An elementary proof of the lack of null controllability for the heat equation on the half line

TL;DR

This paper presents an elementary proof demonstrating the impossibility of null controllability for the heat equation on the half line, using the Fokas method, which also extends to higher dimensions and other boundary value problems.

Contribution

It introduces a novel, elementary proof technique based on the Fokas method for null controllability issues in heat equations, applicable to higher dimensions and various boundary conditions.

Findings

Proves lack of null controllability for the heat equation on the half line.

Extends the proof technique to higher dimensions and different boundary value problems.

Suggests a new methodology for studying controllability problems.

Abstract

In this note, we give an elementary proof of the lack of null controllability for the heat equation on the half line by employing the machinery inherited by the unified transform, known also as the Fokas method. This approach also extends in a uniform way to higher dimensions and different initial-boundary value problems governed by the heat equation, suggesting a novel methodology for studying problems related to controllability.