Boundary Control for Cooperative Elliptic Systems under Conjugation Conditions ()
1. Introduction
In today’s rapidly progressing science and technology, the field of control theory is at the forefront of the creative interplay of mathematics, engineering, and computer science.
Control theory has two objectives:
To understand the fundamental principle of control, and to characterize them mathematically.
The control problem is to choose the control space U to minimize an energy functional J(u), subject to constraints on the control such as Uad (set of admissible control) is a closed convex subset of U.
Various optimal control problems, of systems governed by finite order elliptic, parabolic and hyperbolic operators with finite number of variables have been introduced by Lions [1]. These problems have been extended to non-cooperative systems in [2] [3] and cooperative systems in [4] [5] [6] [7]. The control problems for infinite order hyperbolic operators have been studied in [8] [9].
Some existence results have been established for nonlinear systems in [10] [11] [12] [13] [14].
Some applications for control problems were introduced for example in [2] [15].
New optimal control problems of distributed systems described by an elliptic, parabolic and hyperbolic operators with conjugation conditions and by a quadratic cost functional have been studied by Sergienko and Deineka [16] [17] [18].
In the present work, using the theory of Lions [1], Sergienko and Deineka [16] [17] [18], the boundary control for some cooperative elliptic systems of the form
(1)
under conjugation conditions is discussed, where
(2)
System (1), where (2) is satisfied is called cooperative system. Such systems appear in some biological and physical problems [19].
Our paper is organized as follows: In Section 2, we first prove the existence and uniqueness of the state for 2 × 2 Dirichlet cooperative system under conjugation conditions, then we study the optimal boundary control of this system. Section 3 is devoted to discuss the boundary control for 2 × 2 Neumann cooperative elliptic system under conjugation conditions. In Section 4, we generalize the discussion which has been introduced in Section 2, to n × n Dirichlet cooperative system with conjugation conditions. Finally in Section 5, we generalize the problem which has been established in Section 3 to n × n Neumann cooperative elliptic system under conjugation conditions.
2. Boundary Control for 2 × 2 Dirichlet Elliptic Systems
In this section, we study the boundary control for the following 2 × 2 cooperative Dirichlet elliptic system
(3)
under conjugation conditions:
(4)
(5)
where
a, b, c and d are given numbers such that b, c > 0,
and
(6)
We first prove the existence of the state of systems (3) under the following conditions:
(7)
where
is a positive constant determined by Friedrich inequality:
(8)
Then, we prove the existence of boundary control for this system and we find the set of equations and inequalities that characterizes this boundary control.
Existence and uniqueness of the state
By Cartesian product, we have the following chain of Sobolev spaces:
On
, we define the bilinear form:
(9)
Then, we have
Lemma 2.1 The bilinear form (9) is coercive on
; that is, there exists a positive constant C such that
(10)
Proof.
As in [19], we choose m is large enough such that
and
.
Then,
From (6), we get
By Cauchy Schwartz inequality
From (8), we deduce
Therefore (7) implies
which proves the coerciveness condition of the bilinear form (2.7). Then using Lax-Milgram lemma, we can prove the following theorem:
Theorem 2.1 For a given
there exists a unique solution
for systems (3) with conjugation conditions (4) and (5) if conditions (7) are satisfied.
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
(11)
under conjugation conditions
(12)
(13)
The observation equation is given by:
For a given
, the cost functional is given by
(14)
where N is a hermitian positive definite operator such that:
(15)
The control problem then is to:
(16)
The cost functional (14) can be written as
If we let:
and
Then
From (15),
. Using the theory of Lions [1], there exists a unique optimal control of problem (16), moreover it is characterized by
Theorem 2.2 Let us suppose that (10) holds and the cost functional is given by (14), then the boundary control u is characterized by
together with(11), (12) and (13), where
is the adjoint state.
Proof.
The optimal control u is characterized by [1]
Then
So
(17)
Since the model A of the system is given by
and since
then
and since the adjoint state is defined by:
hence (17) implies
3. Boundary Control for 2 × 2 Neumann Elliptic Systems
In this section, we study the boundary control for 2 × 2 cooperative Neumann elliptic system in the form
(18)
with conjugation conditions (4) and (5), where
,
. For this, we introduce again the bilinear form (9) which is coercive on
, since
Then, by Lax-Milgram lemma, there exists a unique solution
for system (18) such that:
(19)
where
is a continuous linear form defined on
. Then, applying Green’s formula, we get
using (19), the state
of the system is given by the solution of
(20)
under conjugation conditions (12), (13). For a given
, the cost functional is again given by (14), then there exists a unique optimal control
for (16) and we deduce:
Theorem 3.1 If the cost functional is given by (14), there exists a unique boundary control
, such that:
moreover it is characterized by the following equations and inequalities
Together with (20), (12) and (13).
4. Boundary Control for n × n Cooperative Dirichlet Systems
In this section, we generalize the discussion which has been introduced in section 2 to n × n cooperative Dirichlet system of the form
(21)
under conjugation conditions
(22)
and
(23)
To prove the existence of the state of system (21), we assume that:
(24)
where, I is identity matrix and
is a positive constant determined by Friedrich inequality(8).
By Cartesian product, we have the following chain of Sobolev spaces:
On
, the bilinear form is defined by:
(25)
As in lemma 2.1, (24), implies
(26)
Now, let
(27)
be a continuous linear form on
, then using Lax-Milgram lemma, there exists a unique solution
such that:
Then, we have
Theorem 4.1 For
there exists a unique solution
for cooperative Dirichlet system (21) with conjugation conditions (22) and (23) if condition (24) is satisfied.
So, we can formulate the corresponding control problem:
The space
is the space of controls. For a control
, the state
of the system is given by the solution of
(28)
under conjugation conditions:
(29)
The observation may be takes as
the cost functional is given by
(30)
where N is a hermitian positive definite operator such that:
(31)
The control problem then is to find:
(32)
where
is closed convex subset of
. The cost functional (30) can be written as
if we let
and
then
From (31),
using the theory of Lions [1], there exists a unique optimal control of problem(32); moreover it is characterized by
Theorem 4.2 Let us suppose that (26) holds and the cost functional is given by (13), then the boundary control u is characterized by
(33)
together with (28) and (29) where
is the adjoint state.
Proof. The optimal control
is characterized by:
then
(34)
since the adjoint state is defined by (33), (34) implies
Applying Green’s formula, we obtain
Since
we obtain by using equation (28),
hence
5. Boundary Control for n × n Cooperative Neumann Systems
We generalize here, the results which have been established in section (3) to the following n × n Neumann elliptic system
(35)
with conjugation conditions (22) and (23), where
are given functions. Since
the bilinear form (25) is coercive on
.
Then using Lax-Milgram lemma, there exists a unique solution y for system (35) such that:
where
is a continuous linear form defined on
. Let us multiply both sides of first equation of (35) by
and integrate over
, we get
Applying Green’s formula,
then from
we obtain the Neumann conditions
Then, we have the corresponding control problem:
The space
is the space of controls, the state
of the system is given by the solution of
(36)
under conjugation conditions (29), where
is a given control in
. For a given
, the cost functional is again given by (30). As in theorem (4.2), we can prove:
Theorem 5.1 Let us suppose that (26) holds and the cost functional is given by (30), there exists a unique optimal control u, such that:
moreover it is characterized by the following equations and inequalities
together with(36) and (29), where
is the adjoint state.
6. Conclusion
In the present work, we concentrated on optimal control problems for cooperative elliptic systems under conjugation conditions. We proved the existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic system. Then we discussed the existence and uniqueness of the optimal control of boundary type for this system and we gave the set of equations and inequalities that characterizes this control. Also, we studied the problem with Neumann condition. At last, we generalized the discussion to n × n cooperative elliptic systems under conjugation conditions.
Acknowledgements
The authors would like to express their gratitude to Professor Dr. I. M. Gali, Mathematics Department, Faculty of Science, Al-Azhar University, for suggesting the problem and critically reading the manuscript.