Numerical exact controllability of the 1D heat equation: dual algorithms

This work addresses the numerical approximation of distributed null controls (with small support) for the 1D heat equation, with Dirichlet boundary conditions: the goal is to compute a control that drives the solution from a prescribed initial state at t = 0 to zero at t = T . The earlier contribution by Carthel, Glowinski and Lions [3] considers boundary controls and exhibits numerical instability, which is closely related to the regularization effect of the heat equation. There, the controllability problem is solved through a dual reformulation. In this work, we deal with other related methods. We first introduce some constrained extremal problems (each of them corresponding to the minimization of a functional that involves weighted integrals of the state and the control) and we apply appropriate duality techniques that lead to the formulation of equivalent unconstrained extremal problems. Then, we introduce appropriate numerical approximations of the associated dual problems and we solve them by applying conjugate gradient techniques. We present several experiments and we discuss the robustness of this approach with respect to the control support. We also compare the results to those in the previous paper [7], where primal methods (directly connected to the original constrained problem) were considered.