Controllability of Second-Order Equations in L2(Ω)‎

Abstract EN

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 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)⊂L2(Ω)→L2(Ω) is an unbounded linear operator with the following spectral decomposition: Az=∑j=1∞λj∑k=1γj〈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 ϕj,kω=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.