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In the discrete setting of one-dimensional finite-differences we prove aCarleman estimate for a semi-discretization of the parabolic operator$\partial_t-\partial_x (c\partial_x)$ where the diffusion coefficient $c$ has ajump. As a consequence of this Carleman estimate, we deduce consistentnull-controllability results for classes of semi-linear parabolic equations.

Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and application to controllability