Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equation under the Bardos-Lebeau-Rauch geometric control condition

We extend the result of the null controllability property of the heat equation, obtained as limit, when ε tends to zero, of the exact controllability of a singularly perturbed damped wave equation depending on a parameter ε > 0, described in [1], to bounded domains which satisfy the Bardos-Lebeau-Rauch geometric control condition [2]. We add to the method of Lopez, Zhang and Zuazua in [1] an explicit in ε > 0 observability estimate for the singularly perturbed damped wave equation under the Bardos-Lebeau-Rauch geometric control condition. Here the geometric conditions are more optimal than in [1] and the proof is simpler than in [1]. Instead of using global Carleman inequalities as in [1], we apply an integral representation formula.