The earliest theory of optimal control was introduced by Lions .
Majority of the research in this field has focused on discussing the optimal control problem by using several operator types (such as elliptic, parabolic, or hyperbolic operators)   .
The discussion was extended to systems involving different types of operators (such as infinite order  -  or infinite number of variables    ).
In    , the studies continued to develop using different types of systems (cooperative or non-cooperative).
Based on the theories proposed by Lions  and Dubinskii   , the distributed control problem with Dirichlet conditions for 2 × 2 non-cooperative hyperbolic systems involving infinite order operators was discussed in a previous study ; in this study, we extend this problem to n× n cooperative hyperbolic systems.
The system can be defined as
with , .
for all . (This implies that the system (1) is cooperative), (2)
for all , (3)
and with boundary .
This paper is constituted of four sections. Section 1 presents the Sobolev spaces of infinite order, which we refer to later in the paper. In section 2, the state of n× n cooperative system with Dirichlet conditions is studied. In Section 3, the formulation of the distributed control with constraints is introduced. Finally, Section 4 presents some examples for the control problem without constraints.
2. Necessary Spaces:   
The Sobolev spaces of infinite order operators, which are used in this study, have already been presented in Reference .
We will list them briefly below:
* The conjugate space of is defined as,
where and .
Then we have the following chains:
* is a Hilbert space of measurable functions
, , that map an interval (0, T) in to the space , such that: , and
* In a similar manner as that of , we obtain the constructed space , and the following chains:
with the norm:
which is also a Hilbert space.
3. State of the System
We study the following cooperative hyperbolic systems with Dirichlet conditions:
with , .
We have the operators
it is easy to write A as a matrix take the form:
Let M be square coefficients matrix such that
Let , so that S represents square matrix takes the form
Hence, we can rewrite the first equation in system (4) as follows:
The bilinear form is defined on as follows:
where S maps onto , so that
There exists a constant , such that:
that is, (6) is coercive on .
then we deduce
which proves the coerciveness condition on .
If (2), (3) and (7) are hold, then $! for system (4), for .
Let be a continuous linear form defined on by
where and .
Then, by the Lax-Milgram lemma,
$! such that
Now, let us multiply system (4) by , and then integrate it over Q:
By using Green's formula:
from (6), (8) and (9) we have
Then, we deduce that
Thus, the proof is complete.
4. Control Problem with Constraints
The space is the space of controls .
The state of the system is determined by the solution of
with , .
The observation function is given by
The cost function is given by
where and M ≥ 0 is a constant.
Then, the control problem is to minimize J over which is a closed convex subset of .
i.e. to determine such that
Based on the above data and previous results, we have the following theorem:
Assuming that (7),(10) and (11) hold, $! the optimal control such that: , and it is determined by:
where is the adjoint state.
As in , is determined by:
which is equivalent to:
Now, let us define a hyperbolic infinite order operator B as follows:
Since, , from (3), we obtain
Now, let us set the following notation:
According to the form of the adjoint equation in :
and by Lemma 2,
$! Solution for (12).
Now, we transform (14) as follows:
we multiply (12) by and integrating between 0, T, then we obtain:
hence (14) becomes
Thus, the proof is complete.
5. Control Problem without Constraints
1) The case if i.e. (there are no constraints on the control ), then (13) takes the form , hence
Let us consider n=2 in (1), also (2) and (3) are satisfied, the space is the space of controls and the state is determined by:
together with (16), where is the adjoint state.
2) The case if there are no constraints on ,
i.e. , (19)
hence, (13) takes the following form:
If we take n = 2,
then . (21)
So, (13) is equivalent to
so, the optimal control is determined by:
In this paper, we have some important results. First of all we proved the existence and uniqueness of the state for system (4), which is (2 ´ 2) cooperative hyperbolic systems involving infinite order operators (Lemma 2). Then we found the necessary and sufficient conditions of optimality for system (10), that give the characterization of optimal control (Theorem 1).
Finally, we derived the necessary and sufficient conditions of optimality for some cases without control constraints.
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.
The authors thank the anonymous referees for their valuable suggestions which led to the improvement of the manuscript.
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