_{1}

^{*}

The article has been retracted due to the investigation of complaints received against it. The substantial portions of the text came from A. Alsaban's paper, "Optimal control under Conjugation Conditions For elliptic Systems", and Ahmed Mohammed Abdallah’s article has been retraced because of plagiarism. The scientific community takes a very strong view on this matter and we treat all unethical behavior such as plagiarism seriously. This paper published in Vol.3 No.1 1-11, 2014, has been removed from this site.

The necessary and sufficient conditions of optimality for systems governed by elliptic operators have been studi- ed by Lions in [

In this section, we consider the following 2 × 2 elliptic system (see [

where

where Ω is an open subset of

en elliptic equation is specified in bounded, continuous and strictly Lipschitz domains in

Since

we define the following bilinear form on

Lemma 1(see [

Let

Lemma 2(see [

First Eigen values

For

Theorem 1

For

The bilinear form can be written as

We choose m large enough such that a + m > 0 and d + m > 0 and the equivalent norm

By Cauchy Schwartz and Lemma 1 and 2 in the paper

tion

Let

The energy functional

A unique state

are specified on domain

The observation equation is given by

The cost functional is given by:

where

The control problem then is to find:

where

where

Assume that (2), (6) hold. The cost function being given (5), necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied

Outline of proof

Since

Since

then

By Green^{’}s formula or derivative in the sense of distribution

where

From (7), we obtain

Remark 1

Generalization to n × n systems

and conjugation conditions

where

In this case, the bilinear form is given by:

The linear form is given by:

The cost functional is given by:

where

on

In this case the necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied.

If constraints are absent i.e. when

problem of finding the vector-function

We can find last relations at

We can choose

Let

For a control

The observation equation is given by

Since the cost function (5) can be written as:

where

since

The necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied

Outline of proof

Since J(v) is differentiable and U_{ad} is bounded then the optimal control u is characterized

Since

then

By Green’s formula or derivative in the sense of distribution

where

From (6), we obtain

We can add the control in the part of Dirichlet condition, i.e.

The difference appears in characterization

Remark 2

Generalization to n × n systems

and conjugation conditions

where Ω is an open subset of

terior to Ω,

In this case, the bilinear form is given by:

The linear form is given by

The cost functional is given by:

where

on

In this case the necessary and sufficient for u to be an optimal control is that the following equations and inequalities be satisfied.

If constraints are absent i.e. when

problem of finding the vector-function

We can find last relations at n = 2 and q = 0 [

The aim is to find the existence and uniqueness of solution (state of the system) and necessary and sufficient condition for optimality of the system in 2 × 2 and generalization to n × n in the case of distributed and bounda- ry control. In this paper, we connect [