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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 φ = (φ1, . . . , φm) ∗ in terms of a unique localized variable. As a consequence, we will obtain the null controllability property for the system with one control force. Mathematics Subject Classification (2000). 93B05, 93B07, 35K50.

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