Controllability of a 4 x 4 quadratic reaction-diffusion system

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Le Balc'h, Kevin¹ 

[1] Ecole Normale Super Rennes, IRMAR, Campus Ker Lann, F-35170 Bruz, France

关键词: Controllability to stationary states;    Parabolic system;    Nonlinear coupling;    Carleman estimate;    Return method;   

DOI  :  10.1016/j.jde.2018.08.046

来源: Elsevier

【 摘 要 】

We consider a 4 x 4 nonlinear reaction-diffusion system posed on a smooth domain Omega of R-N (N >= 1) with controls localized in some arbitrary nonempty open subset omega of the domain Omega. This system is a model for the evolution of concentrations in reversible chemical reactions. We prove the local exact controllability to stationary constant solutions of the underlying reaction-diffusion system for every N >= 1 in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics. The proof is based on a linearization which uses return method and an adequate change of variables that creates cross diffusion which will be used as coupling terms of second order. The controllability properties of the linearized system are deduced from Carleman estimates. A Kakutani's fixed-point argument enables to go back to the nonlinear parabolic system. Then, we prove a global controllability result in large time for 1 <= N <= 2 thanks to our local controllability result together with a known theorem on the asymptotics of the free nonlinear reaction-diffusion system. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free