Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix. (11th June 2009) - Larez, Hanzel; Leiva, Hugo
Abstract : We prove the interior approximate controllability for the following2 × 2 reaction-diffusion system with cross-diffusion matrixu t = a Δ u − β ( − Δ ) 1 / 2 u + b Δ v + 1 ω f 1 ( t, x ) in( 0, τ ) × Ω, v t = c Δ u − d Δ v − β ( − Δ ) 1 / 2 v + 1 ω f 2 ( t, x ) in( 0, τ ) × Ω, u = v = 0, on( 0, T ) × ∂ Ω, u ( 0, x ) = u 0 ( x ), v ( 0, x ) = v 0 ( x ), x ∈ Ω, whereΩ is a bounded domain inℝ N ( N ≥ 1 ), u 0, v 0 ∈ L 2 ( Ω ), the2 × 2 diffusion matrixD = [ a b c d ] has semisimple and positive eigenvalues0 < ρ 1 ≤ ρ 2, β is an arbitrary constant, ω is an open nonempty subset ofΩ, 1 ω denotes the characteristic function of the setω, and the distributed controlsf 1, f 2 ∈ L 2 ( [ 0, τ ] ; L 2 ( Ω ) ) . Specifically, we prove the following statement: ifλ 1 1 / 2 ρ 1 + β > 0 (whereλ 1 is the first eigenvalue of− Δ ), then for allτ > 0 and all open nonempty subsetω ofΩ the system is approximately controllable on[ 0, τ ] .