Local controllability of reaction-diffusion systems around nonnegative stationary states

TL;DR

This paper proves local exact controllability of nonlinear reaction-diffusion systems near nonnegative stationary states using spectral inequalities, affine transformations, and inverse mapping techniques.

Contribution

It introduces a new null-controllability result for linearized systems and adapts the source term method to the $L^$ setting for reaction-diffusion equations.

Findings

Achieved local controllability for reaction-diffusion systems in any positive time.

Developed a spectral inequality for Neumann Laplacian eigenfunctions.

Revealed invariant quantities that limit controllability in the full $L^$ space.

Abstract

We consider a $n \times n$ nonlinear reaction-diffusion system posed on a smooth bounded domain $\Omega$ 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 $\omega$ of the domain $\Omega$. 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 $L^\infty(\Omega)^n$. 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, Tak\'eo 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.