Control of three heat equations coupled with two cubic nonlinearities

TL;DR

This paper proves the global null controllability of a coupled system of three heat equations with nonlinearities, despite the linearized system being uncontrollable, by leveraging the return method and nonlinear effects.

Contribution

It demonstrates that cubic nonlinearities enable control of a coupled heat system that is not controllable in the linearized case, using advanced control techniques.

Findings

Global null controllability achieved with nonlinearities

Linearized system is not controllable

Return method and algebraic techniques are effective

Abstract

We study the null controllability of three parabolic equations. The control is acting only on one of the three equations. The three equations are coupled by means of two cubic nonlinearities. The linearized control system around 0 is not null controllable. However, using the cubic nonlinearities, we prove the (global) null controllability of the control system. The proof relies on the return method, an algebraic solvability and smoothing properties of the parabolic equations.