Controllability of Second - Order Equations in L 2 Ω

We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space L2 Ω : ÿ Ay 1ωu t , t ∈ 0, τ , y 0 y0, ẏ 0 y1, where Ω is a domain in N 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 ⊂ L2 Ω → L2 Ω is an unbounded linear operator with the following spectral decomposition:Az ∑∞ j 1 λj ∑γj k 1〈z, φj,k〉φj,k , with the eigenvalues λj given by the following formula: λj j2mπ2m, 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 j,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.