Boundary Value Problems

, 2010:281238 | Cite as

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

Open Access
Research Article

Abstract

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives Open image in new window in Open image in new window , Open image in new window in Open image in new window , Open image in new window on Open image in new window , Open image in new window , Open image in new window in Open image in new window where Open image in new window is a smooth bounded domain, Open image in new window , the diffusion matrix Open image in new window has semisimple and positive eigenvalues Open image in new window , Open image in new window , Open image in new window is an open nonempty set, and Open image in new window is the characteristic function of Open image in new window . Specifically, we prove that under some conditions over the coefficients Open image in new window , the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all Open image in new window the system is approximately controllable on Open image in new window .

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 Open image in new window is an open nonempty set of Open image in new window and Open image in new window is the characteristic function of Open image in new window .

We assume the following assumptions.

(H1) Open image in new window is a smooth bounded domain in Open image in new window .

(H2) The diffusion matrix Open image in new window has semisimple and positive eigenvalues Open image in new window

(H3) Open image in new window are real constants, Open image in new window are real constants belonging to the interval Open image in new window

(H4) Open image in new window

(H5) The distributed controls Open image in new window .

Specifically, we prove the following statements.

(i) If Open image in new window and Open image in new window , where Open image in new window is the first eigenvalue of Open image in new window with Dirichlet condition, or if Open image in new window , and Open image in new window then, under the hypotheses (H1)–(H3), the semigroup generated by the linear operator of the system is exponentially stable.

(ii) If Open image in new window and under the hypotheses (H1)–(H5), then, for all Open image in new window and all open nonempty subset Open image in new window of Open image in new window the system is approximately controllable on Open image in new window

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 Open image in new window where Open image in new window is the space of bounded uniformly continuous real valued functions on Open image in new window , if Open image in new window and Open image in new window are locally Lipshitz and under some conditions over the coefficients Open image in new window , and if Open image in new window then Open image in new window for all Open image in new window Moreover, Open image in new window and Open image in new window satisfy the system of ordinary differential equations

with the initial data

The same result holds for Open image in new window

In the work done in [2], the authers studied the system (1.1) with Open image in new window , and Open image in new window They proved that if the diffusion matrix Open image in new window has semi-simple and positive eigenvalues Open image in new window , Open image in new window then if Open image in new window ( Open image in new window is the first eigenvalue of Open image in new window ), the system is approximately controllable on Open image in new window for all open nonempty subset Open image in new window of Open image in new window

2. Notations and Preliminaries

In the following we denote by

Open image in new window the set of Open image in new window matrices with entries from Open image in new window ,

Open image in new window the set of all measurable functions Open image in new window such that Open image in new window ,

Open image in new window the set of all the functions Open image in new window that have generalized derivatives Open image in new window for all Open image in new window ,

Open image in new window the closure of the set Open image in new window in the Hilbert space Open image in new window ,

Open image in new window the set of all the functions Open image in new window that have generalized derivatives Open image in new window for all Open image in new window .

We will use the following results.

Theorem 2.1 (cf. [3]).

Let us consider the following classical boundary-eigenvalue problem for the laplacien:

where Open image in new window is a nonempty bounded open set in Open image in new window and Open image in new window .

This problem has a countable system of eigenvalues Open image in new window and Open image in new window as Open image in new window .

(i) All the eigenvalues Open image in new window have finite multiplicity Open image in new window equal to the dimension of the corresponding eigenspace Open image in new window .

(ii) Let Open image in new window be a basis of the Open image in new window for every Open image in new window then the eigenvectors Open image in new window form a complete orthonormal system in the space Open image in new window Hence for all Open image in new window we have Open image in new window If we put Open image in new window then we get Open image in new window .

(iii) Also, the eigenfunctions Open image in new window , where Open image in new window is the space of infinitely continuously differentiable functions on Open image in new window and compactly supported in Open image in new window .

(iv) For all Open image in new window we have Open image in new window .

(v) The operator Open image in new window generates an analytic semigroup Open image in new window on Open image in new window defined by

Definition 2.2.

Let Open image in new window a real number, the operator Open image in new window is defined by

In particular, we obtain Open image in new window and Open image in new window . Since Open image in new window form a complete orthonormal system in the space Open image in new window then it is dense in Open image in new window , and hence Open image in new window is dense in Open image in new window .

Proposition 2.3 (cf. [4]).

Let Open image in new window be a Hilbert separable space and Open image in new window and Open image in new window two families of bounded linear operators in Open image in new window , with Open image in new window a family of complete orthogonal projections such that Open image in new window

Define the following family of linear operators Open image in new window Then

(a) Open image in new window is a linear and bounded operator if Open image in new window with Open image in new window continiuous for Open image in new window

(b) under the above condition (a), Open image in new window is a strongly continiuous semigroup in the Hilbert space Open image in new window whose infinitesimal generator Open image in new window is given by

Theorem 2.4 (cf. [5]).

Suppose Open image in new window is connected, Open image in new window is a real function in Open image in new window , and Open image in new window on a nonempty open subset of Open image in new window . Then Open image in new window in Open image in new window .

3. Abstract Formulation of the Problem

In this section we consider the following notations.
  1. (i)

    Open image in new window is a Hilbert space with the inner product

     

Open image in new window We define

  1. (iii)

    Let Open image in new window then we can define the linear operator

     

where

Therefore, for all Open image in new window

If we put

then (3.3) can be written as

and we have for all Open image in new window

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

where Open image in new window and Open image in new window is a bounded linear operator from Open image in new window into Open image in new window .

4. Main Results

4.1. Generation of a Open image in new window-Semigroup

Theorem 4.1.

If Open image in new window , then, under hypotheses (H1)–(H3), the linear operator Open image in new window defined by (3.3) is the infinitesimal generator of strongly continuous semigroup Open image in new window given by
Moreover, if
then the Open image in new window -semigoup Open image in new window is exponentially stable, that is, there exist two positives constants Open image in new window such that

Proof.

In order to apply the Proposition 2.3, we observe that Open image in new window can be written as follows:

Therefore, Open image in new window and Open image in new window

Now, we have to verify condition (a) of the Proposition 2.3. We shall suppose that Open image in new window Then, there exists a set Open image in new window of complementary projections on Open image in new window such that
If Open image in new window is the matrix passage from the canonical basis of Open image in new window to the basis composed with the eigenvectors of Open image in new window , then
We have also
From (4.10)-(4.11) into (4.7) we obtain
As Open image in new window as Open image in new window then this implies the existence of a positive number Open image in new window and a real number Open image in new window such that Open image in new window    for every Open image in new window Therefore Open image in new window is a strongly continious semigroup Open image in new window given by (4.1). We can even estimate the constants Open image in new window and Open image in new window as follows.
  1. (i)
     
hence, if we put
we easily obtain
then we find that

Therefore, the linear operator Open image in new window generates a strongly continuous semigroup Open image in new window on Open image in new window given by expression (4.1).

Finally, if Open image in new window we have already proved (4.20). Using (4.20) into (4.1) we get that the Open image in new window -semigoup Open image in new window is exponentially stable. The expression (4.5) is verfied with Open image in new window and Open image in new window is defined by (4.19).

Theorem 4.2.

then, under the hypotheses (H1)–(H3), the linear operator Open image in new window defined by (3.3) is the infinitesimal generator of strongly continuous semigroup exponentially stable Open image in new window defined by (4.1). Specially, there exist two positives constants Open image in new window such that

To prove this result, we need the following lemma.

Lemma 4.3.

For every two real positives constants Open image in new window and Open image in new window , one has for every Open image in new window

Proof of Lemma 4.3.

It is easy to verify that for every Open image in new window , for all Open image in new window .

Hence, we get (4.23).

Also, it is easy to verify that for every Open image in new window , for all Open image in new window . Let Open image in new window and Open image in new window , then we get

Hence, from Open image in new window (4.26) we get Open image in new window for all Open image in new window and Open image in new window , which gives (4.24).

With the same manner we can prove that for every Open image in new window and every Open image in new window we have
and consequently, for every two real positives constants Open image in new window and Open image in new window and every Open image in new window 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
To estimate Open image in new window we have in taking into account Open image in new window
and applying the Lemma 4.3 Open image in new window we get
From (4.31)-(4.33) we get

and Open image in new window

Applying Lemma 4.3 and taking into account (4.21) we get with the same manner that for every Open image in new window
From (4.34)-(4.40) into (4.12) we get
where Open image in new window is defined by (4.17) and

Using (4.41) into (4.1) we get that the Open image in new window -semigoup Open image in new window generated by Open image in new window is exponentially stable. Expression (4.22) is verfied with Open image in new window and Open image in new window 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 Open image in new window and Open image in new window the initial value problem (3.9) admits a unique mild solution given by

This solution is denoted by Open image in new window

Definition 4.4.

System (3.9) is said to be approximately controllable at time Open image in new window whenever the set Open image in new window is densely embedded in Open image in new window ; that is,

The following criteria for approximate controllability can be found in [6].

Criteria 1.

System (3.9) is approximately controllable on Open image in new window 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 Open image in new window and all open subset Open image in new window system (3.9) is approximately controllable on Open image in new window .

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 Open image in new window and from this we get Open image in new window . Under the assumptions of the lemma we get Open image in new window as Open image in new window and so Open image in new window If Open image in new window , we divide Open image in new window by Open image in new window and we pass Open image in new window we get Open image in new window . If Open image in new window we divide Open image in new window by Open image in new window and we pass Open image in new window and get Open image in new window . If Open image in new window , we divide Open image in new window by Open image in new window and we pass Open image in new window and get Open image in new window But in this we case we can integrate under the symbol of sommation over the intervall Open image in new window and we get Open image in new window . Hence Open image in new window . Continuing this way we see that Open image in new window for all Open image in new window

We are now ready to prove Theorem 4.5. For this purpose, we observe that

where Open image in new window is the Open image in new window -semigroup generated by Open image in new window .

Without lose of generality, we suppose that Open image in new window Hence

where Open image in new window

If (4.46) is satisfied, then (4.50) take the form
Then, from lemma 4.6 we obtain that for Open image in new window and all Open image in new window
On the other hand, from Theorem 2.4 we know that Open image in new window are analytic functions, which implies the analticity of Open image in new window and Open image in new window Then we can conclude that for Open image in new window and all Open image in new window

Hence Open image in new window for all Open image in new window which implies that Open image in new window This completes the proof of Theorem 4.5.

References

  1. 1.
    Badraoui S:Asymptotic behavior of solutions to a Open image in new window reaction-diffusion system with a cross diffusion matrix on unbounded domains. Electronic Journal of Differential Equations 2006,2006(61):1-13.MathSciNetMATHGoogle Scholar
  2. 2.
    Larez H, Leiva H:Interior controllability of a Open image in new window a reaction-diffusion system with cross diffusion matrix. to appear in Boundary Value ProblemsGoogle Scholar
  3. 3.
    Zeidler E: Applied Functional Analysis, Applied Mathematical Sciences. Volume 109. Springer, New York, NY, USA; 1995:xvi+404.MATHGoogle Scholar
  4. 4.
    Leiva H:A lemma on Open image in new window-semigroups and applications. Quaestiones Mathematicae 2003,26(3):247-265. 10.2989/16073600309486057MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Axler S, Bourdon P, Ramey W: Harmonic Function Theory, Graduate Texts in Mathematics. Volume 137. Springer, New York, NY, USA; 1992:xii+231.CrossRefMATHGoogle Scholar
  6. 6.
    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.CrossRefMATHGoogle Scholar

Copyright information

© 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.

Authors and Affiliations

  1. 1.Laboratoire LAIGUniversité du 08 Mai 1945GuelmaAlgeria

Personalised recommendations