Small-time global controllability of a class of bilinear fourth-order parabolic equations

TL;DR

This paper demonstrates small-time global controllability for a class of fourth-order nonlinear parabolic equations on a one-dimensional torus, using frequency-localized controls and fixed-point methods.

Contribution

It introduces a novel geometric control approach for fourth-order equations and establishes both approximate and exact controllability results in small time.

Findings

Achieved small-time global approximate controllability between states with the same sign.

Proved small-time global exact controllability to non-zero constant states.

Developed a control strategy using frequency-localized controls and fixed-point arguments.

Abstract

In this work, we investigate the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by a bilinear control posed on the one-dimensional torus. The controls depend only on time and act through a prescribed family of spatial profiles. Our first result establishes the small-time global approximate controllability of the system using three scalar controls, between states that share the same sign. This property is obtained by adapting the geometric control approach to the fourth-order setting, using a finite family of frequency-localized controls. We then study the small-time global exact controllability to non-zero constant states for the concerned system. This second result is achieved by analyzing the null controllability of an appropriate linearized fourth-order system and by deducing the controllability of the nonlinear bilinear model through a fixed-point argument together with the small-time global approximate control property.