Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type

0.37 MB·

by  Olha P. Kupenko

DISCRETEANDCONTINUOUS doi:10.3934/dcdsb.2015.20.xx
DYNAMICALSYSTEMSSERIESB
Volume20,Number9,November2015 pp. X–XX
SHAPE STABILITY OF OPTIMAL CONTROL PROBLEMS
IN COEFFICIENTS FOR COUPLED SYSTEM
7
OF HAMMERSTEIN TYPE
1
0
2
Olha P. Kupenko
b
e DnipropetrovskMiningUniversity
F
DepartmentofSystemAnalysisandControl
7 KarlMarksav.,19,49005Dnipropetrovsk,Ukraine
2 and
InstituteforAppliedSystemAnalysis
] NationalAcademyofSciences andMinistryofEducationandScienceofUkraine
C Peremogyav.,37/35,IPSA,03056Kyiv,Ukraine
O Rosanna Manzo
.
h Universit`adegliStudi diSalerno
t DipartimentodiIngegneriadell’Informazione,IngegneriaElettricaeMatematicaApplicata
a ViaGiovanniPaoloII,132,84084Fisciano(SA),Italy
m
[ (Communicated by Gregoire Allaire)

2
v Abstract. In this paper we consider an optimal control problem (OCP) for
1 the coupled system of a nonlinear monotone Dirichlet problem with matrix-
1 valuedL∞(Ω;RN×N)-controlsincoefficientsandanonlinearequationofHam-
6 merstein type. Since problems of this type have no solutions in general, we
6 make a special assumption on the coefficients of the state equation and in-
0 troduce the class of so-called solenoidal admissiblecontrols. Using the direct
. methodincalculusofvariations,weprovetheexistenceofanoptimalcontrol.
1
Wealsostudythestabilityoftheoptimalcontrolproblemwithrespecttothe
0
domain perturbation. In particular, we derive the sufficient conditions of the
7
Mosco-stabilityforthegivenclassofOCPs.
1
:
v 1. Introduction. The aim of this paper is to prove the existence result for an
Xi optimal control problem (OCP) governed by the system of a nonlinear monotone
elliptic equation with homogeneous Dirichlet boundary conditions and a nonlinear
r
a equationofHammersteintype, andtoprovidesensitivityanalysisofthe considered
optimization problem with respect to the domain perturbations. As controls we
consider the matrix of coefficients in the main part of the elliptic equation. We
assumethatadmissiblecontrolsaremeasurableanduniformlyboundedmatricesof
L∞(Ω;RN×N).
Systems with distributed parameters and optimal control problems for systems
describedby PDE,nonlinear integraland ordinarydifferentialequations havebeen
widely studied by many authors (see for example [21, 25, 26, 27, 36]). However,
systems which contain equations of different types and optimization problems as-
sociated with them are still less well understood. In general case including as well
2010 Mathematics Subject Classification. Primary: 49J20,35J57; Secondary: 49J45, 93C73.
Keywordsandphrases. NonlinearmonotoneDirichletproblem,equationofHammersteintype,
controlincoefficients, domainperturbation.
1
2 OLHA P. KUPENKO AND ROSANNA MANZO
control and state constraints, such problems are rather complex and have no sim-
ple constructive solutions. The system, considered in the present paper, contains
two equations: a nonlinear monotone elliptic equation with homogeneous Dirichlet
boundary conditions and a nonlinear equation of Hammerstein type, which nonlin-
early depends on the solution of the first object. The optimal control problem we
study here is to minimize the discrepancy between a given distribution z ∈Lp(Ω)
d
andasolutionofHammersteinequationz =z(U,y),choosinganappropriatematrix
of coefficients U ∈U , i.e.
ad
I (U,y,z)= |z(x)−z (x)|pdx−→inf (1)
Ω d
ZΩ
subject to constrains
z+BF(y,z)=g in Ω, (2)
−div U(x)[(∇y)p−2]∇y +|y|p−2y =f in Ω, (3)
(cid:0) U ∈Uad, y(cid:1)∈W01,p(Ω), (4)
where U ⊂L∞(Ω;RN×N) is a set of admissible controls, B :Lq(Ω)→Lp(Ω) is a
ad
positive linear operator, F : W1,p(Ω)×Lp(Ω) → Lq(Ω) is an essentially nonlinear
0
and non-monotone operator, f ∈W−1,q(Ω) and g ∈Lp(Ω) are given distributions.
Sincetherangeofoptimalcontrolproblemsincoefficientsisverywide,including
as well optimal shape design problems, optimization of certain evolution systems,
some problems originating in mechanics and others, this topic has been widely
studied by many authors. We mainly could mention Allaire [2], Buttazzo & Dal
Maso [7], Calvo-Jurado & Casado-Diaz [8], Haslinger & Neittaanmaki [19], Lions
[26], Lurie [27], Murat [29], Murat & Tartar [30], Pironneau [31], Raytum [32],
Sokolowski & Zolesio [33], Tiba [34], Melnik & Zgurovsky [36]. In fact (see for
instance[29]), the mostofoptimalcontrolproblemsincoefficientsforlinearelliptic
equations have no solution in general. It turns out that this circumstance is the
characteristicfeatureforthemajorityofoptimalcontrolproblemsincoefficients. To
overcomethisdifficulty,inpresentarticle,byanalogywith[13,22,24],weputsome
additional constrains on the set of admissible controls. Namely, we consider the
matrix-valued controls from the so-called generalized solenoidal set. The elements
ofthissetdonotbelongtoanySobolevspace,butstillarealittlebit“moreregular”
thenthose fromL∞-class. Typically,the matrix ofcoefficientsin the principle part
ofPDEsstandsforanisotropicphysicalpropertiesofmediawheretheprocessesare
studied. Themainreasonweintroducetheclassofgeneralizedsolenoidalcontrolsis
toachievethedesiredwell-posednessofthecorrespondingOCPandavoidthe“over
regularity”ofoptimalcharacteristics. Wegivetheprecisedefinitionofsuchcontrols
inSection3andprovethatinthiscasetheoriginaloptimalcontrolproblemadmits
atleastonesolution. Itshouldbenoticedthatwedonotinvolvethehomogenization
method and the relaxation procedure in this process.
In practice, the equations of Hammerstein type appear as integral or integro-
differential equations. The class of integral equations is very important for theory
and applications, since there are less restrictions on smoothness of the desired so-
lutions involved in comparison to those for the solutions of differential equations.
Appearance of integral equations when solving boundary value problems is quite
natural,sinceequationsofsuchtypebindtogetherthevaluesofknownandunknown
functionsonboundeddomains,incontrasttodifferentialequations,wheredomains
are infinitely small. It should be also mentioned here, that solution uniqueness is
SHAPE STABILITY OF OCPS FOR SYSTEMS OF HAMMERSTEIN TYPE 3
nottypicalfor equations of Hammersteintype or optimizationproblems associated
withsuchobjects(see[1]). Indeed,thispropertyrequiresratherstrongassumptions
on operators B and F, which is rather restrictive in view of numerous applications
(see [35]). The physical motivation of optimal control problems which are similar
to those investigated in the present paper is widely discussed in [1, 37].
As was pointed above, the principal feature of this problem is the fact that an
optimal solution for (1)–(4) does not exist in general (see, e.g., [7], [8], [29], [32]).
So here we have a typical situation for the generaloptimal controltheory. Namely,
the original control object is described by well-posed boundary value problem, but
the associated optimal control problem is ill-posed and requires relaxation.
Since there is no good topology a priori given on the set of all open subsets of
RN,westudythestabilitypropertiesoftheoriginalcontrolproblemimposingsome
constraintsondomainperturbations. Namely,weconsidertwotypesofdomainper-
turbations: so-calledtopologicallyadmissibleperturbations(followingDancer[11]),
and perturbations in the Hausdorff complementary topology (following Bucur and
Zolesio [6]). The asymptoticalbehavior of sets of admissible triplets Ξ — controls
ε
and the corresponding states — under domain perturbation is described in detail
in Section 4. In particular, we show that in this case the sequences of admissible
triplets to the perturbed problems are compact with respect to the weak conver-
gence in L∞(D;RN×N)×W1,p(D)×Lp(D). Section 5 is devoted to the stability
0
properties of optimal controlproblem (1)–(4) under the domain perturbation. Our
treatment of this question is based on a new stability concept for optimal control
problems(see for comparison[13, 14]). We showthat Mosco-stableoptimalcontrol
problemspossess“good” variationalproperties,whichallowusingoptimalsolutions
to the perturbed problems in “simpler” domains as a basis for the construction of
suboptimal controls for the original control problem. As a practical motivation of
this approach we want to point out that the “real” domain Ω is never perfectly
smooth but contains microscopic asperities of size significantly smaller than char-
acteristic length scale of the domain. So a direct numerical computation of the
solutions of optimal control problems in such domains is extremely difficult. Usu-
allyitneedsaveryfinediscretizationmesh,whichmeansanenormouscomputation
time, and such a computation is often irrelevant. In view ofthe variationalproper-
ties of Mosco-stable problems we can replace the “rough” domainΩ by a family of
more “regular” domains {Ω } ⊂ D forming some admissible perturbation and
ε ε>0
toapproximatetheoriginalproblembythe correspondingperturbedproblems[15].
2. Notation and preliminaries. Throughout the paper D and Ω are bounded
open subsets of RN, N ≥ 1 and Ω ⊂⊂ D. Let χ be the characteristic function
Ω
of the set Ω and let LN(Ω) be the N-dimensional Lebesgue measure of Ω. The
spaceD′(Ω)ofdistributions inΩ is the dualofthe spaceC∞(Ω). Forrealnumbers
0
2 ≤ p < +∞, and 1 < q < +∞ such that 1/p+1/q = 1, the space W1,p(Ω) is the
0
closure of C∞(Ω) in the Sobolev space W1,p(Ω) with respect to the norm
0
1/p
N ∂y p
kyk = dx+ |y|pdx , ∀y ∈W1,p(Ω), (5)
W01,p(Ω) ZΩk=1(cid:12)∂xi(cid:12) ZΩ ! 0
X(cid:12) (cid:12)
while W−1,q(Ω) is the dual spa(cid:12)(cid:12)ce of(cid:12)(cid:12)W1,p(Ω).
0
For any vector field v ∈ Lq(Ω;RN), the divergence is an element of the space
W−1,q(Ω) defined by the formula
4 OLHA P. KUPENKO AND ROSANNA MANZO
hdivv,ϕiW1,p(Ω) =− (v,∇ϕ)RN dx, ∀ϕ∈W01,p(Ω), (6)
0 ZΩ
whereh·,·i denotesthedualitypairingbetweenW−1,q(Ω)andW1,p(Ω), and
W1,p(Ω) 0
0
(·,·)RN denotes the scalar product of two vectors in RN. A vector field v is said to
be solenoidal, if divv=0.
Monotone operators. Let α and β be constants such that 0<α≤β <+∞. We
define Mα,β(D) as the set of all square symmetric matrices U(x)=[a (x)]
p ij 1≤i,j≤N
in L∞(D;RN×N) such that the following conditions of growth, monotonicity, and
strong coercivity are fulfilled:
|a (x)|≤β a.e. in D, ∀ i,j ∈{1,...,N}, (7)
ij
U(x)([ζp−2]ζ−[ηp−2]η),ζ −η ≥0 a.e. in D, ∀ζ,η ∈RN, (8)
RN
(cid:0) N (cid:1)
U(x)[ζp−2]ζ,ζ = a (x)|ζ |p−2ζ ζ ≥α|ζ|p a.e in D, (9)
RN ij j j i p
i,j=1
(cid:0) (cid:1) X
1/p
N
where |η| = |η |p is the Ho¨lder norm of η ∈RN and
p k
(cid:18)k=1 (cid:19)
P
[ηp−2]=diag{|η |p−2,|η |p−2,...,|η |p−2}, ∀η ∈RN. (10)
1 2 N
Remark2.1. ItiseasytoseethatMα,β(D)isanonemptysubsetofL∞(D;RN×N).
p
Indeed, as a representative of the set Mα,β(D) we can take any diagonal matrix of
p
the form U(x)=diag{δ (x),δ (x),...,δ (x)}, where functions δ (x)∈L∞(D) are
1 2 N i
such that α≤δ (x)≤β a.e. in D ∀i∈{1,...,N} (see [13]).
i
LetusconsideranonlinearoperatorA:Mα,β(D)×W1,p(Ω)→W−1,q(Ω)defined
p 0
as
A(U,y)=−div U(x)[(∇y)p−2]∇y +|y|p−2y,
or via the paring (cid:0) (cid:1)
N ∂y p−2 ∂y ∂v
hA(U,y),vi = a (x) dx
W01,p(Ω) i,j=1ZΩ ij (cid:12)∂xj(cid:12) ∂xj!∂xi
X (cid:12) (cid:12)
(cid:12) (cid:12)
+ |y|p−2yvdx, ∀v ∈(cid:12)W1,p(cid:12)(Ω).
0
ZΩ
In view of properties (7)–(9), for every fixed matrix U ∈ Mα,β(D), the operator
p
A(U,·) turns out to be coercive, strongly monotone and demi-continuous in the
following sense: y → y strongly in W1,p(Ω) implies that A(U,y ) ⇀ A(U,y )
k 0 0 k 0
weakly in W−1,q(Ω) (see [18]). Then by well-known existence results for nonlinear
elliptic equations with strictly monotone semi-continuous coercive operators (see
[18, 36]), the nonlinear Dirichlet boundary value problem
A(U,y)=f in Ω, y ∈W1,p(Ω), (11)
0
admits a unique weak solution in W1,p(Ω) for every fixed matrix U ∈ Mα,β(D)
0 p
and every distribution f ∈ W−1,q(D). Let us recall that a function y is the weak
SHAPE STABILITY OF OCPS FOR SYSTEMS OF HAMMERSTEIN TYPE 5
solution of (11) if
y ∈W1,p(Ω), (12)
0
U(x)[(∇y)p−2]∇y,∇v dx+ |y|p−2yvdx= fvdx, ∀v ∈W1,p(Ω).
RN 0
ZΩ ZΩ ZΩ
(cid:0) (cid:1) (13)
System of nonlinear operator equations with an equation of Hammerstein type.
LetY andZ be Banachspaces,letY ⊂Y be anarbitraryboundedset, andletZ∗
0
bethedualspacetoZ. Tobeginwithwerecallsomeusefulpropertiesofnon-linear
operators, concerning the solvability problem for Hammerstein type equations and
systems.
Definition 2.1. We say that the operator G : D(G) ⊂ Z → Z∗ is radially con-
tinuous if for any z ,z ∈ X there exists ε > 0 such that z +τz ∈ D(G) for all
1 2 1 2
τ ∈[0,ε] and the real-valuedfunction [0,ε]∋τ →hG(z +τz ),z i is continuous.
1 2 2 Z
Definition 2.2. An operator G : Y ×Z → Z∗ is said to have a uniformly semi-
bounded variation (u.s.b.v.) if for any bounded set Y ⊂ Y and any elements
0
z ,z ∈D(G) such that kz k ≤R, i=1,2, the following inequality
1 2 i Z
hG(y,z )−G(y,z ),z −z i ≥− inf C (R;k|z −z k| ) (14)
1 2 1 2 Z y 1 2 Z
y∈Y0
holds true providedthe function C :R ×R →R is continuous for each element
y + +
1
y ∈Y , and C (r,t)→0 as t→0, ∀r >0. Here, k|·k| is a seminorm on Z such
0 y Z
t
that k|·k| is compact with respect to the norm k·k .
Z Z
It is worth to note that Definition 2.2 gives in fact a certain generalization of
the classical monotonicity property. Indeed, if C (ρ,r) ≡ 0, then (14) implies the
y
monotonicity property for the operator G with respect to the second argument.
Remark 2.2. EachoperatorG:Y ×Z →Z∗ with u.s.b.v. possessesthe following
property (see for comparison Remark 1.1.2 in [1]): if a set K ⊂ Z is such that
kzk ≤ k and hG(y,z),zi ≤ k for all z ∈ K and y ∈ Y , then there exists a
Z 1 Z 2 0
constant C >0 such that kG(y,z)kZ∗ ≤C, ∀z ∈K and ∀y ∈Y0.
Let B :Z∗ →Z and F :Y ×Z →Z∗ be given operators such that the mapping
Z∗ ∋ z∗ 7→ B(z∗)∈ Z is linear. Let g ∈ Z be a given distribution. Then a typical
operator equation of Hammerstein type can be represented as follows
z+BF(y,z)=g. (15)
The following existence result is well-known (see [1, Theorem 1.2.1]).
Theorem 2.3. Let B : Z∗ → Z be a linear continuous positive operator such that
it has the right inverse operator B−1 :Z →Z∗. Let F :Y ×Z →Z∗ be an operator
r
with u.s.b.v such that F(y,·) : Z → Z∗ is radially continuous for each y ∈ Y and
0
the following inequality holds true
hF(y,z)−B−1g,zi ≥0 if only kzk >λ>0, λ=const.
r Z Z
Then the set
H(y)={z ∈Z : z+BF(y,z)=g in the sense of distributions }
is non-empty and weakly compact for every fixed y ∈Y and g ∈Z.
0
6 OLHA P. KUPENKO AND ROSANNA MANZO
Definition 2.4. We say that
(M) the operator F : Y ×Z → Z∗ possesses the M-property if for any sequences
{yk}k∈N ⊂ Y and {zk}k∈N ⊂ Z such that yk → y strongly in Y and zk → z
weakly in Z as k →∞, the condition
lim hF(y ,z ),z i =hF(y,z),zi (16)
k k k Z Z
k→∞
implies that z →z strongly in Z.
k
(A) the operator F : Y ×Z → Z∗ possesses the A-property if for any sequences
{yk}k∈N ⊂ Y and {zk}k∈N ⊂ Z such that yk → y strongly in Y and zk → z
weakly in Z as k →∞, the following relation
liminfhF(y ,z ),z i ≥hF(y,z),zi (17)
k k k Z Z
k→∞
holds true.
In what follows, we set Y =W1,p(Ω), Z =Lp(Ω), and Z∗ =Lq(Ω).
0
2.1. Capacity. There are many ways to define the Sobolev capacity. We use the
notion of local p-capacity which can be defined in the following way:
Definition 2.5. For acompactsetK containedinanarbitraryball B, capacityof
K in B, denoted by C (K,B), is defined as follows
p
C (K,B)=inf |Dϕ|pdx, ∀ϕ∈C∞(B), ϕ≥1 on K .
p 0
(cid:26)ZB (cid:27)
For open sets contained in B the capacity is defined by an interior approxi-
mating procedure by compact sets (see [20]), and for arbitrary sets by an exterior
approximating procedure by open sets.
It is said that a property holds p-quasi everywhere (abbreviated as p-q.e.) if
it holds outside a set of p-capacity zero. It is said that a property holds almost
everywhere(abbreviatedas a.e.) if it holds outside a setof Lebesgue measure zero.
A function y is called p-quasi–continuous if for any δ > 0 there exists an open
set A such that C (A ,B) < δ and y is continuous in D \A . We recall that
δ p δ δ
any function y ∈ W1,p(D) has a unique (up to a set of p-capacity zero) p-quasi
continuous representative. Let us recall the following results (see [3, 20]):
Theorem 2.6. Let y ∈W1,p(RN). Then y| ∈W1,p(Ω) provided y =0 p-q.e. on
Ω 0
Ωc for a p-quasi-continuous representative.
Theorem 2.7. Let Ω be a bounded open subset of RN, and let y ∈ W1,p(Ω). If
y =0 a.e. in Ω, then y =0 p-q.e. in Ω.
For these and other propertieson quasi-continuousrepresentatives,the readeris
referred to [3, 16, 20, 38].
2.2. Convergence of sets. In order to speak about “domain perturbation”, we
have to prescribe a topology on the space of open subsets of D. To do this, for the
family of all open subsets of D, we define the Hausdorff complementary topology,
denoted by Hc, given by the metric:
dHc(Ω1,Ω2)= sup |d(x,Ωc1)−d(x,Ωc2)|,
x∈RN
where Ωc are the complements of Ω in RN.
i i
SHAPE STABILITY OF OCPS FOR SYSTEMS OF HAMMERSTEIN TYPE 7
Definition 2.8. We say that a sequence {Ω } of open subsets of D converges
ε ε>0
to an open set Ω⊆D in Hc-topology, if dHc(Ωε,Ω) converges to 0 as ε→0.
The Hc-topology has some good properties, namely the space of open subsets
of D is compact with respect to Hc-convergence, and if Ω −H→c Ω, then for any
ε
compact K ⊂⊂ Ω we have K ⊂⊂ Ω for ε small enough. Moreover, a sequence of
ε
open sets {Ω } ⊂D Hc-converges to an open set Ω, if and only if the sequence
ε ε>0
ofcomplements{Ωc} convergestoΩc inthesenseofKuratowski. Werecallhere
ε ε>0
thatasequence{C } ofclosedsubsetsofRN issaidtobeconvergenttoaclosed
ε ε>0
set C in the sense of Kuratowskiif the following two properties hold:
(K ) for every x ∈ C, there exists a sequence {x ∈C } such that x → x as
1 ε ε ε>0 ε
ε→0;
(K ) if {ε } is a sequence of indices converging to zero, {x } is a sequence
2 k k∈N k k∈N
such that x ∈ C for every k ∈ N, and x converges to some x ∈ RN, then
k εk k
x∈C.
For these and other properties on Hc-topology, we refer to [17].
It is well known (see [4]) that in the case when p > N, the Hc-convergence of
open sets {Ω } ⊂ D is equivalent to the convergence in the sense of Mosco of
ε ε>0
the associated Sobolev spaces.
Definition 2.9. We say a sequence of spaces W1,p(Ω ) converges in the
0 ε
ε>0
sense of Mosco to W1,p(Ω) (see for comparison[2n8]) if the foollowing conditions are
0
satisfied:
(M ) foreveryy ∈W1,p(Ω)thereexistsasequence y ∈W1,p(Ω ) suchthat
1 0 ε 0 ε
ε>0
y →y strongly in W1,p(RN); n o
ε
(M ) if {ε } is a sequence converging to 0 and y ∈W1,p(Ω ) is a se-
2 qeuenckeeks∈uNch that y → ψ weakly in W1,p(RN)n, tkhen th0ere eεxkisotsk∈aNfunction
k
y ∈W1,p(Ω) such that y = ψ| .
0 Ω
Hereinafter we denote bey y (respect. y) the zero-extension to RN of a function
ε
defined on Ω (respect. on Ω), that is, y =y χ and y =yχ .
ε ε ε Ωε Ω
e e
Following Bucur & Trebeschi (see [5]), we have the following result.
e e e e
Theorem2.10. Let{Ω } beasequenceofopensubsetsofDsuchthatΩ −H→c Ω
ε ε>0 ε
and Ω ∈W (D) for every ε>0, with the class W (D) defined as
ε w w
W (D)={Ω⊆D : ∀x∈∂Ω,∀0<r <R<1;
w
1
R C (Ωc∩B(x,t);B(x,2t)) p−1 dt
p
≥w(r,R,x) , (18)
Zr Cp(B(x,t);B(x,2t)) ! t 
where B(x,t) is the ball of radius t centered at x, and the function 
w:(0,1)×(0,1)×D →R+
is such that
1. lim w(r,R,x)=+∞, locally uniformly on x∈D;
r→0
2. w is a lower semicontinuous function in the third argument.
8 OLHA P. KUPENKO AND ROSANNA MANZO
Then Ω∈W (D) and the sequence of Sobolev spaces W1,p(Ω ) converges
w 0 ε
ε>0
in the sense of Mosco to W1,p(Ω). n o
0
Theorem 2.11. Let N ≥p>N−1 and let {Ω } be a sequence of open subsets
ε ε>0
Hc
of D such that Ω −→ Ω and Ω ∈ O (D) for every ε > 0, where the class O (D)
ε ε l l
is defined as follows
O (D)={Ω⊆D : ♯Ωc ≤l} (19)
l
(here by ♯ one denotes the number of connected components). Then Ω∈O(D) and
l
the sequence of Sobolev spaces W1,p(Ω ) converges in the sense of Mosco to
0 ε
ε>0
W1,p(Ω). n o
0
In the meantime, the perturbation in Hc-topology (without some additional as-
sumptions) may be very irregular. It means that the continuity of the mapping
Ω 7→ y , which associates to every Ω the corresponding solution y of a Dirichlet
Ω Ω
boundary problem (12)–(13), may fail (see, for instance, [4, 10]). In view of this,
we introduce one more concept of the set convergence. Following Dancer [11] (see
also [12]), we say that
Definition 2.12. Asequence{Ω } ofopensubsetsofD topologicallyconverges
ε ε>0
top
toanopensetΩ⊆D(insymbolsΩ −→ Ω)ifthereexistsacompactsetK ⊂Ωof
ε 0
p-capacity zero (C (K ,D)=0) and a compact set K ⊂RN of Lebesgue measure
p 0 1
zero such that
(D ) Ω′ ⊂⊂Ω\K implies that Ω′ ⊂⊂Ω for ε small enough;
1 0 ε
(D ) for any open set U with Ω∪K ⊂U, we have Ω ⊂U for ε small enough.
2 1 ε
Note that without supplementaryregularityassumptions onthe sets, there is no
connectionbetweentopologicalsetconvergence,whichissometimescalled“conver-
gence in the sense of compacts” and the set convergence in the Hausdorff comple-
mentary topology (for examples and details see Remark 6.1 in the Appendix).
3. Setting of the optimal control problem and existence result. Let ξ ,
1
ξ be given functions of L∞(D) such that 0 ≤ ξ (x) ≤ ξ (x) a.e. in D. Let
2 1 2
{Q ,..., Q } be a collection of nonempty compact convex subsets of W−1,q(D).
1 N
To define the class of admissible controls, we introduce two sets
U = U =[a ]∈Mα,β(D) ξ (x)≤a (x)≤ξ (x)a.e. inD, ∀i,j =1,...,N ,
b ij p 1 ij 2
(20)
(cid:8) (cid:12) (cid:9)
(cid:12)
U = U =[u ,...,u ]∈Mα,β(D) divu ∈Q , ∀i=1,...,N , (21)
sol 1 N p i i
assuming tha(cid:8)t the intersection U ∩U ⊂(cid:12)L∞(D;RN×N) is nonempty(cid:9).
b sol (cid:12)
Definition 3.1. We say that a matrix U = [a ] is an admissible control of
ij
solenoidal type if U ∈U :=U ∩U .
ad b sol
Remark 3.1. As was shown in [13] the set U is compact with respect to weak-∗
ad
topology of the space L∞(D;RN×N).
Let us consider the following optimal control problem:
Minimize I (U,y,z)= |z(x)−z (x)|pdx , (22)
Ω d
n ZΩ o
SHAPE STABILITY OF OCPS FOR SYSTEMS OF HAMMERSTEIN TYPE 9
subject to the constraints
U(x)[(∇y)p−2]∇y,∇v dx+ |y|p−2yvdx=hf,vi , ∀v ∈W1,p(Ω),
RN W1,p(Ω) 0
ZΩ ZΩ 0
(cid:0) (cid:1) (23)
U ∈U , y ∈W1,p(Ω), (24)
ad 0
zφdx+ BF(y,z)φdx= gφdx, (25)
ZΩ ZΩ ZΩ
where f ∈W−1,q(D), g ∈Lp(D), and z ∈Lp(D) are given distributions.
d
Hereinafter, Ξ ⊂ L∞(D;RN×N) × W1,p(Ω) × Lp(Ω) denotes the set of all
sol 0
admissible triplets to the optimal control problem (22)–(25).
Definition3.2. Letτ bethetopologyonthesetL∞(D;RN×N)×W1,p(Ω)×Lp(Ω)
0
which we define as a product of the weak-∗ topology of L∞(D;RN×N), the weak
topology of W1,p(Ω), and the weak topology of Lp(Ω).
0
Further we use the following result (see [13, 23]).
Proposition 3.1. For each U ∈Mα,β(D) and every f ∈W−1,q(D), a weak solu-
p
tion y to variational problem (23)–(24) satisfies the estimate
kykp ≤Ckfkq , (26)
W1,p(Ω) W−1,q(D)
0
where C is a constant depending only on p and α.
Proposition 3.2. Let B : Lq(Ω) → Lp(Ω) and F : W1,p(Ω)×Lp(Ω) → Lq(Ω) be
0
operators satisfying all conditions of Theorem 2.3. Then the set
Ξ = (U,y,z)∈L∞(D;RN×N)×W1,p(Ω)×Lp(Ω):
sol 0
(cid:8) A(U,y)=f, z+BF(y,z)=g)
is nonempty for every f ∈W−1,q(D) and g ∈Lp(D). (cid:9)
Proof. The proof is given in Appendix.
Theorem 3.3. Assume the following conditions hold:
• The operators B :Lq(Ω)→Lp(Ω) and F :W1,p(Ω)×Lp(Ω)→Lq(Ω) satisfy
0
conditions of Theorem 2.3;
• The operator F(·,z):W1,p(Ω)→Lq(Ω) is compact in the following sense: if
0
y →y weakly in W1,p(Ω), then F(y ,z)→F(y ,z) strongly in Lq(Ω).
k 0 0 k 0
Then for every f ∈W−1,q(D) and g ∈Lp(D), the set Ξ is sequentially τ-closed,
sol
i.e. if a sequence {(Uk,yk,zk) ∈ Ξsol}k∈N τ-converges to a triplet (U0,y0,z0) ∈
L∞(Ω;RN×N)×W1,p(Ω)×Lp(Ω), then U ∈ U , y = y(U ), z ∈ H(y ), and,
0 0 ad 0 0 0 0
therefore, (U ,y ,z )∈Ξ .
0 0 0 sol
Proof. Let {(Uk,yk,zk)}k∈N ⊂ Ξsol be any τ-convergent sequence of admissible
tripletstotheoptimalcontrolproblem(22)–(25),andlet(U ,y ,z )beitsτ-limitin
0 0 0
thesenseofDefinition3.2. Sincethecontrols{Uk}k∈N belongtothesetofsolenoidal
matrices U (see (21)), it follows from results given in [22, 24] that U ∈U (see
sol 0 ad
alsoRemark3.1) andy =y(U ). Itremains to showthat z ∈H(y ). To this end,
0 0 0 0
we have to pass to the limit in equation
z +BF(y ,z )=g (27)
k k k
10 OLHA P. KUPENKO AND ROSANNA MANZO
ask →∞andgetthelimitpair(y ,z )isrelatedbytheequationz +BF(y ,z )=
0 0 0 0 0
g. With that in mind, let us rewrite equation (27) in the following way
B∗w +BF(y ,B∗w )=g,
k k k
where w ∈ Lq(Ω), B∗ : Lq(Ω) → Lp(Ω) is the conjugate operator for B, i.e.
k
hBν,wi =hB∗w,νi and B∗w = z . Then, for every k ∈N, we have the
Lq(Ω) Lq(Ω) k k
equality
hB∗w ,w i +hF(y ,B∗w ),B∗w i =hg,w i . (28)
k k Lp(Ω) k k k Lp(Ω) k Lp(Ω)
Taking into account the transformation
hg,w i =hBB−1g,w i =hB−1g,B∗w i ,
k Lp(Ω) r k Lp(Ω) r k Lp(Ω)
we obtain
hw ,B∗w i +hF(y ,B∗w )−B−1g,B∗w i =0. (29)
k k Lp(Ω) k k r k Lp(Ω)
The first term in (29) is strictly positive for every w 6= 0, hence, the second one
k
must be negative. In view of the initial assumptions, namely,
hF(y,x)−B−1g,xi ≥0 if only kxk >λ,
r Lp(Ω) Lp(Ω)
we conclude that
kB∗w k =kz k ≤λ. (30)
k Lp(Ω) k Lp(Ω)
Since the linear positive operator B∗ cannot map unbounded sets into bounded
ones, it follows that kw k ≤λ . As a result, see (28), we have
k Lq(Ω) 1
hF(y ,B∗w ),B∗w i =−hB∗w ,w i +hg,w i , (31)
k k k Lp(Ω) k k Lp(Ω) k Lp(Ω)
and, therefore, hF(y ,B∗w ),B∗w i ≤c . Indeed, all terms in the right-hand
k k k Lp(Ω) 1
side of (31) are bounded provided the sequence {wk}k∈N ⊂ Lq(Ω) is bounded and
operator B is linear and continuous. Hence, in view of Remark 2.2, we get
kF(y ,B∗w )k =kF(y ,z )k ≤c as kz k ≤λ.
k k Lq(Ω) k k Lq(Ω) 2 k Lp(Ω)
Sincetheright-handsideof (31)doesnotdependony ,itfollowsthattheconstant
k
c >0 does not depend on y either.
2 k
Taking these argumentsinto account, we may suppose that up to a subsequence
we have the weak convergenceF(y ,z )→ν in Lq(Ω). As a result, passing to the
k k 0
limit in (27), by continuity of B, we finally get
z +Bν =g. (32)
0 0
Itremainsto showthatν =F(y ,z ). Let us takeanarbitraryelementz ∈Lp(Ω)
0 0 0
suchthatkzk ≤λ. Using the fact thatF is anoperatorwith u.s.b.v., we have
Lp(Ω)
hF(y ,z)−F(y ,z ),z−z i ≥− inf C (λ;k|z−z k| ),
k k k k Lp(Ω) yk∈Y0 yk k Lp(Ω)
where Y ={y ∈W1,p(Ω): y satisfies (26)}, or, after transformation,
0 0
hF(y ,z),z−z i −hF(y ,z ),zi
k k Lp(Ω) k k Lp(Ω)
≥hF(y ,z ),−z i − inf C (λ;k|z−z k| ). (33)
k k k Lp(Ω) yk∈Y0 yk k Lp(Ω)
Since −z =BF(y ,z )−g, it follows from (33) that
k k k
hF(y ,z),z−z i −hF(y ,z ),zi
k k Lp(Ω) k k Lp(Ω)
+hF(y ,z ),gi ≥hF(y ,z ),BF(y ,z )i − inf C (λ;k|z−z k| ).
k k Lp(Ω) k k k k Lp(Ω) yk∈Y0 yk k Lp(Ω)
(34)