We present some results concerning the controllability of a quasi-linear parabolic equation (with linear principal part) in a bounded domain of R N with Dirichlet boundary conditions. We analyze the controllability problem with distributed controls (supported on a small open subset) and boundary controls (supported on a small part of the boundary). We prove that the system is null and approximately controllable at any time if the nonlinear term f (y, ∇y) grows slower than |y| log 3/2 (1 + |y| + |∇y|) + |∇y| log 1/2 (1 + |y| + |∇y|) at infinity (generally, in this case, in the absence of control, blow-up occurs). The proofs use global Carleman estimates, parabolic regularity, and the fixed point method.

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

This paper is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional parabolic equations are considered. We show that, in this framework, boundary controllability is not equivalent and is more complex than distributed controllability. In our main result, we provide necessary and sufficient conditions for the null controllability.

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

Abstract.In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂y ∂n + β y = 0. We consider distributed controls with support in a small set and nonregular coefficients β = β(x, t). For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions. Mathematics Subject Classification. 35K20, 93B05.