Strong convergent approximations of null controls for the 1D heat equation

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 approximations to controls that drive the solution from a prescribed initial state at $$t=0$$ to zero at $$t=T$$. Using ideas from Fursikov and Imanuvilov (Controllability of Evolution Equations, Lecture Notes Series, vol. 34. Seoul National University, Korea, pp. 1–163, 1996), we consider the control that minimizes a functional involving weighted integrals of the state and the control, with weights that blow up at $$t=T$$. The optimality system is equivalent to a differential problem that is fourth-order in space and second-order in time. We first address the numerical solution of the corresponding variational formulation by introducing a space-time finite element that is $$C^1$$ in space and $$C^0$$ in time. We prove a strong convergence result for the approximate controls and then we present some numerical experiments. We also introduce a mixed variational formulation and we prove well-posedness through a suitable inf-sup condition. We introduce a (non-conformal) $$C^0$$ finite 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 final conditions are satisfied exactly.