Controllability properties for some semilinear parabolic PDE with a quadratic gradient term

TL;DR

This paper investigates the controllability of certain semilinear parabolic PDEs with quadratic gradient terms, demonstrating approximate and null controllability using the Cole-Hopf transformation.

Contribution

It introduces a novel application of the Cole-Hopf transformation to establish controllability properties for PDEs with quadratic gradient nonlinearities.

Findings

System is approximately controllable

System is null controllable

Cole-Hopf transformation is effective

Abstract

We study several controllability properties for some semilinear parabolic PDE with a quadratic gradient term. For internal distributed controls, it is shown that the system is approximately and null controllable. The proof relies on the Cole-Hopf transformation. The same approach is used to deal with initial controls.