Controllability of Second-OrderEquations in L2(Ω). (30th November 2010)

• Record Type: Journal Article
• Authors: Leiva, Hugo; Merentes, Nelson
• Other Names: Skiadas Christos H. Academic Editor.
• Is Part Of: Mathematical problems in engineering. Volume 2010(2010)
• Publication Date: 2010-11-30
• Subjects: Engineering mathematics -- Periodicals; 510.2462
• DOI: 10.1155/2010/147195
• Languages: English
• ISSNs: 1024-123X

Abstract

Abstract : We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert spaceL 2 ( Ω ) :y ̈ + A y = 1 ω u ( t ), t ∈ ( 0, τ ], y ( 0 ) = y 0, y ̇ ( 0 ) = y 1, whereΩ is a domain inℝ N ( N ≥ 1 ), y 0, y 1 ∈ L 2 ( Ω ), ω is an open nonempty subset ofΩ, 1 ω denotes the characteristic function of the setω, the distributed controlu belongs toL 2 ( 0, τ ; L 2 ( Ω ) ), andA : D ( A ) ⊂ L 2 ( Ω ) → L 2 ( Ω ) is an unbounded linear operator with the following spectral decomposition:A z = ∑ j = 1 ∞ λ j ∑ k = 1 γ j 〈 z, ϕ j, k 〉 ϕ j, k, with the eigenvaluesλ j given by the following formula:λ j = j 2 m π 2 m, j = 1, 2, 3, … andm ≥ 1 is a fixed integer number, multiplicityγ j is equal to the dimension of the corresponding eigenspace, and{ ϕ j, k } is a complete orthonormal set of eigenvectors (eigenfunctions) ofA . Specifically, we prove the following statement: if for an open nonempty setω ⊂ Ω the restrictionsϕ j, k ω = ϕ j, k | ω ofϕ j, k toω are linearly independent functions onω, then for allτ ≥ 2 / π m - 1 the system is approximately controllable on[ 0, τ ] . As an application, we prove the controllability of the 1D wave equation.