On the small-time bilinear control of a nonlinear heat equation: global approximate controllability and exact controllability to trajectories

TL;DR

This paper investigates the ability to control a nonlinear heat equation on a torus in small time using bilinear controls, achieving approximate controllability in higher dimensions and exact controllability to specific states in one dimension.

Contribution

It establishes small-time approximate controllability for nonlinear parabolic equations under certain conditions and proves exact controllability to trajectories in one dimension, advancing control theory for PDEs.

Findings

Approximate controllability between same-sign states in arbitrary dimensions.

Exact controllability to the ground state in one dimension.

Control results depend on saturation hypotheses on control operators.

Abstract

In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension $d$. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case $d=1$, we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.