1. Introduction
The earliest theory of optimal control was introduced by Lions [1].
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) [2] [3] [4].
The discussion was extended to systems involving different types of operators (such as infinite order [5] - [11] or infinite number of variables [12] [13] [14] ).
In [3] [15] [16] [17], the studies continued to develop using different types of systems (cooperative or non-cooperative).
Based on the theories proposed by Lions [1] and Dubinskii [18] [19] [20], the distributed control problem with Dirichlet conditions for 2 × 2 non-cooperative hyperbolic systems involving infinite order operators was discussed in a previous study [17]; in this study, we extend this problem to n× n cooperative hyperbolic systems.
The system can be defined as
(1)
with
,
.
Where
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: [18] [19] [20]
The Sobolev spaces of infinite order operators, which are used in this study, have already been presented in Reference [17].
We will list them briefly below:
*
,
* The conjugate space of
is defined as,
*
,
where
and
.
Then we have the following chains:
*
,
*
,
where
*
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:
*
,
*
.
Finally,
*
,
with the norm:
which is also a Hilbert space.
3. State of the System
We study the following cooperative hyperbolic systems with Dirichlet conditions:
(4)
with
,
.
We have the operators
such that
it is easy to write A as a matrix take the form:
i.e.
(5)
Let M be
square coefficients matrix such that
i.e.
.
Let
, so that S represents
square matrix takes the form
Therefore,
.
Hence, we can rewrite the first equation in system (4) as follows:
Definition 1:
The bilinear form
is defined on
as follows:
where S maps
onto
, so that
(6)
Lemma 1:
There exists a constant
, such that:
(7)
that is, (6) is coercive on
.
Proof:
We have:
Thus,
then we deduce
then,
which proves the coerciveness condition on
.
Lemma 2:
If (2), (3) and (7) are hold, then $!
for system (4), for
.
Proof:
Let
be a continuous linear form defined on
by
,
, (8)
where
and
.
Then, by the Lax-Milgram lemma,
$!
such that
(9)
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
(10)
with
,
.
The observation function is given by
.
The cost function
is given by
(11)
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:
Theorem 1:
Assuming that (7),(10) and (11) hold, $! the optimal control
such that:
, and it is determined by:
(12)
with
,
and
(13)
where
is the adjoint state.
Proof:
As in [1],
is determined by:
i.e.
which is equivalent to:
(14)
Now, let us define a hyperbolic infinite order operator B as follows:
Since,
, from (3), we obtain
then
.
Now, let us set the following notation:
According to the form of the adjoint equation in [1]:
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
i.e.
.
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
. (15)
Example 1:
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:
(16)
.
(17)
,
,
(18)
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:
(20)
Example 2:
If we take n = 2,
then
. (21)
So, (13) is equivalent to
(22)
so, the optimal control is determined by:
(23)
Further
(24)
6. Conclusion
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.
Acknowledgements
The authors thank the anonymous referees for their valuable suggestions which led to the improvement of the manuscript.