Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force

Abstract

In this paper we will analyze the controllability properties of a linear coupled parabolic system of $m$ equations when a unique distributed control is exerted on the system. We will see that, when a cascade system is considered, we can prove a global Carleman inequality for the adjoint system which bounds the global integrals of the variable $\phi =({\phi }_1,\dots ,{\phi }_m{)}^{\ast }$ in terms of a unique localized variable. As a consequence, we will obtain the null controllability property for the system with one control force.