Local controllability of reaction-diffusion systems around nonnegative\n stationary states

Abstract

We consider a $n \\times n$ nonlinear reaction-diffusion system posed on a\nsmooth bounded domain $\\Omega$ of $\\mathbb{R}^N$. This system models reversible\nchemical reactions. We act on the system through $m$ controls ($1 \\leq m < n$),\nlocalized in some arbitrary nonempty open subset $\\omega$ of the domain\n$\\Omega$. We prove the local exact controllability to nonnegative (constant)\nstationary states in any time $T >0$. A specificity of this control system is\nthe existence of some invariant quantities in the nonlinear dynamics that\nprevents controllability from happening in the whole space\n$L^\\infty(\\Omega)^n$. The proof relies on several ingredients. First, an\nadequate affine change of variables transforms the system into a cascade system\nwith second order coupling terms. Secondly, we establish a new\nnull-controllability result for the linearized system thanks to a spectral\ninequality for finite sums of eigenfunctions of the Neumann Laplacian operator,\ndue to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability\ninequalities for a family of finite dimensional systems. Thirdly, the source\nterm method, introduced by Yuning Liu, Tak\\'eo Takahashi and Marius Tucsnak, is\nrevisited in a $L^{\\infty}$-context. Finally, an appropriate inverse mapping\ntheorem enables to go back to the nonlinear reaction-diffusion system.\n