Controllability of some semilinear shadow reaction-diffusion systems

Article type: Research Article

Authors: Hernández-Santamaría, Víctor^{; *} | Peña-García, Alberto

Affiliations: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, Mexico

Correspondence: [*] Corresponding author. E-mail: victor.santamaria@im.unam.mx.

Abstract: The shadow limit is a versatile tool used to study the reduction of reaction-diffusion systems into simpler PDE-ODE models by letting one of the diffusion coefficients tend to infinity. This reduction has been used to understand different qualitative properties and their interplay between the original model and its reduced version. The aim of this work is to extend previous results about the controllability of linear reaction-diffusion equations and how this property is inherited by the corresponding shadow model. Defining a suitable class of nonlinearities and improving some uniform Carleman estimates, we extend the results to the semilinear case and prove that the original model is null-controllable and that the shadow limit preserves this important feature.

Keywords: Shadow limit, semilinear reaction-diffusion equations, uniform null-controllability, Carleman estimates

DOI: 10.3233/ASY-241930

Journal: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-39, 2024