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

Jose Antonio Villa

TL;DR

The paper addresses null controllability for the heat equation with Fourier boundary conditions when the control acts on a small boundary portion. It develops a boundary Carleman inequality for the adjoint system with tailored weights and uses duality to construct a boundary follower control that enforces $y(T)=0$; this framework extends to a Lipschitz semi-linear term via a fixed-point argument. A key contribution is Theorem 1, which provides the Carleman estimate in the boundary Fourier setting, enabling observability-based control and an explicit iterative scheme for the resulting coupled parabolic system. The results offer a rigorous boundary-control strategy with potential numerical realization, including a fixed-point–based method to solve the semi-linear and coupled problems.

Abstract

In this work, we establish a Carleman inequality for the heat equation with Fourier boundary conditions of the form $\partial_νy+by=f1_γ$, where the control acts on a small portion $γ$ 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.

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

TL;DR

The paper addresses null controllability for the heat equation with Fourier boundary conditions when the control acts on a small boundary portion. It develops a boundary Carleman inequality for the adjoint system with tailored weights and uses duality to construct a boundary follower control that enforces ; this framework extends to a Lipschitz semi-linear term via a fixed-point argument. A key contribution is Theorem 1, which provides the Carleman estimate in the boundary Fourier setting, enabling observability-based control and an explicit iterative scheme for the resulting coupled parabolic system. The results offer a rigorous boundary-control strategy with potential numerical realization, including a fixed-point–based method to solve the semi-linear and coupled problems.

Abstract

In this work, we establish a Carleman inequality for the heat equation with Fourier boundary conditions of the form , where the control acts on a small portion 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.
Paper Structure (9 sections, 15 theorems, 130 equations)

This paper contains 9 sections, 15 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminary
  4. Proof of theorem \ref{['carleman1']}.
  5. Null controllability problem
  6. The semi-linear case
  7. Fixed point iterations.
  8. Conclusions and further problems.
  9. Appendix: proof of Lemma \ref{['lem1']}

Key Result

Lemma 1

Given a non-empty open set $\gamma \subset \Gamma$, there exists a function $\eta \in C^{2}(\overline{\Omega})$ such that

Theorems & Definitions (23)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2: Dubovitski-Milyoutin, alexeev2017commande
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 13 more