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

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## 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

(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]).

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 .

Definition 2.2.

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

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

- (i)
Open image in new window is a Hilbert space with the inner product

Open image in new window We define

- (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.

Proof.

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

- (i)

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.

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 Open image in new window , for all Open image in new window .

Hence, we get (4.23).

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).

Now, we are ready to prove Theorem 4.2.

Proof of Theorem 4.2.

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.

*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.

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 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

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

where 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.

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