Advances in Pure Mathematics, 2012, 2, 114-118
http://dx.doi.org/10.4236/apm.2012.22016 Published Online March 2012 (http://www.SciRP.org/journal/apm)
Sufficient Conditions of Opt im a li ty f or Convex Differential
Inclusions of Elliptic Type and Duality
Elimhan N. Mahmudov1,2
1Department of Industrial Engineering, Faculty of Management, Istanbul Technical University, Istanbul, Turkey
2Azerbaijan National Academy of Sciences, Institute of Cybernetics, Baku, Azerbaijan
Email: elimhan22@yahoo.com
Received October 10, 2011; revised December 2, 2011; accepted December 10, 2011
ABSTRACT
This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic typ e. On the basis of con-
jugacy correspondence the dual problems are constructed. Using the new concepts of locally adjoint mappings in the
form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality
problems are formulated for convex problems and duality theorems are p roved. The results obtained are gene ralized to
the multidimensional case with a second order elliptic operator.
Keywords: Differential Inclusion; Dirichlet; Locally Adjoint; Sufficient Conditions; Duality
1. Introduction
The present paper is devoted to an optimal control prob-
lems described by so-called discrete and differential in-
clusions of elliptic type. A lot of problems in economic
dynamics, as well as classical problems on optimal con-
trol in vibrations, chemical engineering, heat, diffusion
processes, differential games, and so on, can be reduced
to such investigations with ordinary and partial differen-
tial inclusions [1-15]. We refer the reader to the survey
papers [11,16-20]. The present paper is organized as fol-
lows.
In Section 2 first are given some suitable definitions,
supplementary notions and results considered by author
in [18,19]. Then a certain ex tremal Dirichlet’s problem is
formulated for so-called elliptic differential (PC) inclu-
sions with Laplace’s operator and with second order el-
liptic operator in the multidimensional case. In the re-
viewed results for optimality the arisen adjo int inclusions
using the locally adjoint multivalued (LAM) functions
are stated in the Euler-Lagrange form [9,18,19]. It turn
out that such form of optimality conditions automatically
implies the Weierstrass-Pontryagin maximum condition.
Apparently it happens because the LAM is more applica-
ble apparat in different type of problems governed by
differential inclusion s [16-20].
In Section 3 the main problem is to formulate and
study the dual problems to the stated problems with con-
vex structures. Convexity is a crucial marker in classify-
ing optimization problems, and it’s often accompanied
by interesting phenomena of duality. It is well known
that duality theory by virtue of its applications is one of
the central directions in convex optimality problems. In
mathematical economics duality theory is interpreted in
the form of prices, in mechanics the potential energy and
complementary energy are in a mutually dual relation,
the displacement field and the stress field are solutions of
the direct and the dual problems, respectively.
To establish the dual problem we use the duality theo-
rems of operations of addition and infimal convo lution of
convex functions. Here a remarkable specific feature of
second order elliptic partial differential inclusions in com-
parisons with ordinary ones is that they admit valuable
results in the case of multidimensional domain. Our ap-
proach to establish duality th eory for continuou s problem
is based on the passage to the formal limit from duality
problem in approximating problem. But to avo id difficult
and fatiguing calculations we omit it and announce only
dual problem constructed for continuous problems (PC)
and then (PM). Consequently construction of duality
problem in our paper is an unforeseen part of the “ice-
berg”. Further it is shown that direct and duality prob-
lems are connected to each other by the duality relations.
The proved duality theo rems allow one to con clude that a
sufficient condition for an extremum is an extremal rela-
tion for the primary and dual problems. It means that if
some pair of feasible solutions (u(.),u*(.)) satisfy this re-
lation, then u(.) and u*(.) are solutions of the primary and
dual problem, respectively. We note that a considerable
part of the investigations of Ekeland and Temam [7] for
simple variational problem is devoted to such problems.
Besides there are similar results for ordinary differential
C
opyright © 2012 SciRes. APM
E. N. MAHMUDOV 115
inclusions in [17-19]. Some duality relations and opti-
mality conditions for an extremum of different control
problems with partial differential inclu sions can be found
in [18,19]. At the end of Section 3 we consider a linear
optimal control problem of elliptic type.
Furthermore, observe that in elliptic differential inclu-
sions for simplicity of the exposition the solution is taken
in the space of classical solutions. Apparently by passing
to more general function spaces of generalized solutions
the most natural approach for elliptic differential inclu-
sions is the use of single-valued selections of a multi-
valued mapping [1,3,8,9].
2. Necessary Concepts and Problems
Statements
Throughout this section and the next sections we use
special notation conventional in the [18,19]. Let be
the n-dimensional Euclidian space, 12
is a pair of
elements 12 and
n
R
uu
,
, n
uu R12
,uu
nn
is their inner product.
A multivalued mapping
:
F
RPR (
n
PR de-
notes the family of all subsets of ) is convex if its
graph
n
R


,:
g
ph FuvvFu
is a convex subset of
. It is convex-valued if
2n
R
F
u
is a convex set for
each . Let us introduce the
notations:

: uFuudomF 


**
,sup,
v: ,
M
uvvvvF u



*
,,,Muv
**
*
,:
n
Fuvv Fuvv
vR

For convex F we let
if Fu

*
,
*
, Muv  .
Obviously the function M and the sets
F
uv
n
Rn
R


*
,
gph F
can be
interpreted as Hamiltonian function and argmaximum
sets, respectively.
Definition 2.1. For a convex mapping F a multivalued
mapping from into defined by


** *
,,: ,
**
F
vuvuu vK uv

uv

*,
is called the locally adjoint mapping (LAM) to F at the
point where
gph F
, ,gphF
uv
is the dual to
the basic cone
,
gph F
uv . We refer to [1,6,8,9] for
various definitions in this direction.
It is clear that for a convex F the Hamiltonian is con-
cave on u and convex on function. Let us denote
*
v


*** *
,inf,,:
H
uvuuvvu,vgphF .
It is clear that by the conjugacy correspondence of
convex analysis [4],[6-9]:
 




* *
*
**
,
.
**
,inf,
.,
u
H
uv uu
Mv 
Muv
u


Corollary 2.1.
***
,uFvuvand equality The inclusion
** **
,, ,
H
uvuu Muv are equivalent.
In Section 3 we study the following problem for ellip-
tic differential inclusion with homogeneous boundary
value conditions:
minimize


.:, duguxxx
,
J
subject to

,, uxFux xxR
, (1)
and
0, uxx B
, (2)
is a Laplace’s operator, where
.,: nn
F
xR PR is multivalued mapping for all
,
12
x
xx11
RR
B
1
:n
in the bounded region , a closed
piecewise-smooth simple curve is its boundary,
g
RRu
ddd
is a continuous convex function on
and 12
x
xx
. We label this continuous problem
P
C and call it Dirichlet problem for elliptic differen-
tial inclusions. The problem is to find a solution
ux
of the boundary v a lu e prob lem (1 ) , (2) th at mini mizes th e
cost functional
..Ju Here, a feasible solution is
understood to be a classical solution for simplicity of the
exposition.
The subject of the research in Section 6 in the follow-
ing multidimensional optimal control problem
M
P for
elliptic differential inclusions:
minimize


.:, d
G
uguxxx
J
,
subject to
,, ,Lu xF u xxxG (3)
and
0,uxx S
(4)
where
11
., :
F
xR PR is a convex closed multi-
valued mapping for all n-dimensional vectors
,,
1n
x
xxn
GR
S
11
:
in the bounded set , a closed
piecewise-smooth surface is its boundary,
g
RG R
12
dddd
n
is a continuous and convex on u func-
tion,
x
xx x
. L is a second-order elliptic ope-
rator:
 
,1 1
:,
nn
ij i
ij i
iji
uu
Luabxcx u
xx x


 



 




11
,,
ij i
ax CGbx CGcxCG
ij
ax
where is a positively definite matrix, ux
Copyright © 2012 SciRes. APM
E. N. MAHMUDOV
116
and

1
CG G
are the spaces of continuous functions and
functions having a continuous derivative in , respec-
tively.
A function in

ux


,CG CG


M
P

2

ux
.,
that satisfies
the inclusion (3) in G and the boundary condition (4) on
S we call a classical solution of the problem posed, where
is the space of functions ux having conti-
nuous all second-order derivatives. It is required to find a
classical solution of the boundary value problem
that minimizes the cost functional J(u(.)).
2
CG

M
P
In the next theorem is referred sufficient conditions for
optimality for problems and of Mahmu-
dov [18].
C
P
Theorem 2.1. Assume that a continuous function g is
convex with respect to u, and
F
x

ux

C
P

*
ux



,ux x



, ,
is a convex map-
ping for all fixed x. Then for the optimality of the solu-
tion among all feasible solutions in convex prob-
lem it is sufficient that there exist a classical solu-
tion such that the following condition:
a)

*
uxx u

**
Fu

,,x
g
ux x

0, xB

12
,xx 

P


,u xx



,

*
ux ,
b) .
 
*
,,,uxFux uxx


x

,,xL
For a problem the Euler-Lagrange type inclu-
sion (a) and argmaximum condition (b) consist of the
following conditions, respectively:
i)
 
*** *
Lu xF u xu
g
ux x

0,xxS
L L


**
,d
,
ii)
 
**
,,,Lu xFu xuxu

*

x

*
,,
where is the operator adjoint to .
3. On Duality in Elliptic Differential
Inclusions
According to the definition in [4,8,9,18,19 ]
Let us denote
 



**
*
**
,Jux zx
H
ux zxxg 

uxzxxx


where
H
is a Hamiltonian function and
**
,
g
zx is
conjugate function to function

,
g
x for every fixed
11
x
RR


**
,uxzx

C
P
. Then the problem of determining the maxi-
mum
 
**
*
*
()
, (),,
() 0,
max i miz e
Duxzxx
ux xB
PJ

is called the dual problem to the primaryconvex problem
. It is assumed that

 
*2 *
, ux CCzx C  

,uxx

P
.
Theorem 3.1. Assume that is an arbitrarily
feasible solution of the primary problem C and
**
,uxzx is a feasible solution of the du al problem
D
P




*
*
. Then the inequality
J
uxJ ux is va-
lid.
Proof. It is clear from the definitions of the functions
*
H
and
g
that the following inequ a lities hold:

 
***
** *
,,
,,
Huxzxuxx
ux zxuxuxux

 
 

** *
,, ,
g
zxxzxux guxx
Therefore;




*****
**
,, ,
,, ,.
Huxzxuxxgzxx
uxuxux uxguxx
 
  (5)
*0, 0,ux uxxB
Then since

 
, by the familiar
Green theorem [21] we have
 
  
**
**
,,d
,,d0
B
uxuxux uxx
ux ux
uxuxs
nn












ux
(6)
where n is outher normal for a curve B. Then integrating
both sides of the inequality (5) due to (6) we obtain the
required inequality.
Theorem 3.2. If the feasible solutions and
**
,uxzx,


*,zx guxx satisfy the condi-
tions of Theorem 2.1, then they are optimal solutions of
the primary
P

C and dual
D
P problems, respec-
tively, and their values are equal .
Proof. To proceed, first note that by Theorem 2.1
ux
is a solution of the primary problem
P
 

**
,uxzx C. We
need to prove that the pair is a solution
to problem
D
P. By Definition 2.1 of a LAM, the con-
dition (a) of the Theorem 2.1 is equivalent to the ine-
quality
 
 
**
*
,
,0,
ux zxuux
uxv ux
 


,,,u vgphFx.

The latter yields

***
,,,ux zxuxdomHx


, (7)
where

** **
,,:,:, ,domHxu vHuvx
 .
Copyright © 2012 SciRes. APM
E. N. MAHMUDOV 117
Further, since [4,6,8,9]
 
*
,,
g
u xdomg x
 
**
,domg x
it is clear
that

zx . (8)
Consequently, it can be concluded from (7.3), (7.4)
that the indicated pair of functions

**
,uxzx

is a
feasible solutions, i.e. the set of feasible solutions to
D
P

*
,ux
is nonempty. Let us now justify the optimality of
the solution
to problem

*
zx
D
P. By the
Corollary 2.1



**
***
,,,
:,, ,
Fvuvx
uHuvx uu

* *
,,.Muvx
Using this formula and the condition (a) of the Theo-
rem 2.1 we get


 

*
,
,.x uxx
***
**
,,
,
Huxzxuxx
uxuxzxMu


Now based on the condition (c) of Theorem 2.1 we
have the following equ ality
  
**
,,,xuxx
ux uxMu
.
Thus
 

 
***
**
,,Huxzxuxx
uxux zx



*
,,
.uxux (9)
On the other hand the inclusion

,guxx
*
zx
is equivalent with the eq uality




,
** *
,,
g
zxxuxzxguxx

 

**
,
. (10)
Then in view of (8)-(10) as in the proof of Theorem
3.1 it is not hard to show that


*
J
ux J
ux z x

P
 
. This completes the proof
of the theorem.
Now let us formulate the dual problem to the convex
problem with homogeneous boundary conditions.
In this case the duality problem consists in the following
 
**
, ,uxzx


*,
**
*
*
()
, (),,
()0,
maximiz e
MD uxzxxG
ux xS
PJ

Here
 

 

**
*
*** **
,
,,
G
Juxzx
d
H
Luxzxuxx g

zxx
x
 


*2
1,, .
n
u xC GCGz
xx x

 
*
,,x CG
Now by replacing the Laplace operator with the
second order elliptic operator L and using the idea sug-
gested in the proofs of Theorems 3.1 and 3.2 it is easy to
get the following theorem.
ux
and pair of functio ns Theorem 3.3. If
**
,uxzx, are feasible solutions to the primary
convex problem
P

with homogeneous boundary
value conditions and dual problem
M
D, respectively,
then P

**
,
*
J
uxJ ux zx
. In addition, if the
assertions (i), (ii) for sufficiency of optimality are valid
here and
*,zx guxx
, then the values of the
cost functionals are equal and

**
,uxzx is solu-
tion of the dual problem
M
D
Let us consider the following example:
.
P



minimize, d
LD R
PJuxguxxx

,uxAuxBwx wxV, subject to
nn
where
A
is
matrix, is a rectangular Bnr
matrix, is a closed convex set and
r
VR
g
is con-
tinuously differentiable function on u. It is required to
find a controlling parameter such that the fea-
sible solution corresponding to it minimizes

wx V
Ju
.

F
Let us introduce a convex mapping uAuBV.
By elementary calculations, it can be shown, that

** **
,
*** **
***
,inf, ,:
,,
,
uw
V
H
uvuuAu BwvwV
MBvu Av
uAv


 
** **
sup ,.
VwV
M
Bw wBw
where
Then obviously the duality problem for primary prob-
lem
L
D
P
 
has a form:
**
*,,
J
uxzx maximize

** **
,,ux zxAuxxR
 
*0, ,ux xB
 

**
*
*** *
((),())
(,d.
V
Juxzx
where
Buxgzxxx
M


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E. N. MAHMUDOV
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