Numerical null controllability of the 1D linear Schrödinger equation

This paper deals with the numerical approximation to boundary controls that drive the solution to the 1D linear Schrödinger equation to a prescribed state at a final time. Using ideas from Fursikov and Imanuvilov, we consider the control that minimizes over the class of admissible controls a functional that involves weighted integrals, with weights that blow up at  T. We will see that this extremal problem is equivalent to a differential problem that is fourth order in space and second order in time. Adapting some numerical techniques applied by the first author and Münch to the heat equation, we approximate the variational formulation by introducing appropriate space-time finite elements that are C1 in space and C0 in time. We present two approaches; the second one relies on a change of variable which leads to a lower condition number for the stiffness matrix. The results of some experiments show the efficiency of these methods.