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 u<inf>t</inf> =aΔu- β(-Δ)^1/2 u+bΔv+ 1 <inf>ω</inf>f<inf>1</inf> (t,x) in (0,τ)×Ω, v<inf>t</inf> =cΔu-dΔv- β(-Δ)^1/2 v+ 1 <inf>ω</inf>f<inf>2</inf> (t,x) in (0,τ)×Ω, u=v=0, on (0,T)×δΩ, u(0,x)= u<inf>0</inf> (x), v(0,x)= v<inf>0</inf> (x)x ∈ Ω, x, where Ω is a bounded domain in R^N (N≥1), u<inf>0</inf>, v<inf>0</inf> L² (Ω), the 2×2 diffusion matrix D=[<inf>c d</inf>^{a 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 f<inf>1</inf>, f<inf>2</inf> ∈ L² ([ 0, τ]; L² (Ω)). Specifically, we prove the following statement: if λ<inf>1</inf>^1/2 p1 + β >0 (where λ<inf>1</inf> 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.

Año de publicación:

2009

Keywords:

Fuente:

scopus

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