No abstract

International audienceThis paper tries to summarize recent results on the controllability of systems of (several) parabolic equations. The emphasis is placed on the extension of the Kalman rank condition (for finite dimensional systems of differential equations) to parabolic systems. This question is itself tied with the proof of global Carleman estimates for systems and leads to a wide field of open problems

International audienceThis paper is devoted to prove the controllability to trajectories of a system of $n$ one-dimensional parabolic equations when the control is exerted on a part of the boundary by means of $m$ controls. We give a general \textit{Kalman condition} (necessary and sufficient) and also present a construction and sharp estimates of a biothorgonal family in $L^{2}(0,T; \C)$ to $\{ t^{j}e^{-\Lambda_{k}t}\}$.Cet article a pour but de démontrer la contrôlabilité des trajectoires d’un système de $n$ équations paraboliques en une dimension d’espace par $m$ contrôles exercés sur une partie du bord. On obtient une condition de Kalman (nécessaire et suffisante). La démonstration passe par la construction dans $L^{2}(0,T; \C)$ d’une famille biorthogonale de la suite $\{ t^{j}e^{-\Lambda_{k}t}\}$ et par une estimation de la norme de ses éléments

We present a generalization of the Kalman rank condition to the case of n×n linear parabolic systems with constant coefficients and diagonalizable diffusion matrix. To reach the result, we are led to prove a global Carleman estimate for the solutions of a scalar 2n−order parabolic equation and deduce from it an observability inequality for our adjoint system. AMS 2000 subject classification: 93B05, 93B07, 35K05, 35K55, 35R30.