Numerical approximation of bang–bang controls for the heat equation: An optimal design approach

This work is concerned with the numerical approximation of null controls of the minimal L∞L∞-norm for the linear heat equation with a bounded potential. Both the cases of internal and boundary controls are considered. Dual arguments typically allow to reduce the search of controls to the unconstrained minimization of a conjugate function with respect to the initial condition of a backward heat equation. However, as a consequence of the regularization property of the heat operator, this condition lives in a huge space that cannot be approximated with robustness. For this reason the minimization is severally ill-posed. On the other hand, the optimality conditions for this problem show that the unique control vv of the minimal L∞L∞-norm has a bang–bang structure as it takes only two values: this allows to reformulate the problem as an optimal design problem where the new unknowns are the amplitude of the bang–bang control and the space–time regions where it takes its two possible values. This second optimization variable is modeled through a characteristic function. Since this new problem is not convex, we obtain a relaxed formulation of it which, in particular, lets the use of a gradient method for the numerical resolution. Numerical experiments are described within this new approach.