We consider the following data completion problem for the Laplace equation in the cylindrical domain: = ]0, a[×O,O ⊂ Rn−1 (O is a smooth bounded open set anda > 0), limited by the faces 0 = {0}×O and a = {a}×O. The Neumann and Dirichlet boundary conditions are given on 0 while no condition is given on a. The completion data problem consists in recovering a boundary condition on a. This problem has been known to be ill-posed since Hadamard [12]. The problem is set as an optimal control problem with a regularized cost function. To obtain directly an approximation of the missing data on a we use the method of factorization of elliptic boundary value problems. This method allows us to factorize a boundary value problem in the product of two parabolic problems. Here it is applied to the optimality system (i.e. jointly on the state and adjoint state equations).