This paper deals with the numerical computation of distributed null controls for the 1D heat equation, with Dirichlet boundary conditions. The goal is to compute a control that drives (a numerical approximation of) the solution from a prescribed initial state at t = 0 to zero at t = T. Using ideas from Fursikov and Imanuvilov [18], we consider the control that minimizes over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near T. The optimality system is equivalent to a di erential problem that is fourth-order in space and second-order in time. We rst address the numerical solution of the corresponding variational formulation by introducing a space-time nite element that is C1 in space and C0 in time. We prove a strong convergence result for the approximate controls and then we present some numerical experiments. IWe also introduce a mixed variational formulation and we prove well-posedness through a suitable inf-sup condition. We introduce a (non-conformal) C0 nite element approximation and we provide new numerical results. In both cases, thanks to an appropriate change of variable, we observe a polynomial dependance of the condition number with respect to the discretization parameter. Furthermore, with this second method, the initial and nal conditions are satis ed exactly.
