Null controllability of the heat equation in pseudo-cylinders by an internal control

Sorbonne Université, Laboratoire Jacques-Louis Lions, 75005 Paris, France.

^* Corresponding author: barcena@ljll.math.upmc.fr

Received: 1 June 2020
Accepted: 16 July 2020

Abstract

In this paper we prove the null controllability of the heat equation in domains with a cylindrical part and limited by a Lipschitz graph. The proof consists mainly on getting a Carleman estimate which presents the usual absorption properties. The main difficulty we face is the loss of existence of the usual weighted function in C² smooth domains. In order to deal with this, we use its cylindrical structure and approximate the system by the same system stated in regular domains. Finally, we show some applications like the controllability of the semi-linear heat equation in those domains.