Boundary null controllability for the heat equation with dynamic boundary conditions

TL;DR

This paper proves that the heat equation with dynamic boundary conditions can be driven to zero using boundary controls on any subboundary, employing a new Carleman estimate and regularity analysis.

Contribution

It introduces a novel boundary Carleman estimate and regularity estimates to establish null controllability for the heat equation with dynamic boundary conditions.

Findings

Null controllability at any positive time.

Boundary control on arbitrary subboundary suffices.

New boundary Carleman estimate developed.

Abstract

In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control supported on an arbitrary subboundary. The proof of the main result combines a new boundary Carleman estimate and some regularity estimates for the adjoint system, with an explicit dependence with respect to the final time. This technique allows us to overcome a new difficulty that arises when absorbing a normal derivative term.