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

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.