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

Jean Cauvin-Vila (1, 2) , Virginie Ehrlacher (1, 2) , Amaury Hayat (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS

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 volumic 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.

Mots clés: Cross-diffusion systems; Parabolic PDEs; Feedback stabilization; Boundary control; Exponential stability; Backstepping

Domaines: Equations aux dérivées partielles [math.AP]

Jean Cauvin-Vila, Virginie Ehrlacher, Amaury Hayat. Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system. Journal of Differential Equations, 2023, 350, pp.251-307. ⟨10.1016/j.jde.2022.12.021⟩. ⟨hal-03706841v2⟩