Research Article
Open Access
Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains
- Salah Badraoui1Email author
Received: 11 July 2009
Accepted: 5 January 2010
Published: 19 January 2010
Abstract
We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives
in
,
in
,
on
,
,
in
where
is a smooth bounded domain,
, the diffusion matrix
has semisimple and positive eigenvalues
,
,
is an open nonempty set, and
is the characteristic function of
. Specifically, we prove that under some conditions over the coefficients
, the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all
the system is approximately controllable on
.
1. Introduction
In this paper we prove controllability for the following reaction-diffusion system with cross diffusion matrix:
(1.1)
where
is an open nonempty set of
and
is the characteristic function of
.
We assume the following assumptions.
(H1)
is a smooth bounded domain in
.
(H2) The diffusion matrix
has semisimple and positive eigenvalues
(H3)
are real constants,
are real constants belonging to the interval
(H4)
(H5) The distributed controls
.
Specifically, we prove the following statements.
(i) If
and
, where
is the first eigenvalue of
with Dirichlet condition, or if
, and
then, under the hypotheses (H1)–(H3), the semigroup generated by the linear operator of the system is exponentially stable.
(ii) If
and under the hypotheses (H1)–(H5), then, for all
and all open nonempty subset
of
the system is approximately controllable on
This paper has been motivated by the work done in [1] and the work done by H. Larez and H. Leiva in [2]. In the work [1], the auther studies the asymptotic behavior of the solution of the system
(1.2)
supplemeted with the initial conditions
(1.3)
The author proved that in the Banach space
where
is the space of bounded uniformly continuous real valued functions on
, if
and
are locally Lipshitz and under some conditions over the coefficients
, and if
then
for all
Moreover,
and
satisfy the system of ordinary differential equations
(1.4)
with the initial data
(1.5)
The same result holds for
In the work done in [2], the authers studied the system (1.1) with
, and
They proved that if the diffusion matrix
has semi-simple and positive eigenvalues
,
then if
(
is the first eigenvalue of
), the system is approximately controllable on
for all open nonempty subset
of
2. Notations and Preliminaries
In the following we denote by
the set of
matrices with entries from
,
the set of all measurable functions
such that
,
the set of all the functions
that have generalized derivatives
for all
,
the closure of the set
in the Hilbert space
,
the set of all the functions
that have generalized derivatives
for all
.
We will use the following results.
Theorem 2.1 (cf. [3]).
Let us consider the following classical boundary-eigenvalue problem for the laplacien:
(2.1)
where
is a nonempty bounded open set in
and
.
This problem has a countable system of eigenvalues
and
as
.
(i) All the eigenvalues
have finite multiplicity
equal to the dimension of the corresponding eigenspace
.
(ii) Let
be a basis of the
for every
then the eigenvectors
form a complete orthonormal system in the space
Hence for all
we have
If we put
then we get
.
(iii) Also, the eigenfunctions
, where
is the space of infinitely continuously differentiable functions on
and compactly supported in
.
(iv) For all
we have
.
(v) The operator
generates an analytic semigroup
on
defined by
generates an analytic semigroup
on
defined by
(2.2)
Definition 2.2.
Let
a real number, the operator
is defined by
a real number, the operator
is defined by
(2.3)
In particular, we obtain
and
. Since
form a complete orthonormal system in the space
then it is dense in
, and hence
is dense in
.
Proposition 2.3 (cf. [4]).
Let
be a Hilbert separable space and
and
two families of bounded linear operators in
, with
a family of complete orthogonal projections such that
Define the following family of linear operators
Then
(a)
is a linear and bounded operator if
with
continiuous for
(b) under the above condition (a),
is a strongly continiuous semigroup in the Hilbert space
whose infinitesimal generator
is given by
is a strongly continiuous semigroup in the Hilbert space
whose infinitesimal generator
is given by
(2.4)
Theorem 2.4 (cf. [5]).
Suppose
is connected,
is a real function in
, and
on a nonempty open subset of
. Then
in
.
3. Abstract Formulation of the Problem
In this section we consider the following notations.
- (i)
is a Hilbert space with the inner product
(3.1)
We define
(3.2)
- (iii)
Let
then we can define the linear operator
(3.3)
where
(3.4)
Therefore, for all
(3.5)
If we put
(3.6)
then (3.3) can be written as
(3.7)
and we have for all
(3.8)
Consequently, system (1.1) can be written as an abstract differential equation in the Hilbert space
in the following form:
(3.9)
where
and
is a bounded linear operator from
into
.
4. Main Results
4.1. Generation of a
-Semigroup
Theorem 4.1.
If
, then, under hypotheses (H1)–(H3), the linear operator
defined by (3.3) is the infinitesimal generator of strongly continuous semigroup
given by
, then, under hypotheses (H1)–(H3), the linear operator
defined by (3.3) is the infinitesimal generator of strongly continuous semigroup
given by
(4.1)
where
(4.2)
(4.3)
Moreover, if
(4.4)
then the
-semigoup
is exponentially stable, that is, there exist two positives constants
such that
-semigoup
is exponentially stable, that is, there exist two positives constants
such that
(4.5)
Proof.
In order to apply the Proposition 2.3, we observe that
can be written as follows:
can be written as follows:
(4.6)
where
(4.7)
Therefore,
and
Now, we have to verify condition (a) of the Proposition 2.3. We shall suppose that
Then, there exists a set
of complementary projections on
such that
Then, there exists a set
of complementary projections on
such that
(4.8)
If
is the matrix passage from the canonical basis of
to the basis composed with the eigenvectors of
, then
is the matrix passage from the canonical basis of
to the basis composed with the eigenvectors of
, then
(4.9)
Hence,
(4.10)
We have also
(4.11)
From (4.10)-(4.11) into (4.7) we obtain
(4.12)
where
(4.13)
As
we get
we get
(4.14)
As
as
then this implies the existence of a positive number
and a real number
such that
for every
Therefore
is a strongly continious semigroup
given by (4.1). We can even estimate the constants
and
as follows.
as
then this implies the existence of a positive number
and a real number
such that
for every
Therefore
is a strongly continious semigroup
given by (4.1). We can even estimate the constants
and
as follows. - (i)If
As
, then there exist constants 
(4.15)
hence, if we put
(4.16)
(4.17)
we easily obtain
(4.18)
If
. If we put
(4.19)
then we find that
(4.20)
Therefore, the linear operator
generates a strongly continuous semigroup
on
given by expression (4.1).
Finally, if
we have already proved (4.20). Using (4.20) into (4.1) we get that the
-semigoup
is exponentially stable. The expression (4.5) is verfied with
and
is defined by (4.19).
Theorem 4.2.
If
(4.21)
then, under the hypotheses (H1)–(H3), the linear operator
defined by (3.3) is the infinitesimal generator of strongly continuous semigroup exponentially stable
defined by (4.1). Specially, there exist two positives constants
such that
defined by (3.3) is the infinitesimal generator of strongly continuous semigroup exponentially stable
defined by (4.1). Specially, there exist two positives constants
such that
(4.22)
To prove this result, we need the following lemma.
Lemma 4.3.
For every two real positives constants
and
, one has for every
and
, one has for every
(4.23)
and for every
(4.24)
Proof of Lemma 4.3.
It is easy to verify that for every
, for all
.
Let
and
, then we get
and
, then we get
(4.25)
Hence, we get (4.23).
Also, it is easy to verify that for every
, for all
. Let
and
, then we get
, for all
. Let
and
, then we get
(4.26)
Hence, from
(4.26) we get
for all
and
, which gives (4.24).
With the same manner we can prove that for every
and every
we have
and every
we have
(4.27)
and consequently, for every two real positives constants
and
and every
we have
and
and every
we have
(4.28)
Now, we are ready to prove Theorem 4.2.
Proof of Theorem 4.2.
By applying Proposition 2.3 we start from formula (4.12) and we put
(4.29)
where
(4.30)
To estimate
we have in taking into account
we have in taking into account
(4.31)
and applying the Lemma 4.3
we get
we get
(4.32)
for all
and
. But we have
, for all
Then we get for every
that
and
. But we have
, for all
Then we get for every
that
(4.33)
From (4.31)-(4.33) we get
(4.34)
where
(4.35)
and
Applying Lemma 4.3 and taking into account (4.21) we get with the same manner that for every
(4.36)
where
(4.37)
and or every
(4.38)
where
(4.39)
(4.40)
From (4.34)-(4.40) into (4.12) we get
(4.41)
where
is defined by (4.17) and
is defined by (4.17) and
(4.42)
Using (4.41) into (4.1) we get that the
-semigoup
generated by
is exponentially stable. Expression (4.22) is verfied with
and
is defined by (4.42).
4.2. Approximate Controllability
Befor giving the definition of the approximate controllabiliy for the sytem (3.9), we have the following known result: for all
and
the initial value problem (3.9) admits a unique mild solution given by
(4.43)
This solution is denoted by
Definition 4.4.
System (3.9) is said to be approximately controllable at time
whenever the set
is densely embedded in
; that is,
whenever the set
is densely embedded in
; that is,
(4.44)
The following criteria for approximate controllability can be found in [6].
Criteria 1.
System (3.9) is approximately controllable on
if and only if
if and only if
(4.45)
Now, we are ready to formulate the third main result of this work.
Theorem 4.5.
If the following condition
(4.46)
is satisfied; then, under hypotheses (H1)–(H5), for all
and all open subset
system (3.9) is approximately controllable on
.
Proof.
The proof of this theorem relies on the Criteria 1 and the following lemma.
Lemma 4.6.
Let
and
be sequences of real numbers such that
,
and
, for all
, then for any
one has
and
be sequences of real numbers such that
,
and
, for all
, then for any
one has
(4.47)
Proof of Lemma 4.6.
By analyticity we get
and from this we get
. Under the assumptions of the lemma we get
as
and so
If
, we divide
by
and we pass
we get
. If
we divide
by
and we pass
and get
. If
, we divide
by
and we pass
and get
But in this we case we can integrate under the symbol of sommation over the intervall
and we get
. Hence
. Continuing this way we see that
for all
We are now ready to prove Theorem 4.5. For this purpose, we observe that
(4.48)
where
is the
-semigroup generated by
.
Without lose of generality, we suppose that
Hence
Hence
(4.49)
where
Now, suppose for
that
, for all
Then
that
, for all
Then
(4.50)
If (4.46) is satisfied, then (4.50) take the form
(4.51)
Then, from lemma 4.6 we obtain that for
and all
and all
(4.52)
Since
we get that all
we get that all
(4.53)
On the other hand, from Theorem 2.4 we know that
are analytic functions, which implies the analticity of
and
Then we can conclude that for
and all
are analytic functions, which implies the analticity of
and
Then we can conclude that for
and all
(4.54)
Hence
for all
which implies that
This completes the proof of Theorem 4.5.
Authors’ Affiliations
(1)
Laboratoire LAIG, Université du 08 Mai 1945
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Copyright
© Salah Badraoui. 2010
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
