Year: 2018
Type: preprint
Abstract: We consider a $n \times n$ nonlinear reaction-diffusion system posed on a smooth bounded domain $Ω$ of $\mathbb{R}^N$. This system models reversible chemical reactions. We act on the system through $m$ controls ($1 \leq m < n$), 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 $L^\infty(Ω)^n$. 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 $L^{\infty}$-context. Finally, an appropriate inverse mapping theorem enables to go back to the nonlinear reaction-diffusion system.
Source: arXiv (Cornell University)
Author: Balc'H, Kévin Le
Institution: Institut de recherche mathématique de Rennes
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: Université de Rennes 1
OpenAlex ID: W4289543827