Null controllability of a coupled degenerate system with the first and zero order terms by a single control

¹ School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P.R. China.
² School of Science, East China University of Technology, Nanchang 330013, P.R. China.

^* Corresponding author. binwu@nuist.edu.cn

Received: 15 April 2018
Accepted: 8 July 2020

Abstract

This paper concerns the null controllability of a system of m linear degenerate parabolic equations with coupling terms of first and zero order, and only one control force localized in some arbitrary nonempty open subset ω of Ω. The key ingredient for proving the null controllability is to obtain the observability inequality for the corresponding adjoint system. Due to the degeneracy, we transfer to study an approximate nondegenerate adjoint system. In order to deal with the coupling first order terms, we first prove a new Carleman estimate for a degenerate parabolic equation in Sobolev spaces of negative order. Based on this Carleman estimate, we obtain a uniform Carleman estimate and then an observation inequality for this approximate adjoint system.