Interior Controllability of a Reaction-Diffusion System with Cross-Diffusion Matrix
© 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 .
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,
is verified for all , with , and .
whose eigenvalues are and .
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.
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]).
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]).
(i)There exists a complete orthonormal set of eigenvectors of .
So, is a family of complete orthogonal projections in and ,
is a family of complete orthogonal projections in .
where , , and , is a bounded linear operator.
Now, we will use the following Lemma from  to prove the following theorem.
Then the following hold.
(a) is a linear and bounded operator if , , with , continuous for .
In fact, .
Therefore, generates a strongly continuous semigroup given by (2.14).
and using (2.14) we obtain (2.16).
3. Proof of the Main Theorem
Definition 3.1 (approximate controllability).
where , are Hilbert spaces, is the infinitesimal generator of strongly continuous semigroup in , the control function belongs to .
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 .
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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