Copyright © 2009 Hugo Leiva and Yamilet Quintana. 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.

Abstract

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z=L2(Ω) given by z′=−Az+1ωu(t), t∈[0,τ], where Ω is a domain in ℝn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u∈L2(0,t1;L2(Ω)) and A:D(A)⊂Z→Z is an unbounded linear operator with the following spectral decomposition: Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k. The eigenvalues 0<λ1<λ2<⋯<⋯λn→∞ of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace, and {ϕj,k} is a complete orthonormal set of eigenvectors of A. The operator −A generates a strongly continuous semigroup {T(t)} given by T(t)z=∑j=1∞e−λjt∑k=1γj〈z,ϕj,k〉ϕj,k. Our result can be applied to the nD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.