Controllability of second-order equations in L2(ω)

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): ÿ + Ay = 1<inf>ω</inf>u(t),t ∈ (0, τ], y(0) = y0, ẏ(0) = y1, where Ω is a domain in R^N (N ≥ 1), y<inf>0</inf>,y<inf>1</inf> ∈ L²(Ω), ω is an open nonempty subset of Ω, 1<inf>ω</inf> denotes the characteristic function of the set ω, the distributed control u belongs to L²(0,τ; L²(Ω)), and A : D(A) ⊂ L ²(Ω) → L²(Ω) is an unbounded linear operator with the following spectral decomposition: Az = σ ^∞<inf>j</inf>=1 λj Σ ^γj <inf>k</inf>=1 (Z,φj,k)φj,k, with the eigenvalues λ<inf>j</inf> given by the following formula: λ<inf>j</inf> = j^2mπ ^2m, j = 1,2,3,... and m ≥ 1 is a fixed integer number, multiplicity γj is equal to the dimension of the corresponding eigenspace, and {φj,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 <inf>j,k</inf> = φ<inf>j,k|ω</inf> of φ<inf>j,k</inf> 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. Copyright © 2010 H. Leiva and N. Merentes.

Año de publicación:

2010

Keywords:

Fuente:

scopus