Local controllability of reaction-diffusion systems around nonnegative stationary states

Kévin Le Balc’h (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We consider a n × n nonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝN. 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.

Mots clés: Controllability; reaction-diffusion system; nonlinear coupling

Domaines: Mathématiques [math]

Kévin Le Balc’h. Local controllability of reaction-diffusion systems around nonnegative stationary states. ESAIM: Control, Optimisation and Calculus of Variations, 2020, 26, pp.55. ⟨10.1051/cocv/2019033⟩. ⟨hal-01871478v2⟩