%0 Journal Article
%A Doubova, Anna
%A Fernández Cara, E.
%A González Burgos, Manuel
%A Zuazua Iriondo, Enrique
%T On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient
%D 2002
%@ 0363-0129
%U https://hdl.handle.net/20.500.14352/56999
%X We present some results concerning the controllability of a quasi-linear parabolic equation (with linear principal part) in a bounded domain of ${\mathbb R}^N$ 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 $f( y, \nabla y)$ grows slower than $|y| \log^{3/2}(1+ |y| + |\nabla y|) + |\nabla y| \log^{1/2}(1+ |y| + |\nabla y|)$ 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.
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