# Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension

## TL;DR

This paper demonstrates that nonlinear parabolic PDEs on a torus of any dimension can be approximately controlled using only finitely many Fourier modes, extending controllability results to more complex, higher-dimensional systems.

## Contribution

It establishes approximate controllability for nonlinear parabolic PDEs in arbitrary dimensions with polynomial growth nonlinearities, using geometric control theory techniques.

## Key findings

- Approximate controllability is achieved with finitely many Fourier modes.
- The results apply to PDEs with polynomial growth nonlinearities.
- The approach extends controllability theory to higher-dimensional, nonlinear PDEs.

## Abstract

In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the corresponding Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.