Interior Controllability of a Reaction-Diffusion System with Cross-Diffusion Matrix

We prove the interior approximate controllability for the following reaction-diffusion system with cross-diffusion matrix in , in , , on , , , , where is a bounded domain in , , the diffusion matrix has semisimple and positive eigenvalues , is an arbitrary constant, is an open nonempty subset of , denotes the characteristic function of the set , and the distributed controls . Specifically, we prove the following statement: if (where is the first eigenvalue of ), then for all and all open nonempty subset of the system is approximately controllable on .

In this paper we prove the interior approximate controllability for the following reaction-diffusion system with cross-diffusion matrix

where is a bounded domain in (), , the diffusion matrix

has semisimple and positive eigenvalues, is an arbitrary constant, is an open nonempty subset of , denotes the characteristic function of the set , and the distributed controls . Specifically, we prove the following statement: if (the first eigenvalue of ), then for all and all open nonempty subset of , the system is approximately controllable on .

When this system takes the following particular form:

This paper has been motivated by the work done Badraoui in [1], where author studies the asymptotic behavior of the solutions for the system (1.3) on the unbounded domain . That is to say, he studies the system:

supplemented with the initial conditions:

where the diffusion coefficients and are assumed positive constants, while the diffusion coefficients , and the coefficient are arbitrary constants. He assume also the following three conditions:

(H1), and ,

(H2) where is the space of bounded and uniformly continuous real-valued functions,

(H3) and , for all and . Moreover, and are locally Lipshitz; namely, for all and all constant , there exist a constant such that

is verified for all , with , and .

We note that the hypothesis (H1) implies that the eigenvalues of the matrix are simple and positive. But, this condition is not necessary for the eigenvalues of to be positive, in fact we can find matrices with and been negative and having positive eigenvalues. For example, one can consider the following matrix:

whose eigenvalues are and .

The system (1.1) can be written in the following matrix form:

where , the distributed controls , and is the identity matrix of dimension .

Our technique is simple and elegant from mathematical point of view, it rests on the shoulders of the following fundamental results.

Theorem 1.1.

The eigenfunctions of with Dirichlet boundary condition are real analytic functions.

Theorem 1.2 (see [2, Theorem  1.23, page 20]).

Suppose is open, nonempty, and connected set, and is real analytic function in with on a nonempty open subset of . Then, in .

Lemma 1.3 (see [3, Lemma  3.14, page 62]).

Let and be two sequences of real numbers such that . Then

if and only if

Finally, with this technique those young mathematicians who live in remote and inhospitable places, far from major research centers in the world, can also understand and enjoy the interior controllability with a minor effort.

In this section we choose a Hilbert space where system (1.8) can be written as an abstract differential equation; to this end, we consider the following notations:

Let us consider the Hilbert space and the eigenvalues of , each one with finite multiplicity equal to the dimension of the corresponding eigenspace. Then, we have the following well-known properties (see [3, pages 45-46]).

(i)There exists a complete orthonormal set of eigenvectors of .

(ii)For all , we have

where is the inner product in and

So, is a family of complete orthogonal projections in and ,

(iii) generates an analytic semigroup given by

Now, we denote by the Hilbert space and define the following operator:

with . Therefore, for all , we obtain

where

is a family of complete orthogonal projections in .

Consequently, system (1.8) can be written as an abstract differential equation in :

where , , and , is a bounded linear operator.

Now, we will use the following Lemma from [4] to prove the following theorem.

Lemma 2.1.

Let be a Hilbert separable space and , two families of bounded linear operator in , with a family of complete orthogonal projection such that

Define the following family of linear operators:

Then the following hold.

(a) is a linear and bounded operator if , , with , continuous for .

(b)Under the above (a), is a strongly continuous semigroup in the Hilbert space , whose infinitesimal generator is given by

with

(c)The spectrum of is given by

where .

Theorem 2.2.

The operator define by (2.5) is the infinitesimal generator of a strongly continuous semigroup given by:

where and with

Moreover, if , then there exists such that

Proof.

In order to apply the foregoing Lemma, we observe that can be written as follows:

with

Therefore, with

Clearly that is a bounded linear operator (linear and continuous). That is, there exists such that

In fact, .

Now, we have to verify condition (a) of Lemma 2.1. To this end, without loss of generality, we will suppose that . Then, there exists a set of complementary projections on such that

Hence,

This implies the existence of positive numbers such that

Therefore, generates a strongly continuous semigroup given by (2.14).

Finally, if , then

and using (2.14) we obtain (2.16).

In this section we will prove the main result of this paper on the controllability of the linear system (2.8). But, before we will give the definition of approximate controllability for this system. To this end, for all and , the initial value problem

where the control function belonging to admits only one mild solution given by

Definition 3.1 (approximate controllability).

The system (2.8) is said to be approximately controllable on if for every , there exists such that the solution of (3.2) corresponding to verifies:

The following result can be found in [5] for the general evolution equation:

where , are Hilbert spaces, is the infinitesimal generator of strongly continuous semigroup in , the control function belongs to .

Theorem 3.2.

System (3.4) is approximately controllable on if and only if

Now, one is ready to formulate and prove the main theorem of this work.

Theorem 3.3 (main theorem).

If , then for all and all open nonempty subset of the system, (2.8) is approximately controllable on .

Proof.

We will apply Theorem 3.2 to prove the approximate controllability of system (2.8). With this purpose, we observe that

On the other hand,

Without lose of generality, we will suppose that . Then, there exists a set of complementary projections on such that

Hence,

Therefore,

where .

Now, suppose for that , for all . Then,

Clearly that, is a decreasing sequence. Then, from Lemma 1.3, we obtain for all x that

Since , we get that

On the other hand, from Theorem 1.1 we know that are analytic functions, which implies the analyticity of , . Then, from Theorem 1.2 we get that

Hence , , which implies that . This completes the proof of the main theorem.

This work was supported by the CDHT-ULA-project: 1546-08-05-B.

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