We consider a n × n nonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝ$$. This system models reversible chemical reactions. We act on the system through m controls (1 ≤ m < n), localized in some arbitrary nonempty open subset ω of the domain Ω. We prove the local exact controllability to nonnegative (constant) stationary states in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space L$$(Ω)$$. The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak, is revisited in a L$$-context. Finally, an appropriate inverse mapping theorem in suitable spaces enables to go back to the nonlinear reaction-diffusion system.
Keywords: Controllability, reaction-diffusion system, nonlinear coupling
@article{COCV_2020__26_1_A55_0,
author = {Le Balc{\textquoteright}h, K\'evin},
title = {Local controllability of reaction-diffusion systems around nonnegative stationary states},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019033},
mrnumber = {4147583},
zbl = {1446.93013},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019033/}
}
TY - JOUR AU - Le Balc’h, Kévin TI - Local controllability of reaction-diffusion systems around nonnegative stationary states JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019033/ DO - 10.1051/cocv/2019033 LA - en ID - COCV_2020__26_1_A55_0 ER -
%0 Journal Article %A Le Balc’h, Kévin %T Local controllability of reaction-diffusion systems around nonnegative stationary states %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019033/ %R 10.1051/cocv/2019033 %G en %F COCV_2020__26_1_A55_0
Le Balc’h, Kévin. Local controllability of reaction-diffusion systems around nonnegative stationary states. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 55. doi: 10.1051/cocv/2019033
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