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Controllability of a parabolic system with a diffusive interface

Abstract : We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain or on a Riemannian manifold, where the transmission conditions involve an additional parabolic operator on the interface. This system is an idealization of a three-layer model in which the central layer has a small thickness $\delta$. We prove a Carleman estimate in the neighborhood of the interface for an associated elliptic operator by means of partial estimates in several microlocal regions. In turn, from the Carleman estimate, we obtain a spectral inequality that yields the null-controllability of the parabolic system. These results are uniform with respect to the small parameter $\delta$.
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Jérôme Le Rousseau, Matthieu Léautaud, Luc Robbiano. Controllability of a parabolic system with a diffusive interface. Journal of the European Mathematical Society, European Mathematical Society, 2013, 15 (4), pp.1485-1574. ⟨10.4171/JEMS/397⟩. ⟨hal-00618234⟩

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