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 minimal 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 can not 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 v of minimal 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.