Boundary Value Problems
Volume 2010 (2010), Article ID 281238, 14 pages
doi:10.1155/2010/281238

Approximate controllability of a reaction-diffusion system with a cross-diffusion matrix and fractional derivatives on bounded domains

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 x∈Ω, where Ω⊂RN (N≥1) is a smooth bounded domain, u0,v0∈L2(Ω), the diffusion matrix M=(a1a2b1b2) has semisimple and positive eigenvalues 0<ρ1≤ρ2, 0<α1,α2,β1,β2<1, ω⊂Ω 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*].