Controllability for a 2x2 nonlinear degenerate parabolic system via one boundary control force

TL;DR

This paper investigates the boundary controllability of a nonlinear 2x2 degenerate parabolic system with a single boundary control, providing new estimates on control cost and establishing local exact controllability results.

Contribution

It introduces a novel approach combining moment method and iterative techniques to achieve controllability for a nonlinear degenerate system with minimal control input.

Findings

Boundary null controllability for the linear system established.

Control cost estimates derived for the nonlinear system.

Local exact boundary controllability to zero proved.

Abstract

In this paper we study the local boundary controllability for a non linear system of two degenerate parabolic equations with a control acting on only one equation. We analyze boundary null controllability properties for the linear system via the moment method by Fattorini and Russell, together with some results on biorthogonal families. Moreover, we provide an estimate on the null-control cost. This estimate let us prove a local exact boundary controllability result to zero of the nonlinear system following the iterative method from Lebeau and Robbiano as in \cite{Burgos_2020, Liu_2012}.