Copyright © 2009 Hanzel Larez and Hugo Leiva. 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 prove the interior approximate controllability for the following 2×2 reaction-diffusion system with cross-diffusion matrix ut=aΔu−β(−Δ)1/2u+bΔv+1ωf1(t,x) in (0,τ)×Ω, vt=cΔu−dΔv−β(−Δ)1/2v+1ωf2(t,x) in (0,τ)×Ω, u=v=0, on (0,T)×∂Ω, u(0,x)=u0(x), v(0,x)=v0(x), x∈Ω, where Ω is a bounded domain in ℝN (N≥1), u0,v0∈L2(Ω), the 2×2 diffusion matrix D=[abcd] has semisimple and positive eigenvalues 0<ρ1≤ρ2, β is an arbitrary constant, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, and the distributed controls f1,f2∈L2([0,τ];L2(Ω)). Specifically, we prove the following statement: if λ11/2ρ1+β>0 (where λ1 is the first eigenvalue of −Δ), then for all τ>0 and all open nonempty subset ω of Ω the system is approximately controllable on [0,τ].