A Generalization of the Kalman Rank Condition for Time–dependent Coupled Linear Parabolic Systems

In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of n coupled parabolic equations (posed in the time interval (0,T ) with T > 0) where the coupling matrices A and B depend on the time variable t . To be precise, we will prove that the Kalman rank condition rank [A|B](t0) = n , with t0 ∈ [0,T ] , is a sufficient condition (but not necessary) for obtaining the exact controllability to the trajectories of the considered parabolic system. In the case of analytic matrices A and B (and, in particular, constant matrices), we will see that the Kalman rank condition characterizes the controllability properties of the system. When the matrices A and B are constant and condition rank [A |B] = n holds, we will be able to state a Carleman inequality for the corresponding adjoint problem. Mathematics subject classification (2000): 93B05, 93B07, 35K05, 35K55, 35R30.