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.
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机译：这项工作涉及对具有有限电势的线性热方程的最小$ L ^ {\ infty} $-范数的空控制的数值计算。内部和边界（Dirichlet和Neumann）控制的情况都被考虑。对偶参数允许将控件的搜索减少到相对于逆向热方程的初始条件的共轭函数的无约束最小化。然而，由于热操作员的正则化性质，该初始（最终）条件存在于巨大的空间中，无法用鲁棒性来近似。因此，对于抛物线情况非常具体，最小化有时会引起不适。另一方面，此问题的最优性条件表明，通常，最小$ L ^ {\ infty} $-norm的唯一控制$ v $具有爆炸结构，因为他仅采用两个值：这允许将问题重新表述为最佳设计问题，其中新的未知数是爆炸控制的幅度和控制采用其两个可能值的时空区域。该第二优化变量通过特征函数建模。由于针对该新控制问题设置的可容许性不是凸的，因此我们得到了它的宽松表述，这导致了一个定位良好的松弛问题，并允许使用梯度下降法求解该问题。在这种新方法中描述了内部和边界可控情况的数值实验。
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