Ω

∫
∂Ω

gε ds+ ϕ|∂Ω

∫
∂Ω

(1− χΓ0,ε)θ ds.



EJDE-2022/12 OPTIMAL CONTROL PROBLEMS 21

Passing to the limit in the above eq as ε→ 0, and using (5.3), (H4), (H5) and (H9),
we obtain

−
∫

Ω

div(A0∇u0)ϕdx+

∫
Ω

χ0u0ϕdx = ϕ|∂Ω g0 +

∫
∂Ω

(1− χ)θϕ ds.

Integrating by parts

−ϕ
∫
∂Ω

(A0∇u0) ·ndx+

∫
Ω

A0∇u0∇ϕ+

∫
Ω

χ0u0ϕdx = ϕ|∂Ω g0 +

∫
∂Ω

(1−χ)θϕ ds.

Comparing the coefficients of ϕ, then again integrate by parts and combine with
(5.4) we obtain ∫

∂Ω

A0∇u0 · nds = g0 +

∫
∂Ω

(1− χ)θ ds.

Now we show that u0 is constant on ∂Ω. Since P εuε ⇀ u0 weakly in H1(Ω). As
in the proof of [14, Theorem 1, p. 354], we have

χΓ0,εP εuε
∣∣
∂Ω

⇀ χu0|∂Ω weakly L2(∂Ω).

On the other hand, since uε = cε on Γ0,ε, and

χΓ0,εuε|Γ0,ε = χΓ0,εcε ⇀ c · χ

where c is just the limit of sequence of real numbers cε. Comparing both the limits
of χΓ0,εuε|Γ0,ε , we obtain

χu0|∂Ω = χ · c =⇒ u0 = c on ∂Ω, as χ 6= 0 on ∂Ω.

Similarly P εpε ⇀ p0 weakly in H1(Ω).
Set zε = tAε∇pε −∇uε. Then up to a subsequence

zε ⇀ z0 weakly in [L2(Ω)]N ,

where z0 satisfies

− div(z0) + χ0p0 = 0 in Ω. (5.5)

Arguing as we did for the state equation, we prove that

p0|∂Ω = d0 (constant), and

∫
∂Ω

z0 · nds = 0.

To identify the limits z0 and p0, we introduce the auxiliary functions µεk and ψεk,
k = 1, . . . , N as done in [16, p. 572].

µεk − xk ⇀ 0 weakly in H1(Ω),

div(Aε∇µεk) = (P ε)∗ div(A0ek) in Ωε,(
Aε∇(−µεk + xk)

)
· nε = 0 on ∂Tε,

˜(Aε∇µεk) ⇀ A0ek weakly in [L2(Ω)]N ,

(5.6)

where (Pε)
∗ is the adjoint of P ε, and xk is kth co-ordinate function and ek is the

kth standard basis vector. Similarly introducing ψεk ∈ Vε, k = 1, . . . , N satisfy

div(tAε∇ψεk +∇µεk) = 0 in Ωε,

(tAε∇ψεk +∇µεk) = 0 on ∂Tε.
(5.7)



22 I. MISHRA EJDE-2022/12

It follows that P εψεk ⇀ ψ0
k weakly in H1(Ω). Let ϕ ∈ D(Ω), then multiplying the

first equation of (4.1) by ϕψεk and adjoint equation (4.3) by ϕµεk, integrating by
parts, and subtracting the second identity from the first, we obtain

0 = −
∫

Ω

z̃εµ
ε
k∇ϕdx−

∫
Ω

∇(P εpε)(Aε∇µεk )̃ϕ+

∫
Ω

ξ̃ε∇ϕ(P εψεk) dx

−
∫

Ω

(P εuε)b
ε
k∇ϕdx+

∫
Ω

χΩε
ϕP εψεkP

εuε dx.

Let us denote by b0k the weak limit in [L2(Ω)]N of
(
tAε∇ψεk +∇µεk

)
. It is clear that

div(b0k) = 0 in Ω. (5.8)

Passing to the limit in (5.8) using div-curl lemma [3, Lemma 1.1], we obtain

0 = −
∫

Ω

z0xk∇ϕdx−
∫

Ω

∇p0(A0ek)ϕdx+

∫
Ω

(A0∇u0)∇ϕψk0 dx

−
∫

Ω

u0b
0
k∇ϕdx+

∫
Ω

χ0ϕψ
0
ku0 dx−

∫
Ω

χ0p0ϕxk dx

From (5.4), (5.5), and (5.8), integrating by parts, we obtain

z0 · ek = tA0∇p0 · ek + tA0∇ψ0
k∇u0 − b0k · ∇u0. (5.9)

Hence z0 = tA0∇p0−B#
I ∇u0, and B#ek = b0k−tA0∇ψ0

k. Combining with (5.5), we
obtain second equation of (4.6). Thus we obtain the homogenized equation (4.6).

Acknowledgements: The author would like to thank the anonymous referee for
his/her useful comments which really improved the manuscript. This research was
supported by Women Scientist Scheme (WOS-A), DST, file no: SR/WOS-A/PM-
77/2017.

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Available from: www.math.univ-toulouse.fr.

Indira Mishra

Indian Institute of Information Technology Lucknow, Chak Ganjaria, C. G. City, Luc-
know, Uttar Pradesh 226002, India

Email address: indira.mishra1@gmail.com, indira@iiitl.ac.in


	1. Introduction
	1.1. Notation and problem setting
	1.2. Optimality conditions

	2. The periodic unfolding method for perforated domain
	3. Main results
	3.1. Convergence of optimal controls

	4. Boundary homogenization of optimal control problems
	4.1. Optimal controls

	5. Appendix by C. Conca
	Acknowledgements:

	References