An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator ()
1. Introduction
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 [1] , which discusses 2 × 2 cooperative systems of elliptic operator. Further research in this area developed the problem by studying different operator types (el- liptic, parabolic, or hyperbolic) or higher system degree as in [2] - [6] . Many boundary control problems have been introduced in [7] - [10] .
In [3] , we discussed distributed control problem for 2 × 2 cooperative hyperbolic systems involving Schrö- dinger operator.
Here, using the theory of [11] , we consider the following 2 × 2 cooperative hyperbolic systems involving Schrö- dinger operator:
(1)
with
.
where
,
,
and
are given numbers such that
,
,
i.e. the system (1) is called cooperative (2)
is a positive function and tending to
at infinity, (3)
and
with boundary
.
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.
2. Some Concepts and Results
Here we shall consider some results about the following eigenvalue problem which introduced in [1] and [12] :
(4)
The associated space is
, with respect to the norm:
(5)
Since the imbedding of
into
is compact, then the operator
considered as an
Operator in
is positive self-adjoint with compact inverse. Hence its spectrum consists of an infinite se- quence of positive eigenvalues, tending to infinity; moreover the smallest one which is called the principal ei- genvalue denoted by
is simple and is associated with an eigenfunction which does not change sign in
. It is characterized by:
(6)
We have:
![]()
which is continuous and compact.
Let us introduce the space
of measurable function
which is defined on open interval
and the variable
,
denotes the time.
On
with Lebesgue measure
we have the norm:
![]()
and the scalar product
,
the space
with the scalar product and the norm above is a Hilbert space.
Analogously, we can define the spaces
,
with the scalar product:
![]()
then we have:
![]()
3. The Existence and Uniqueness for the State of the System (1)
We have the bilinear form:
(7)
For all
the function
is measurable on
.
The coerciveness condition of the bilinear form (7) in
has been proved by Serag [1] , by using the
conditions for having the maximum principle for cooperative system (1) which have been obtained by Fleckinger [13] , and take the form:
(8)
that means:
(9)
Theorem (3.1):
Under the hypotheses (2) and (9), if
,
,
and
,
, then there exists a unique solution:
for system (1).
Proof:
Let
be a continuous linear form defined on
by:
(10)
then by Lax-Milgram lemma, there exists a unique element
such that:
(11)
Now, let us multiply both sides of first equation of system (1) by
, and the second equation by:
then integration over
, we have:
![]()
![]()
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.
4. Formulation of the Control Problem
The space
is the space of controls. For a control
, the state
of the system is given by the solution of
(12)
with
.
The observation equation is given by
.
For a given
, the cost function is given by:
. (13)
where
is hermitian positive definite operator:
(14)
The control problem then is to find
such that
, where
is a closed con-
vex subset of
.
Since the cost function (14) can be written as (see [11] ):
![]()
where
is a continuous coercive bilinear form and
is a continuous linear form on
.
Then there exists a unique optimal control
such that
for all
by using the general theory of Lions [11] . Moreover, we have the following theorem which gives the necessary and sufficient conditions of optimality:
Theorem (4.1):
Assume that (9) and (14) hold. If the cost function is given by (13), the optimal control
is then characterized by the following equations and inequalities:
(15)
with ![]()
(16)
together with (12) , where
is the adjoint state.
Proof:
The optimal control
is characterized by [11]
,
Which is equivalent to:
![]()
i.e.
(17)
this inequality can be written as:
(18)
Now, since:
![]()
where
![]()
by using Green formula and (12), we have:
![]()
then
![]()
and ![]()
since the adjoint equation takes the form [11] : ![]()
and from theorem (3.1), we have a unique solution
which satisfies
,
,
,
.
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.
5. Formulation of the Problem When the Observation Is on the Boundary
The observation equation is given by:
![]()
.
This is interpreted as follows [11] : we take the trace of
on
, which is particular in
. Let this be denoted by
.
For a given
, the cost function is given by:
. (19)
where
is defined as in (14).
The control problem then is to find
such that
, where
is a closed con-
vex subset of
.
Since the cost function (19) can be written as [11] :
,
where
is a continuous coercive bilinear form and
is a continuous linear form on
. Then using the general theory of Lions [11] , there exists a unique optimal control
such that
for all
. Moreover, we have the following theorem which gives the necessary and suf-
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:
(20)
with
together with (16) and (12).
Proof:
The optimal control
is characterized by [11] :
![]()
Which is equivalent to:
![]()
i.e.
(21)
this inequality can be written as:
(22)
since the adjoint system takes the form [11] :
![]()
and from theorem (3.1), we get a unique solution
which satisfies:
.
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.
6. Conclusions
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
(Theorem 3.1). Then we found the necessary and sufficient conditions of optimality for system (1), that give the characterization of optimal control (Theorem 4.1). Finally, we also find the necessary and sufficient conditions of optimal control when the observation is on the boundary (Theorem 5.1).
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.