Boundary Control Problems for 2 × 2 Cooperative Hyperbolic Systems with Infinite Order Operators ()
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] - [11], and by varying the nature of control (such as distributed control [6] [11] [12] [13] and boundary control [3] [5] [8] ).
References [14] [15] were among the first studies that presented the control problems of systems including infinite order operators. These problems were then extended in different ways, such as for higher system degrees [16] [17], and for parabolic and hyperbolic systems [14] [17] [18] [19] [20] [21].
Based on the theories proposed by Lions [1] and Dubinskii [22] [23] [24], the distributed control problem with Dirichlet conditions for 2 × 2 non-cooperative hyperbolic systems involving infinite order operators was discussed in a previous study [13]; in this study, we extend this problem to cooperative hyperbolic systems of the boundary type with Neumann conditions for different observation functions.
The system can be defined as
(1)
where a, b, c,and d are constant such that
.
(This implies that the system (1) is cooperative.)
(2)
and
with the boundary as
.
The rest of this paper is organized into four sections. Section 2 presents the Sobolev spaces of infinite order, which we refer to later in the paper. In Section 3, the state of the cooperative system with Neumann conditions is discussed. In Section 4, the nascency and sufficient conditions for optimal boundary control are derived. Finally, in Section 5, the formulation of the control problem for boundary observation function is studied.
2. Necessary Spaces
The Sobolev spaces of infinite order operators, which are used in this study, have already been presented in Reference [13]. We list them briefly below:
·
· The formal conjugate space to the space
is defined as
where
and
.
Then, we have the following chain:
·
,
·
denotes the space of measurable functions
, such that
,
.
is a Hilbert space.
· 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 2 × 2 cooperative hyperbolic systems with Neuman conditions:
(3)
with
.
We have the following bilinear form:
(4)
where A maps from
onto
, and
(5)
since
is an infinite order operator.
Then,
(6)
Lemma (1):
There exists a constant
, such that
(7)
that is,
is coercive on
.
Proof:
We have
then,
By the Cauchy Schwarz inequality, we have
Hence,
Moreover, we assume that
Lemma (2):
By satisfying (7), system (3) has a unique solution:
.
Proof:
Let
is defined on
by
(8)
.
Then, by the Lax-Milgram lemma,
such that
(9)
.
Now, let us multiply system (2) by
and
as follows, and then integrate it over Q:
Hence,
By applying Green’s formula, we obtain
By summing the two equations, and from (6), (8), and (9), we obtain
Then, we deduce that
Thus, Equation (9) is equivalent to system (2), thereby completing the proof.
4. Control Problem When the Observation Function Is Given on Q
The space
is the space of controls
.
The state
of the system is given by the solution of
(10)
with
.
The observation equation is given by
(11)
The cost function is given by
(12)
where
, and
is a Hermitian positive definite operator:
(13)
Then, the control problem is to minimize J over
, which is a closed convex subset of
.
i.e., to determine
such that
,
.
Moreover, we have the following theorem:
Theorem 1:
Assuming that (7), (12), and (13) hold, $! the optimal control
, such that
if the following equations and inequalities are satisfied:
(14)
with
,
, (15)
together with (10), where
and
is the adjoint state.
Proof:
Since
is characterized by
,
, which is equivalent to
(16)
Now, since
where
from (10), we obtain
Thus,
According to the form of the adjoint equation in [1] we have proved system (14).
Now, we transform (16) by using (14) as follows:
Then, we obtain
Using (10), we have
which is equivalent to
.
Thus, the proof is complete.
5. Boundary Observation Function
Let us define the operator
as follows:
;
therefore,
is the observation equation on
.
The cost function
is defined by
, (17)
where
is defined as in (13), and
.
Then, the control problem is to minimize J over
, which is a closed convex subset of
, i.e., to determine
such that
.
Since the cost function (16) can be written as [14]
,
such that
,
.
Based on the above considerations, we obtain the following theorem.
Theorem 2:
Assuming that (7), (13), and (17) hold, the optimal control
is determined by the following systems:
(18)
with
together with (10) and (15).
Proof:
The optimal control
is described by [14]
which is equivalent to
(19)
According to the form of the adjoint equation in [1],
Then, by using theorem 1, we have a unique solution
, which satisfies
.
This proves system (18).
Now, from (20) and (18), we have
Using the Green formula, we obtain
Using (10), 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 (2), 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 studied the control problem when the observation function is given on the boundary (Theorem 2).
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 author would like to express sincere gratitude to the editor and the anonymous reviewers for their helpful comments and suggestions.