Numerical null controllability of parabolic PDEs using Lagrangian methods

TL;DR

This paper develops a novel numerical methodology using Lagrangian techniques to solve null controllability problems for linear parabolic PDEs, enabling exact state control in multi-dimensional settings.

Contribution

It introduces a new approach combining classical Lagrangian and Augmented Lagrangian methods with Carleman inequalities for numerical controllability of parabolic equations.

Findings

Successful numerical control in 2D and 3D problems

Method effectively computes controls driving states to zero

Validation through numerical experiments

Abstract

In this paper, we study several theoretical and numerical questions concerning the null controllability problems for linear parabolic equations and systems for several dimensions. The control is distributed and acts on a small subset of the domain. The main goal is to compute numerically a control that drives a numerical approximation of the state from prescribed initial data exactly to zero. We introduce a methodology for solving numerical controllability problems that is new in some sense. The main idea is to apply classical Lagrangian and Augmented Lagrangian techniques to suitable constrained extremal formulations that involve unbounded weights in time that make global Carleman inequalities possible. The theoretical results are validated by satisfactory numerical experiments for spatially 2D and 3D problems.