Asymptotic behavior of Carleman weight functions and application to controllability

In the development of controllability and inverse problem results for semi-discrete systems, by using Carleman estimates, it is required to estimate of the discrete operators applied to Carleman weight functions. This work aims to establish the asymptotic behavior of Carleman weight functions under these discrete operators. We provide a characterization of the error term in arbitrary order and dimension, extending previously known results. This generalization is of independent interest due to its applications in deriving Carleman estimates for semi-discrete stochastic operators. The aforementioned estimates hold for Carleman weight functions used for parabolic, hyperbolic, and elliptic operators, which are applied to obtain control and inverse problems results for those operators. We apply these results to obtain $\phi$-controllability result for a fully discrete parabolic operator, which is based on a Carleman estimate for a fully-discrete parabolic operator.