Local null controllability of coupled degenerate systems with nonlocal terms and one control force

TL;DR

This paper proves local null controllability for a class of coupled nonlinear parabolic systems with nonlocal, degenerate diffusion, using a novel Carleman estimate and inverse mapping theorem, extending previous single-equation results.

Contribution

It establishes the first local null controllability result for coupled degenerate systems with nonlocal terms using only one control force.

Findings

Achieved local null controllability with one control for coupled systems.

Developed a new Carleman estimate tailored for degenerate coupled systems.

Extended previous single-equation controllability results to coupled systems.

Abstract

In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanivilov, is obtained from the global null controllability of the linearized system provided by \textit{Lyusternik's Inverse Mapping Theorem}. This work extends the results previously treated by the authors for just one equation. For the system, the main issue is to obtain similar results with just one internal control, which requires a new Carleman estimate with the local term just depending on one of the state function.