Numerical approximation of bang-bang controls for the heat equation: an optimal design approach - Université Clermont Auvergne Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2012

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

Résumé

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.
Fichier principal
Vignette du fichier
BB_Munch_periago_28feb.pdf (1.51 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00684214 , version 1 (30-03-2012)
hal-00684214 , version 2 (06-04-2012)
hal-00684214 , version 3 (02-05-2013)

Identifiants

  • HAL Id : hal-00684214 , version 3

Citer

Arnaud Munch, Francisco Periago. Numerical approximation of bang-bang controls for the heat equation: an optimal design approach. 2012. ⟨hal-00684214v3⟩
171 Consultations
474 Téléchargements

Partager

Gmail Facebook X LinkedIn More