Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 147195, 11 pages
doi:10.1155/2010/147195
Research Article

Controllability of Second-Order Equations in 𝐿 𝟐 ( Ω )

Hugo Leiva1 and Nelson Merentes2

1Departamento de Matemática, Facultad de Ciencias, Universidad de los Andes, Mérida 5101, Venezuela
2Departamento de Matemática, Facultad de Ciencias, Universidad Central de Venezuela, Caracas 1020, Venezuela

Received 24 August 2010; Accepted 13 November 2010

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.

Abstract

We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space 𝐿 2 ( Ω ) : ̈ 𝑦 + 𝐴 𝑦 = 1 𝜔 𝑢 ( 𝑡 ) , 𝑡 ∈ ( 0 , 𝜏 ] , 𝑦 ( 0 ) = 𝑦 0 , ̇ 𝑦 ( 0 ) = 𝑦 1 , where Ω is a domain in ℝ 𝑁 ( 𝑁 ≥ 1 ) , 𝑦 0 , 𝑦 1 ∈ 𝐿 2 ( Ω ) , 𝜔 is an open nonempty subset of Ω , 1 𝜔 denotes the characteristic function of the set 𝜔 , the distributed control 𝑢 belongs to 𝐿 2 ( 0 , 𝜏 ; 𝐿 2 ( Ω ) ) , and 𝐴 ∶ 𝐷 ( 𝐴 ) ⊂ 𝐿 2 ( Ω ) → 𝐿 2 ( Ω ) is an unbounded linear operator with the following spectral decomposition: ∑ 𝐴 𝑧 = ∞ 𝑗 = 1 𝜆 𝑗 ∑ 𝛾 𝑗 𝑘 = 1 ⟨ 𝑧 , 𝜙 𝑗 , 𝑘 ⟩ 𝜙 𝑗 , 𝑘 , with the eigenvalues 𝜆 𝑗 given by the following formula: 𝜆 𝑗 = 𝑗 2 𝑚 𝜋 2 𝑚 , 𝑗 = 1 , 2 , 3 , … and 𝑚 ≥ 1 is a fixed integer number, multiplicity 𝛾 𝑗 is equal to the dimension of the corresponding eigenspace, and { 𝜙 𝑗 , 𝑘 } is a complete orthonormal set of eigenvectors (eigenfunctions) of 𝐴 . Specifically, we prove the following statement: if for an open nonempty set 𝜔 ⊂ Ω the restrictions 𝜙 𝜔 𝑗 , 𝑘 = 𝜙 𝑗 , 𝑘 | 𝜔 of 𝜙 𝑗 , 𝑘 to 𝜔 are linearly independent functions on 𝜔 , then for all 𝜏 ≥ 2 / 𝜋 𝑚 − 1 the system is approximately controllable on [ 0 , 𝜏 ] . As an application, we prove the controllability of the 1D wave equation.