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 = 1ωu(t),t ∈ (0, τ], y(0) = y0, ẏ(0) = y1, where Ω is a domain in RN (N ≥ 1), y0,y1 ∈ L2(Ω), ω is an open nonempty subset of Ω, 1ω 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 ∞j=1 λj Σ γjk=1 (Z,φj,k)φj,k, with the eigenvalues λj given by the following formula: λj = j2mπ 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 φωj,k = φj,k|ω of φj,k to ω are linearly independent functions on ω, then for all τ ≥ 2/πm-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