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

This work is concerned with the numerical computation of null controls of minimal $L^{\infty}$-norm for the linear heat equation with a bounded potential. Both, the cases of internal and boundary (Dirichlet and Neumann) controls are considered. Dual arguments 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 regularizing property of the heat operator, this initial (final) condition lives in a huge space, that can not be approximated with robustness. For this reason, very specific to the parabolic situation, the minimization is severally ill-posed. On the other hand, the optimality conditions for this problem show that, in general, the unique control $v$ of minimal $L^{\infty}$-norm has a bang-bang structure as he 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 the control takes its two possible values. This second optimization variable is modeled through a characteristic function. Since the admissibility set for this new control problem is not convex, we obtain a relaxed formulation of it which leads to a well-posed relaxed problem and lets use a gradient descent method for the numerical resolution of the problem. Numerical experiments, for the inner and boundary controllability cases, are described within this new approach.