Academic Editor: Christos H. Skiadas
Copyright © 2010 Hugo Leiva and Nelson Merentes. 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 present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space : , , , , where is a domain in , , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control belongs to and is an unbounded linear operator with the following spectral decomposition: , with the eigenvalues given by the following formula: , and is a fixed integer number, multiplicity is equal to the dimension of the corresponding eigenspace, and is a complete orthonormal set of eigenvectors (eigenfunctions) of . Specifically, we prove the following statement: if for an open nonempty set the restrictions of to are linearly independent functions on , then for all the system is approximately controllable on . As an application, we prove the controllability of the 1D wave equation.