_{1}

^{*}

In this paper, we consider cooperative hyperbolic systems involving Schr
?dinger operator defined on R
^{n}. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.

The optimal control problems of distributed systems involving Schrödinger operator have been widely discussed in many papers. One of the first studies was introduced by Serag [

In [

Here, using the theory of [

with

where

i.e. the system (1) is called cooperative (2)

and

The model of the system (1) is given by:

since

We first prove the existence and uniqueness of the state for these systems, then we introduce the optimality conditions of boundary control, we also discuss them when the observation is on the boundary.

Here we shall consider some results about the following eigenvalue problem which introduced in [

The associated space is

Since the imbedding of

Operator in

We have:

which is continuous and compact.

Let us introduce the space

On

and the scalar product

the space

Analogously, we can define the spaces

with the scalar product:

then we have:

We have the bilinear form:

For all

The coerciveness condition of the bilinear form (7) in

conditions for having the maximum principle for cooperative system (1) which have been obtained by Fleckinger [

that means:

Theorem (3.1):

Under the hypotheses (2) and (9), if

Proof:

Let

then by Lax-Milgram lemma, there exists a unique element

Now, let us multiply both sides of first equation of system (1) by

By applying Green’s formula:

By sum the two equations we get:

by comparing the previous equation with (7), (10) and (11) we deduce that:

then the proof is complete.

The space

with

The observation equation is given by

For a given

where

The control problem then is to find

vex subset of

Since the cost function (14) can be written as (see [

where

Then there exists a unique optimal control

Theorem (4.1):

Assume that (9) and (14) hold. If the cost function is given by (13), the optimal control

with

together with (12) , where

Proof:

The optimal control

Which is equivalent to:

i.e.

this inequality can be written as:

Now, since:

where

by using Green formula and (12), we have:

then

and

since the adjoint equation takes the form [

and from theorem (3.1), we have a unique solution

This proves system (15).

Now, we transform (18) by using (15) as follows:

Using Green formula, we obtain:

Using (12), we have:

Thus the proof is complete.

The observation equation is given by:

This is interpreted as follows [

For a given

where

The control problem then is to find

vex subset of

Since the cost function (19) can be written as [

where

ficient conditions of optimality:

Theorem (5.1):

Assume that (9) and (14) hold. If the cost function is given by (19), the optimal control

is then characterized by the following equations and inequalities:

with

Proof:

The optimal control

Which is equivalent to:

i.e.

this inequality can be written as:

since the adjoint system takes the form [

and from theorem (3.1), we get a unique solution

This proves system (20).

Now, we transform (22) by using (20) as follows:

Using Green formula, we obtain:

Using (12), we have:

which is equivalent to:

Thus the proof is complete.

In this paper, we have some important results. First of all we proved the existence and uniqueness of the state for system (1), which is (2 ´ 2) cooperative hyperbolic system involving Schrödinger operator defined on

Also it is evident that by modifying:

-the nature of the control (distributed, boundary(,

-the nature of the observation (distributed, boundary(,

-the initial differential system,

-the type of equation (elliptic, parabolic and hyperbolic),

-the type of system (non-cooperative, cooperative),

-the order of equation,

many of variations on the above problem are possible to study with the help of Lions formalism.