Carleman Inequalities for the Heat Equation with Fourier Boundary Conditions: Applications to Null Controllability Problems

TL;DR

This paper develops a Carleman inequality for the heat equation with Fourier boundary conditions and applies it to achieve null controllability with boundary control on a small boundary region, including an explicit solution and a numerical method.

Contribution

It introduces a new Carleman inequality for the heat equation with Fourier boundary conditions and demonstrates its application to null controllability with boundary control.

Findings

Established a Carleman inequality for the heat equation with Fourier boundary conditions.

Achieved explicit null controllability solution via coupled parabolic systems.

Proposed an iterative numerical method for solving the coupled control system.

Abstract

In this work, we establish a Carleman inequality for the heat equation with Fourier boundary conditions of the form $\partial_\nu y+by=f1_\gamma$, where the control acts on a small portion $\gamma$ of the boundary. We apply this inequality to address the null controllability problem with boundary control supported on this small region. An explicit solution to this problem is obtained via a system of coupled parabolic equations. Based on these results, we propose an iterative numerical method to solve the coupled system.