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

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

(2.2)

Definition 2.2.

Let 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

(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

(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

(4.5)

Proof.

In order to apply the Proposition 2.3, we observe that 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

(4.8)

If 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

(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.

• (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

(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

(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

(4.25)

Hence, we get (4.23).

Also, it is easy to verify that for every , 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

(4.27)

and consequently, for every two real positives constants 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

(4.31)

and applying the Lemma 4.3 we get

(4.32)

for all 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

(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,

(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

(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

(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

(4.49)

where

Now, suppose for 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

(4.52)

Since 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

(4.54)

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