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

In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator ∂t − ∂x(c∂x) where the diffusion coefficient c has a jump. As a consequence of this Carleman estimate, we deduce consistent nullcontrollability results for classes of semi-linear parabolic equations.