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Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains
Boundary Value Problems volume 2010, Article number: 281238 (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:

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

supplemeted with the initial conditions

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

with the initial data

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:

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

Definition 2.2.
Let a real number, the operator
is defined by

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

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

We define

-
(iii)
Let
then we can define the linear operator

where

Therefore, for all

If we put

then (3.3) can be written as

and we have for all

Consequently, system (1.1) can be written as an abstract differential equation in the Hilbert space in the following form:

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

where


Moreover, if

then the -semigoup
is exponentially stable, that is, there exist two positives constants
such that

Proof.
In order to apply the Proposition 2.3, we observe that can be written as follows:

where

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

If is the matrix passage from the canonical basis of
to the basis composed with the eigenvectors of
, then

Hence,

We have also

From (4.10)-(4.11) into (4.7) we obtain

where

As we get

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.
-
(i)
If
As
, then there exist constants
(4.15)
hence, if we put


we easily obtain

If
. If we put

then we find that

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

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

To prove this result, we need the following lemma.
Lemma 4.3.
For every two real positives constants and
, one has for every

and for every

Proof of Lemma 4.3.
It is easy to verify that for every , for all
.
Let and
, then we get

Hence, we get (4.23).
Also, it is easy to verify that for every , for all
. Let
and
, then we get

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 consequently, for every two real positives constants and
and every
we have

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

where

To estimate we have in taking into account

and applying the Lemma 4.3 we get

for all and
. But we have
, for all
Then we get for every
that

From (4.31)-(4.33) we get

where

and
Applying Lemma 4.3 and taking into account (4.21) we get with the same manner that for every

where

and or every

where


From (4.34)-(4.40) into (4.12) we get

where is defined by (4.17) and

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

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,

The following criteria for approximate controllability can be found in [6].
Criteria 1.
System (3.9) is approximately controllable on if and only if

Now, we are ready to formulate the third main result of this work.
Theorem 4.5.
If the following condition

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

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

where is the
-semigroup generated by
.
Without lose of generality, we suppose that Hence

where
Now, suppose for that
, for all
Then

If (4.46) is satisfied, then (4.50) take the form

Then, from lemma 4.6 we obtain that for and all

Since we get that all

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

Hence for all
which implies that
This completes the proof of Theorem 4.5.
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Badraoui, S. Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains. Bound Value Probl 2010, 281238 (2010). https://doi.org/10.1155/2010/281238
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Keywords
- Hilbert Space
- Real Number
- Positive Constant
- Linear Operator
- Bounded Domain