Boundary Value Problems
Volume 2009 (2009), Article ID 560407, 9 pages
doi:10.1155/2009/560407

Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix

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].