Null controllability of the 1D heat equation using flatness

TL;DR

This paper demonstrates a straightforward flatness-based method for achieving null controllability of the 1D heat equation with boundary control, providing explicit control laws and error estimates validated by numerical experiments.

Contribution

It introduces a flatness approach for boundary control of the 1D heat equation, offering explicit control laws and error analysis for practical implementation.

Findings

Explicit control law derived using flatness approach

Error estimates for series truncation provided

Numerical experiments confirm effectiveness

Abstract

We derive in a straightforward way the null controllability of a 1-D heat equation with boundary control. We use the so-called {\em flatness approach}, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control law achieving the exact steering to zero. We also give accurate error estimates when the various series involved are replaced by their partial sums, which is paramount for an actual numerical scheme. Numerical experiments demonstrate the relevance of the approach.