Tracking controllability on moving targets for parabolic equations

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 at 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.