Controllability Results for a Cross Diffusion System with a Free Boundary by a Flatness Approach

Abstract

We study a one-dimensional cross diffusion system with a free boundary modeling the Physical Vapor Deposition. Using the flatness approach, we show several results of boundary controllability for this system in spaces of Gevrey class functions. One of the main difficulties consists in the physical constraints on the state and on the control. More precisely, the state corresponds to volume fractions of the \(n+1\) chemical species and to the thickness of the film produced in the process, whereas the controls are the fluxes of the chemical species. We obtain the local controllability in the case where we apply \(n+1\) nonnegative controls and a controllability result for large time in the case where we apply \(n\) controls without any sign constraints. We also show in this last case that the controllability may fail for small times. We illustrate these results with some numerical simulations.