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This article deals with the problem of computing numerical approximations of  null-controls for parabolic equations or systems by using the Hilbert Uniqueness Method  (HUM). We mainly review recent results on this subject but we also provide new elements to  emphasize the main ideas underlying the penalised HUM approach which is at the heart of  the methods used in practice. We give many numerical illustrations.Cet article est consacré à l’étude du problème du calcul approché de contrôles à zéro  pour des équations ou des systèmes paraboliques par le biais de la méthode HUM (Hilbert  Uniqueness Method). On donne un aperçu des résultats récents sur le sujet tout en mettant  en lumière certains aspects fondamentaux de la méthode HUM pénalisée, qui se trouve au  coeur des algorithmes proposés. De nombreuses illustrations numériques sont également  données

On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems