`Boundary Value ProblemsVolume 2010 (2010), Article ID 281238, 14 pagesdoi:10.1155/2010/281238`
Research Article

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

Laboratoire LAIG, Université du 08 Mai 1945, BP. 401, Guelma 24000, Algeria

Received 11 July 2009; Accepted 5 January 2010

Academic Editor: Ugur Abdulla

Copyright © 2010 Salah Badraoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives ut=a1Δu+a2Δv-c1(-Δ)α1u-c2(-Δ)α2v+1ωf1(x,t) in Ω×]0,t*[, vt=b1Δu+b2Δv-d1(-Δ)β1u-d2(-Δ)β2v+1ωf2(x,t) in Ω×]0,t*[, u=v=0 on Ω×]0,t*[, u(x,0)=u0(x), v(x,0)=v0(x) in xΩ, where ΩRN (N1) is a smooth bounded domain, u0,v0L2(Ω), the diffusion matrix M=(a1a2b1b2) has semisimple and positive eigenvalues 0<ρ1ρ2, 0<α1,α2,β1,β2<1, ωΩ is an open nonempty set, and 1ω is the characteristic function of ω. Specifically, we prove that under some conditions over the coefficients ai,bi,ci,di(i=1,2), the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all t*>0 the system is approximately controllable on [0,t*].