Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system

TL;DR

This paper develops a boundary feedback stabilization method for linearized one-dimensional cross-diffusion systems in moving domains, achieving exponential and finite-time stabilization using backstepping techniques tailored for time-dependent geometries.

Contribution

It introduces a novel backstepping approach for stabilizing cross-diffusion systems in moving domains, including finite-time stabilization, with applications to physical vapor deposition processes.

Findings

Achieved exponential stabilization of the linearized system.

Established finite-time stabilization in small time.

Successfully applied backstepping to time-dependent domains.

Abstract

We study the boundary stabilization of one-dimensional cross-diffusion systems in a moving domain. We show first exponential stabilization and then finite-time stabilization in arbitrary small-time of the linearized system around uniform equilibria, provided the system has an entropic structure with a symmetric mobility matrix. One example of such systems are the equations describing a Physical Vapor Deposition (PVD) process. This stabilization is achieved with respect to both the volume fractions and the thickness of the domain. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time.