Controllability of Second-Order Equations in L2(Ω). (28th December 2010) - Leiva, Hugo; Merentes, Nelson
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 ∑ k = 1 γ j 〈 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 j, k ω = 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.