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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)1/2 u+bΔv+ 1 ωf1 (t,x) in (0,τ)×Ω, vt =cΔu-dΔv- β(-Δ)1/2 v+ 1 ωf2 (t,x) in (0,τ)×Ω, u=v=0, on (0,T)×δΩ, u(0,x)= u0 (x), v(0,x)= v0 (x)x ∈ Ω, x, where Ω is a bounded domain in RN (N≥1), u0, v0 L2 (Ω), the 2×2 diffusion matrix D=[c da b] has semisimple and positive eigenvalues 0< p1 ≤ p2, β is an arbitrary constant, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set , and the distributed controls f1, f2 ∈ L2 ([ 0, τ]; L2 (Ω)). Specifically, we prove the following statement: if λ11/2 p1 + β >0 (where λ1 is the first eigenvalue of -Δ), then for all τ >0 and all open nonempty subset of the system is approximately controllable on [ 0, τ]. Copyright © 2009 H. Larez and H. Leiva.