- Research Article
- Open Access

# Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains

- Salah Badraoui
^{1}Email 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 .

## Keywords

- Hilbert Space
- Real Number
- Positive Constant
- Linear Operator
- Bounded Domain

## 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]).

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 .

Definition 2.2.

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

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

- (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.

Proof.

Therefore, and

- (i)

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.

To prove this result, we need the following lemma.

Lemma 4.3.

Proof of Lemma 4.3.

It is easy to verify that for every , for all .

Hence, we get (4.23).

Hence, from (4.26) we get for all and , which gives (4.24).

Now, we are ready to prove Theorem 4.2.

Proof of Theorem 4.2.

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.

*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.

Now, we are ready to formulate the third main result of this work.

Theorem 4.5.

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.

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

where is the -semigroup generated by .

where

Hence for all which implies that This completes the proof of Theorem 4.5.

## Authors’ Affiliations

## References

- Badraoui S:
**Asymptotic behavior of solutions to a****reaction-diffusion system with a cross diffusion matrix on unbounded domains.***Electronic Journal of Differential Equations*2006,**2006**(61):1-13.MathSciNetMATHGoogle Scholar - Larez H, Leiva H:Interior controllability of a a reaction-diffusion system with cross diffusion matrix. to appear in Boundary Value ProblemsGoogle Scholar
- Zeidler E:
*Applied Functional Analysis, Applied Mathematical Sciences*.*Volume 109*. Springer, New York, NY, USA; 1995:xvi+404.MATHGoogle Scholar - Leiva H:
**A lemma on****-semigroups and applications.***Quaestiones Mathematicae*2003,**26**(3):247-265. 10.2989/16073600309486057MathSciNetView ArticleMATHGoogle Scholar - Axler S, Bourdon P, Ramey W:
*Harmonic Function Theory, Graduate Texts in Mathematics*.*Volume 137*. Springer, New York, NY, USA; 1992:xii+231.View ArticleMATHGoogle Scholar - Curtain RF, Zwart H:
*An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics*.*Volume 21*. Springer, New York, NY, USA; 1995:xviii+698.View ArticleMATHGoogle Scholar

## Copyright

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