# Exact controllability of semilinear heat equations in spaces of analytic functions

### Camille Laurent

CNRS, Sorbonne Université, Laboratoire Jacques-Louis Lions, UMR 7598, Boite courrier 187, 75252, Paris Cedex 05, France### Lionel Rosier

ULCO, LMPA J. Liouville, BP 699, F-62228 Calais, France, CNRS FR 2956, France

## Abstract

It is by now well known that the use of Carleman estimates allows to establish the controllability 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 semilinear 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 $∂_{x}y$. 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.

## Cite this article

Camille Laurent, Lionel Rosier, Exact controllability of semilinear heat equations in spaces of analytic functions. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 4, pp. 1047–1073

DOI 10.1016/J.ANIHPC.2020.03.001