# Exact Controllability of Nonlinear Heat Equations in Spaces of Analytic Functions

## TL;DR

This paper establishes exact controllability for a nonlinear heat equation with analytic nonlinearity in one dimension, using a direct approach with control inputs in Gevrey spaces, extending previous methods based on Carleman estimates.

## Contribution

It introduces a novel direct method for analyzing controllability of nonlinear parabolic equations with analytic nonlinearities, focusing on the relationship between spatial and temporal derivatives.

## Key findings

- Achieved exact controllability for small analytic initial and final states.
- Extended controllability results to functions analytic in a complex domain.
- Provided a new approach bypassing traditional Carleman estimate techniques.

## Abstract

It is by now well known that the use of Carleman estimates allows to establish the control-lability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. In this paper, we pursue the study of the reachable states of parabolic equations based on a direct approach using control inputs in Gevrey spaces by considering a nonlinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable x, the unknown y, and its derivative $\partial$ x y. By investigating carefully a nonlinear Cauchy problem in the spatial variable and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane. 2010 Mathematics Subject Classification: 35K40, 93B05