© H. Larez and H. Leiva. 2009
Received: 30 December 2008
Accepted: 27 May 2009
Published: 11 June 2009
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 .
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 .
Our technique is simple and elegant from mathematical point of view, it rests on the shoulders of the following fundamental results.
Theorem 1.2 (see [2, Theorem 1.23, page 20]).
Lemma 1.3 (see [3, Lemma 3.14, page 62]).
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.
2. Abstract Formulation of the Problem
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]).
Now, we will use the following Lemma from  to prove the following theorem.
Then the following hold.
and using (2.14) we obtain (2.16).
3. Proof of the Main Theorem
Definition 3.1 (approximate controllability).
Now, one is ready to formulate and prove the main theorem of this work.
Theorem 3.3 (main theorem).
This work was supported by the CDHT-ULA-project: 1546-08-05-B.
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