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

TL;DR

This paper proves small-time local null-controllability of a nonlinear cascade reaction-diffusion system with odd coupling terms, overcoming linear controllability obstacles through nonlinear control strategies and fixed-point methods.

Contribution

It introduces a novel control approach for nonlinear systems with odd couplings, extending linear control techniques to nonlinear settings.

Findings

The nonlinear system is small-time locally null-controllable.

Odd controls enable null-controllability of the heat equation with source terms.

A new fixed-point argument handles nonlinearity in control design.

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 small-time locally null-controllable. 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 by the first author 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.