Discrete Carleman estimates and application to controllability for a fully-discrete parabolic operator with dynamic boundary conditions

TL;DR

This paper develops discrete Carleman estimates for a fully-discrete 1-D heat equation with dynamic boundary conditions and demonstrates controllability results using these estimates, linking discrete parameters with Carleman weights.

Contribution

It introduces a novel Carleman estimate tailored for fully-discrete parabolic operators with dynamic boundary conditions, enabling controllability analysis.

Findings

Establishment of a relaxed observability inequality for the adjoint system.

Proof of controllability for the fully-discrete heat equation with dynamic boundary conditions.

Connection between discrete parameters and Carleman weights for the estimates.

Abstract

We consider a fully-discrete approximations of 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding adjoint system. This is done by using a suitable Carleman estimate for such models where the discrete parameters $h$ and $\Delta t$ are connected to the one of the large Carleman parameters.