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 ${\mathrm{\partial }}_t-{\mathrm{\partial }}_x(c{\mathrm{\partial }}_x)$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.