Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations

In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabolic equations. Sub-linear and super-linear cases are considered.