Tracking controllability on moving targets for parabolic equations

TL;DR

This paper investigates boundary control strategies for 1D parabolic equations to track solutions at moving points, combining theoretical controllability results with numerical validation.

Contribution

It introduces a method to control solutions at moving targets by transforming the problem and provides both theoretical proofs and numerical approximations.

Findings

Controlling one boundary point allows control at one target point.

Controlling two boundary points allows control at up to two target points.

Numerical simulations validate the theoretical controllability results.

Abstract

In this paper, we study the tracking controllability of a 1D parabolic type equation. Notably, with controls acting on the boundary, we seek to approximately control the solution of the equation on specific points of the domain. We prove that acting on one boundary point, we control the solution on one target point, whereas acting on two boundary points, we can control the solution on up to two target points. In order to do so, when the target is fixed, we study the controllability by minimizing the corresponding problem with duality results. Afterwards, we study the controllability on moving points by applying a transformation that takes the problem to a fixed target. Lastly, we also solve some of these control problems numerically and compute approximations of the solutions and the desired targets, which validates our theoretical methodology.