no. 2
Null-controllability of cascade reaction-diffusion systems with odd coupling terms
Kevin Le Balc’h1 ; Takéo Takahashi2
1 Sorbonne Université, CNRS, Inria, Laboratoire Jacques-Louis Lions, Paris, France
2 Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
Annales mathématiques Blaise Pascal, Tome 31 (2024) no. 2, pp. 239-271.

In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is locally null-controllable for any arbitrary small time. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced in [18] that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical L 2 parabolic Carleman estimate, conjugated with maximal regularity results, a weighted L p observability inequality for the nonhomogeneous heat equation. Secondly, we perform a duality argument, close to the well-known Hilbert Uniqueness Method in a reflexive Banach setting, to prove that the heat equation perturbed by a source term is null-controllable thanks to odd controls. Finally, the nonlinearity is handled with a Schauder fixed-point argument.

Publié le :
DOI : 10.5802/ambp.430
Classification : 35K45, 35K58, 93B05, 93C10
Mots-clés : Null-controllability, parabolic system, nonlinear coupling, Carleman estimate

Kevin Le Balc’h 1 ; Takéo Takahashi 2

1 Sorbonne Université, CNRS, Inria, Laboratoire Jacques-Louis Lions, Paris, France
2 Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Kevin Le Balc’h; Takéo Takahashi. Null-controllability of cascade reaction-diffusion systems with odd coupling terms. Annales mathématiques Blaise Pascal, Tome 31 (2024) no. 2, pp. 239-271. doi : 10.5802/ambp.430. https://ambp.centre-mersenne.org/articles/10.5802/ambp.430/

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