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

TL;DR

This paper establishes Carleman estimates for semi-discrete parabolic operators with discontinuous diffusion coefficients and applies these estimates to demonstrate null-controllability of certain semi-linear parabolic equations.

Contribution

It introduces a novel Carleman estimate for semi-discrete parabolic operators with discontinuous coefficients and uses it to prove controllability results.

Findings

Proved a Carleman estimate for semi-discrete parabolic operators with jump coefficients.

Derived null-controllability results for semi-linear parabolic equations.

Extended controllability theory to cases with discontinuous diffusion coefficients.

Abstract

In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x)$ 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.