Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 708516, 8 pages
Research Article

Interior Controllability of a Broad Class of Reaction Diffusion Equations

1Departamento de Matemáticas, Universidad de Los Andes, Mérida 5101, Venezuela
2Departamento de Matemáticas, Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080 A, Venezuela

Received 4 December 2008; Revised 30 March 2009; Accepted 24 May 2009

Academic Editor: John Burns

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.


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 n, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control uL2(0,t1;L2(Ω)) and A:D(A)ZZ is an unbounded linear operator with the following spectral decomposition: Az=j=1λjk=1γjz,ϕj,kϕj,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=1eλjtk=1γjz,ϕj,kϕj,k. Our result can be applied to the nD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.