Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains
© Salah Badraoui. 2010
Received: 11 July 2009
Accepted: 5 January 2010
Published: 19 January 2010
We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives in , in , on , , in where is a smooth bounded domain, , the diffusion matrix has semisimple and positive eigenvalues , , is an open nonempty set, and is the characteristic function of . Specifically, we prove that under some conditions over the coefficients , the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all the system is approximately controllable on .
In this paper we prove controllability for the following reaction-diffusion system with cross diffusion matrix:
We assume the following assumptions.
Specifically, we prove the following statements.
(i) If and , where is the first eigenvalue of with Dirichlet condition, or if , and then, under the hypotheses (H1)–(H3), the semigroup generated by the linear operator of the system is exponentially stable.
supplemeted with the initial conditions
The author proved that in the Banach space where is the space of bounded uniformly continuous real valued functions on , if and are locally Lipshitz and under some conditions over the coefficients , and if then for all Moreover, and satisfy the system of ordinary differential equations
with the initial data
In the work done in , the authers studied the system (1.1) with , and They proved that if the diffusion matrix has semi-simple and positive eigenvalues , then if ( is the first eigenvalue of ), the system is approximately controllable on for all open nonempty subset of
2. Notations and Preliminaries
In the following we denote by
We will use the following results.
Theorem 2.1 (cf. ).
Proposition 2.3 (cf. ).
Theorem 2.4 (cf. ).
3. Abstract Formulation of the Problem
If we put
then (3.3) can be written as
4. Main Results
To prove this result, we need the following lemma.
Proof of Lemma 4.3.
Hence, we get (4.23).
Now, we are ready to prove Theorem 4.2.
Proof of Theorem 4.2.
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 and the initial value problem (3.9) admits a unique mild solution given by
The following criteria for approximate controllability can be found in .
Now, we are ready to formulate the third main result of this work.
The proof of this theorem relies on the Criteria 1 and the following lemma.
Proof of Lemma 4.6.
By analyticity we get and from this we get . Under the assumptions of the lemma we get as and so If , we divide by and we pass we get . If we divide by and we pass and get . If , we divide by and we pass and get But in this we case we can integrate under the symbol of sommation over the intervall and we get . Hence . Continuing this way we see that for all
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