A new formulation for the limit matrix occurring in the cost functional of an optimal control problem on homogenization is obtained. It is used to obtain an upper bound for this matrix (in the sense of positive definite matrices)

Kesavan, S.

Rajesh, M.

English

Publications of the IAS Fellows

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 112, No. 2, May 2002, pp. 337–346.
© Printed in India
On the limit matrix obtained in the homogenization
of an optimal control problem
S KESAVAN and M RAJESH
The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113,
India
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
E-mail: kesh@imsc.ernet.in; rajesh@math.iisc.ernet.in
MS received 6 January 2001
Abstract. A new formulation for the limit matrix occurring in the cost functional of
an optimal control problem on homogenization is obtained. It is used to obtain an upper
bound for this matrix (in the sense of positive definite matrices).
Keywords. Homogenization; optimal control; elliptic equations.
1. Introduction
Let  RN be a bounded domain. Let 0 < m < M . We denote byM.m; M;/ the
set of all N N matrices A D A.x/, with coefficients in L1./, such that
mj j2  A.x/    M j j2; a.e. x; for all  2 RN:
Given a family of matrices A" 2 M.m; M;/, let v" 2 H 10 ./ be the unique (weak)
solution of the problem
−div.A"rv"/ D f in 
v" D 0 on @

(1.1)
where f 2 L2./ is given. Then fv"g is bounded inH 10 ./ and if v" * v0 in H 10 ./, we
have
−div.A0rv0/ D f in 
v0 D 0 on @

(1.2)
whenA" H -converges toA0 (cf. Murat [7]). We know that fv"g does not converge strongly
in H 10 ./. Nevertheless,
R

jrv"j2 dx is bounded and hence (at least for a subsequence)
converges. We would like to know if this limit can be expressed in terms of the function
v0. More generally, if B" 2 M.m; M;/, be another family of matrices, consider the
‘energy’ defined by
Z

B"rv"  rv" dx:
337
338 S Kesavan and M Rajesh
Again, this is a bounded sequence and we would like to express its limit (when it
converges) in terms of v0. More precisely, we would like to know if there exists a matrix
B# 2M. Qm; QM;/ such thatZ

B"rv"  rv" dx !
Z

B#rv0  rv0 dx
and, if so, identify that matrix and estimate the constants Qm and QM .
WhenB" D A", it is well-known (cf. Murat [7]) that indeedB# D A0, theH -limit ofA".
It turns out that the solution to this problem is closely related to the question of homog-
enizing an associated optimal control problem.
Let Uad  L2./ be a closed convex set (called the set of admissible controls) and let
f 2 L2./ be given. Given  2 Uad, let u" 2 H 10 ./ be the unique (weak) solution of
the ‘state equation’:
−div .A"ru"/ D f C  in 
u" D 0 on @

: (1.3)
Then, there exists a unique ‘optimal control’ " 2 Uad such that
J".

" / D min
2Uad
J"./; (1.4)
where, for  2 Uad, and u" D u"./ solution of (1.3),
J"./ D 12
Z

B"ru"  ru" dx C N2
Z

2 dx; (1.5)
N > 0 being a fixed constant, called the ‘cost of the control’.
The homogenization of the optimal control problem (1.3)–(1.5) was first studied in
the periodic case by Kesavan and Vanninathan [5] and then in the geneal case under the
framework of H -convergence by Kesavan and Saint Jean Paulin [3]. They also extended
these results (cf. [4]) to the ‘perforated case’ where is replaced by a family of ‘perforated
domains’ "  . In all these cases, it was shown that there exists a matrix B# such
that, in the limit, there is an optimal control problem with A0 and B# replacing A" and B"
respectively in (1.3)–(1.5).
The expression derived for B# is complicated and the symmetry of this matrix, when
all the B" are symmetric, requires a detailed proof (cf. [3,4]). Further, while the ellipticity
could be proved, no upper bound, i.e. an estimate for QM , could be derived.
In this paper, a new formula forB# is obtained and, in the symmetric case, the symmetry
can be read off directly from it. An upper bound is also derived.
The paper is organized as follows. In §2, the equivalence of the two problems stated
above is studied and the existence and uniqueness of the matrix B# is established based on
the results of Kesavan and Saint Jean Paulin [3]. In §3, the new formula for B# is derived
and its properties are studied. In §4, the corresponding results for the perforated case are
stated.
2. Two equivalent problems
Let  RN be a bounded domain and letA" 2M.m; M;/ andB" 2M.m; M;/
be given. We now consider two statements.
Homogenization and optimal control 339
(S1) There exists a matrix B# 2 M. Qm; QM;/ such that, given a strongly convergent
sequence fg"g in H−1./ and the corresponding sequence fv"g of solutions in H 10 ./ of
the problem
−div.A"rv"/ D g" in 
v" D 0 on @

; (2.1)
then, for a subsequence, v" * v0 weakly in H 10 ./ andZ

B"rv"  rv" dx !
Z

B#rv0  rv0 dx; (2.2)
B"rv"  rv" * B#rv0  rv0 in D0./:  (2.3)
Remark 2.1. If g" ! g in H−1./, then (cf. Murat [7])
−div.A0rv0/ D g in 
v0 D 0 on @

; (2.4)
where A0 is the H -limit of A". Also
A"rv" * A0rv0 (2.5)
weakly in L2./N . 
In order to make the second statement, we need to introduce the ‘adjoint state’ function.
Let g" and v" be as above. Then we denote by p" 2 H 10 ./, the adjoint state, which is the
solution of
div .tA"rp" − B"rv"/ D 0 in 
p" D 0 on @

; (2.6)
where, we have denoted the transpose of A" by tA".
Remark 2.2. From the hypotheses, it is evident that fp"g is bounded in H 10 ./. 
Remark 2.3. The system consisting of (2.1) and (2.6) is of the type used by Lions [6] to
construct the optimality system to solve the optimal control problem (1.3)–(1.5), using a
duality argument. The system consisting of (2.1) and (2.6) was used by Kesavan and Saint
Jean Paulin [3] to homogenize the optimal control problem (1.3)–(1.5). 
(S2) There exists a matrix B# 2M. Qm; QM;/ such that, given g" strongly convergent
in H−1./ and v" solution of (2.1) and p" solution of (2.6), then, for a subsequence,
v" * v0; p" * p0 weakly inH 10 ./ and z" D tA"rp"−B"rv" * zweakly inL2./N ,
where
z D tA0rp0 − B#rv0 (2.7)
and A0 is the H -limit of A". 
Remark 2.4. The pair .v0; p0/ will satisfy the homogenized system:
−div.A0rv0/ D g in 
div.tA0rp0 − B#rv0/ D 0 in 
v0 D p0 D 0 on @
9=
; :  (2.8)
We now prove the equivalence of these two statements.
340 S Kesavan and M Rajesh
Theorem 2.1. If B# 2M. Qm; QM;/ is such that (S2) is true, then it also verifies (S1).
The converse is true if B" and B# are symmetric.
Proof. Let (S2) hold for B#. Now, by virtue of (2.1), (2.4), (2.6) and (2.8), we have, for
the relevant subsequence,Z

B"rv"  rv" dx D
Z

tA"rp"  rv" dx
D
Z

A"rv"  rp" dx
D hg"; p"iH−1./;H 10 ./
! hg; p0iH−1./;H 10 ./
D
Z

A0rv0  rp0 dx
D
Z

B#rv0  rv0 dx:
This proves (2.2). Now,
B"rv"  v" D −.tA"rp" − B"rv"/  rv" C .A"rv"/  rp":
Now, by virtue of (2.1) and (2.6), the divergences of the expressions within parantheses
in each of the two terms in the right-hand side are strongly convergent in H−1./. Also,
rv" and rp" converge weakly in L2./N . Thus, by the div-curl lemma of compensated
compactness theory (cf. Murat [7], Murat and Tartar [8] or Tartar [10]), we conclude that,
in view of (2.5) and (2.7),
B"rv"  rv" * −.tA0rp0 − B#rv0/  rv0 C A0rv0  rp0
D B#rv0  rv0
in D0./. This proves (2.3).
Conversely, let B" and B# be symmetric and assume that (S1) holds. Let !b be a
(relatively compact) open subset and let  2 D./ be such that   1 in a neighborhood
of !. Define k" 2 H 10 ./; 1  k  N , to be the unique solution of the problem
−div.A"rk"/ D −div .A0r.xk// in 
k" D 0 on @

: (2.9)
Then, by H -convergence, k" * xk weakly in H 10 ./ and A"rk" * A0r.xk/ weakly
in L2./N . By superposition of the solutions of (2.1) and (2.9), we get
−div.A"r.v"  k"// D g"  .−div .A0r.xk// in :
Hence, by (S1), for a subsequence,
B"r.v"  k"/  r.v"  k"/ * B#r.v0  xk/  r.v0  xk/
in D0./. Hence, using the polarization identity which requires the symmetry of B" and
B#, we get that
B"rv"  rk" * B#rv0  r.xk/ (2.10)
Homogenization and optimal control 341
inD0./. We can now apply the div-curl lemma to the pair .z";rk"/, since div z" D 0, to
get
z"  rk" * z  r.xk/
in D0./. On the other hand
z"  rk" D A"rk"  rp" − B"rv"  rk"
* A0r.xk/  rp0 − B#rv0  r.xk/
again by applying the div-curl lemma to the first term on the right-hand side and by also
using (2.10). Thus, on !, we have
z  ek D tA0rp0  ek − B#rv0  ek
where ek is the standard k-th basis vector of RN . This proves (2.7) on ! and as !b was
arbitrary, we have the result on . 
Remark 2.5. We can use the test functions j" to prove the uniqueness of the matrix B#,
when it exists. Indeed, if we have two matrices B#i ; i D 1; 2, satisfying (S2), (or (S1), in
the symmetric case), setting v" D j" , we then have that (cf. (2.7))
tA0rp0 − B#1 r.xj / D tA0rp0 − B#2 r.xj /:
Thus on !, we have that B#1 D B#2 and the result follows for all of  since !b is
arbitrary. 
The existence of a B# satisfying (S2) was proved in the general case by Kesavan and
Saint Jean Paulin [3]. We recall their formulation and also give another, shorter, proof of
their result.
We first need to define some test functions. First of all, we recall the existence of functions
Xk" 2 H 1./, for 1  k  N , with the following properties (cf. Murat [7]).
Xk" * xk weakly in H 1./
A"rXk" * A0ek weakly in L2./N
fdiv.A"rXk"/g converges strongly in H−1./
9=
; : (2.11)
We now define another set of test functions  k" 2 H 10 ./ for 1  k  N , which verify
−div.tA"r k" C tB"rXk"/ D 0 in 
 k" D 0 on @

: (2.12)
Then, up to a subsequence, f k" g converges weakly in H 10 ./ to  k0 and ftA"r k" C
tB"rXk" g converges weakly in L2./N . Then, (cf. [3]), we define
t .B#/ek D lim
"!0
.tA"r k" C tB"rXk"/− tA0r k0 : (2.13)
Theorem 2.2. B# defined by (2.13) satisfies (S2).
342 S Kesavan and M Rajesh
Proof. We extract a subsequence such that all the bounded sequences that occur below
are convergent in the relevant weak topologies. Let g" ! g strongly in H−1./ and
.v"; p"/ 2 H 10 ./H 10 ./ be the solution of (2.1) and (2.6). Let v" * v0 and p" * p0
weakly in H 10 ./ and let z" * z weakly in L
2./N . Now
z"  rXk" D rp"  A"rXk" − tB"rXk"  rv"
D .A"rXk"/  rp" − .tA"r k" C tB"rXk"/  rv"
C .A"rv"/  r k"
We can pass to the limit, using the div-curl lemma, in each term to get
z  ek D A0ek  rp0 − lim
"!0
.tA"r k" C tB"rXk"/  rv0 C A0rv0  r k0
using (2.13) from which (2.7) follows. 
Remark 2.6. In the statements (S1) and (S2), we have required that the relevant conver-
gences occur for a subsequence. It must be noted that the subsequence is independent of
the strongly convergent sequence g". Indeed, it depends only on the convergences implied
in (2.11) and (2.13) (cf. Rajesh [9]). 
3. Properties of B#
We are now interested in properties like the symmetry and ellipticity of the matrix B#
defined in the previous section. Kesavan and Saint Jean Paulin [3] proved that it is symmet-
ric when all the B" are symmetric and that Qm D m. However, the problem of estimatingQM was left open. In this section, a new formula for B# will be given from which the
symmetry can be just read off and which will also enable us to estimate QM .
First of all, we recall the corrector matrices occurring in the study of H -convergence
of A", as introduced by Murat [7]. If Xk" are the test functions introduced in the previous
section (cf. (2.11)), the corrector matrices are defined by
M"ek D rXk" ; 1  k  N: (3.1)
Then, the following properties hold (cf. Murat [7] or Murat and Tartar [8]):
M" * I weakly in L2./N
2
A"M" * A0 weakly in L2./N
2
tM"A"M" * A0 in D0./N2
fdiv.A"M"/g converges strongly in H−1./N
9>>>=
>>>;
: (3.2)
We now prove the main result of this section.
Theorem 3.1. B# defined by (2.13) is the limit, in the sense of distributions, of tM"B"M".
Proof. It is enough to show that, for 1  i; j  N ,
tM"B"M"ek  ej * B#ek  ej
Homogenization and optimal control 343
in D0./. Now
tM"B"M"ek  ej D tB"M"ej M"ek
D .tA"r j" C tB"M"ej / M"ek − tA"r j" M"ek
D .tA"r j" C tB"rXj" /  rXk" − A"rXk"  r j" :
We can pass to the limit in each of the two terms on the right-hand side using the div-curl
lemma to get
tM"B"M"ek  ej * tB#ej  ek D B#ek  ej : 
COROLLARY 3.1
If the B" are symmetric, then so is B#. 
Theorem 3.2. B# 2M. Qm; QM;/, where Qm D m and
QM D M

M
m
2
:
Proof. That QmDm has already been proved in [3]. Let ’ 2 D./; ’  0 and let  2 RN .
Then Z

B"M" M"’ dx  BM
Z

jM" j2’ dx
 M
m
Z

A"M" M"’ dx
D M
m
Z
G
tM"A"M"  ’ dx:
Passing to the limit, using Theorem 3.1 and (3.2), we get
Z

B#  ’ dx  M
m
Z

A0  ’ dx  M
m
2M
m
j j2
Z

’ dx;
since we know that A0 2 M. Qm; QM;/, where Qm D m and QM D 2M=m (cf.
Murat [7]). Since ’ was arbitrary, this proves the theorem. 
Remark 3.1. Consider the one-dimensional case. Let 0 < M  a".x/  M and 0 < M
 b".x/  M . Let
− d
dx
.a"
du"
dx
/ D f in .0; 1/
u".0/ D u".1/ D 0:
Then, it has been shown by Kesavan and Saint Jean Paulin [3] that
Z 1
0
b"
du"
dx
du"
dx
dx !
Z 1
0
b#
du0
dx
du0
dx
dx
344 S Kesavan and M Rajesh
with
b# D a
2
0
g0
;
where
1
a"
*
1
a0
and
1
g"
 b"
a2"
*
1
g0
in L1.0; 1/ weak -:
This yields precisely the bound obtained above for b#. 
4. The perforated case
We now briefly describe the problem in the perforated case and state the results without
proofs, since those of the corresponding results in the previous case carry over mutatis
mutandis.
Let   RN be a bounded domain and for " > 0, let S"   be a closed set (the set of
perforations). We call " D nS" the perforated domain. Following Briane, Damlamian
and Donato [1], we say that a family fS"g of holes is admissible if the following conditions
are fulfilled.
H1 If " is the characteristic function of S", then every weak- limit of f"g in L1./ is
positive a.e. 
H2 Let
V" D fu 2 H 1."/ju D 0 on @g:
Then, there exists an extension operator P" : V" ! H 10 ./ and a constant C0 > 0,
independent of " such that, for any u 2 V",
P"uj" D u and jjrP"ujjL2./N  C0jjrujjL2."/N :  (4.1)
Analogous to the theory ofH -convergence, we have a theory ofH0-convergence (cf. [1]).
We now assume that we have two families of matrices A" 2M.m; M;/ and B" 2
M.m; M;/. We denote by P " : H−1./ ! V " , the adjoint of P". For functions in
L2."/, we also have the trivial extension operator Q" which extends the functions by
zero across the holes to give a function inL2./. We denote the unit outward normal (with
respect to ") on @S" be n".
(S3) There exists a matrix B# 2 M. Qm; QM;/ such that, given a strongly convergent
sequence fg"g in H−1./ and the corresponding sequence of solutions v" 2 V" of
−div.A"rv"/ D P " g" in "
A"rv"  n" D 0 on @S"
v" D 0 on @
9=
; ; (4.2)
then, for a subsequence, P"v" * v0 weakly in H 10 ./ andR
"
B"rv"  rv" dx *
R

B#rv0  rv0 dx
"B"r.P"v"/  r.P"v"/ * B#rv0  rv0 in D0./

:  (4.3)
(S4) There exists a matrix B# 2 M. Qm; QM;/ such that given a strongly convergent
sequence fg"g in H−1./ and the sequence fv"g of solutions of (4.2) and the sequence
fp"g in V" of solutions of
Homogenization and optimal control 345
div .tA"rp" − B"rv"/ D 0 in "
.tA"rp" − B"rv"/  n" D 0 on @S"
p" D 0 on @
9=
; ; (4.4)
then, for a subsequence, P"v" * v0; P"p" * p0 weakly in H 10 ./; z" D Q".tA"rp" −
B"rv"/ * z weakly in L2./N where
z D tA0rp0 − B#rv0; (4.5)
A0 being the H0-limit of fA"g. 
Theorem 4.1. If B# 2 M. Qm; QM;/ is such that (S4) is true, then so is (S3). The
converse holds when the B" and B# are symmetric. 
Kesavan and Saint Jean Paulin [4] gave a formula for a matrix B# such that (S4) holds.
If we define the corrector matrices QM" by
QM"ek D r.P" QXk"/;
where the QXk" are test functions with properties analogous to those mentioned in (2.11)
(cf. [1] or [4]), it can be shown that (cf. Rajesh [9])
"
t QM"B" QM" * B#
in the sense of distributions. Thus, if B" are symmetric, so is B# and we can show that
QM D M

M
m
2
:
Kesavan and Saint Jean Paulin [4] proved that for all w 2 H 10 ./
Z

B#rw  rw dx  mC−20 jjwjj2H 10 ./:
This, in fact, implies that Qm D mC−20 . We give a proof of this below. It is adapted
from a similar proof by Casado-D´iaz [2], but with different test functions.
Lemma 4.1. LetA D A.x/ be a symmetricNN matrix with coefficients inL1./ such
that
Z

Arw  rw dx  0 (4.6)
for all w 2 H 10 ./. Then
A.x/    0 (4.7)
a.e. in  for all  2 RN .
346 S Kesavan and M Rajesh
Proof. Let  2 RN and let ’  0 be a C10 -function. Define
v".x/ D " cos."−1  x/’.x/;
w".x/ D " sin."−1  x/’.x/:
Then v"; w" 2 H 10 ./. Applying (4.6) to both v" andw" and adding the resulting inequal-
ities, we get
"2
Z

Ar’  r’ dx C
Z

.A  /’2 dx  0:
Passing to the limit as " ! 0, using the arbitrariness of ’, we get (4.7) by standard
arguments. 
References
[1] Briane M, Damlamian A and Donato P,H -convergence for perforated domains, in: Non-
linear Partial Differential Equations and their Applications, Colle`ge de France Seminar,
Vol. XIII, Pitman Research Notes in Mathematics, 1996
[2] Casado-D´iaz J, Personal Communication
[3] Kesavan S and Saint Jean Paulin J, Homogenization of an optimal control problem, SIAM
J. Control Optim. 35 (1997) 1557–1573
[4] Kesavan S and Saint Jean Paulin J, Optimal control on perforated domains, J. Math. Anal.
Appl. 229 (1999) 563–586
[5] Kesavan S and Vanninathan M, L’homoge´ne´isation d’un proble`me de controˆle optimal,
C. R. Acad. Sci., Paris, Se´rie A, 285 (1977) 441–444
[6] Lions J L, Optimal Control of Systems Governed by Partial Differential Equations (Berlin:
Springer-Verlag) (1971)
[7] Murat F, H -convergence, Mimeographed notes, Se´minaire d’Analyse Fonctionnelle et
Nume´rique, Universite´ d’Alger, 1977/78
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Composite Materials (eds) A Cherkaev and R Kohn (Birkhauser) (1997) 21–43
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cutta) (2000)
[10] Tartar L, Compensated compactness and applications to partial differential equations, in:
Nonlinear Analysis and Mechanics, Heriott Watt Symposium (ed) R J Knops, Pitman
Research Notes in Mathematics, 39 (1979) 136–212.


On the limit matrix obtained in the homogenization of an optimal control problem

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On the limit matrix obtained in the homogenization of an optimal control problem

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