Work

Null-controllability of cascade reaction-diffusion systems with odd coupling terms

Year: 2025

Type: article

Abstract: 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.

Source: Annales mathématiques Blaise Pascal

DOI: 10.5802/ambp.430

Authors: Kevin Le Balc’h, Takéo Takahashi

Institutions: Centre National de la Recherche Scientifique, Université Clermont Auvergne, Laboratoire de Mathématiques Blaise Pascal

Language: English

Topic: Stability and Controllability of Differential Equations

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: diamond

Funder: Agence Nationale de la Recherche

Award: New Trends in Control and Stabilization: Constraints and Non-local Terms