Local controllability of reaction-diffusion systems around nonnegative stationary states

ENS Rennes, Université de Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France.

^* Corresponding author: kevin.lebalch@ens-rennes.fr

Received: 28 September 2018
Accepted: 26 April 2019

Abstract

We consider a n × n nonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝ^N. This system models reversible chemical reactions. We act on the system through m controls (1 ≤ m < n), localized in some arbitrary nonempty open subset ω of the domain Ω. We prove the local exact controllability to nonnegative (constant) stationary states in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space L^∞(Ω)ⁿ. The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak, is revisited in a L^∞-context. Finally, an appropriate inverse mapping theorem in suitable spaces enables to go back to the nonlinear reaction-diffusion system.