Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on B
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Research Article Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains Salah Badraoui Laboratoire LAIG, Universit´e du 08 Mai 1945, BP. 401, Guelma 24000, Algeria Correspondence should be addressed to Salah Badraoui, [email protected] Received 11 July 2009; Accepted 5 January 2010 Academic Editor: Ugur Abdulla Copyright q 2010 Salah Badraoui. 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. We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives ut a1 Δu a2 Δv − c1 −Δα1 u − c2 −Δα2 v 1ω f1 x, t in Ω× 0, t∗ , vt b1 Δu b2 Δv − d1 −Δβ1 u − d2 −Δβ2 v 1ω f2 x, t in Ω× 0, t∗ , u v 0 on ∂Ω× 0, t∗ , ux, 0 u0 x, vx, 0 N 2 v0 x in x ∈ Ω, Ω ⊂ R N ≥ 1 is a smooth bounded domain, u0 , v0 ∈ L Ω, the diffusion awhere 1 a2 has semisimple and positive eigenvalues 0 < ρ1 ≤ ρ2 , 0 < α1 , α2 , β1 , β2 < 1, matrix M b1 b2 ω ⊂ Ω is an open nonempty set, and 1ω is the characteristic function of ω. Specifically, we prove that under some conditions over the coefficients ai , bi , ci , di i 1, 2, the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all t∗ > 0 the system is approximately controllable on 0, t∗ .
1. Introduction In this paper we prove controllability for the following reaction-diffusion system with cross diffusion matrix: ut a1 Δu a2 Δv − c1 −Δα1 u − c2 −Δα2 v 1ω f1 x, t
in Ω × 0, t∗ ,
vt b1 Δu b2 Δv − d1 −Δβ1 u − d2 −Δβ2 v 1ω f2 x, t in Ω × 0, t∗ , u v 0 on ∂Ω × 0, t∗ , ux, 0 u0 x,
vx, 0 v0 x
in x ∈ Ω,
where ω is an open nonempty set of Ω and 1ω is the characteristic function of ω.
1.1
2
Boundary Value Problems We assume the following assumptions. H1 Ω is a smooth bounded domain in RN N ≥ 1. a1 a2 H2 The diffusion matrix M b b has semisimple and positive eigenvalues 0 < ρ1 ≤ 1 2 ρ2 . H3 cj , dj j 1, 2 are real constants, αj , βj j 1, 2 are real constants belonging to the interval 0, 1 . H4 u0 , v0 ∈ L2 Ω. H5 The distributed controls f1 , f2 ∈ L2 0, t∗ ; L2 Ω. Specifically, we prove the following statements. 1−β
1 i If c2 d1 0 and min{c1 λ1−α ρ1 , d2 λ1 2 ρ1 } > 0, where λ1 is the first eigenvalue 1 0, d1 / 0, c1 ≥ 0, and d2 ≥ 0; then, under of −Δ with Dirichlet condition, or if c2 / the hypotheses H1–H3, the semigroup generated by the linear operator of the system is exponentially stable.
ii If c2 d1 0 and under the hypotheses H1–H5, then, for all t∗ > 0 and all open nonempty subset ω of Ω the system is approximately controllable on 0, t∗ . This paper has been motivated by the work done in 1 and the work done by H. Larez and H. Leiva in 2 . In the work 1 , the auther studies the asymptotic behav
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