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On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient

Doubova, Anna and Fernández Cara, E. and González Burgos, Manuel and Zuazua Iriondo, Enrique (2002) On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient. SIAM Journal on Control and Optimization, 41 (3). pp. 798-819. ISSN 0363-0129

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Official URL: http://epubs.siam.org/sicon/resource/1/sjcodc/v41/i3/p798_s1

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Abstract

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.

Item Type:Article
Uncontrolled Keywords:Controllability, Parabolic equations, Nonlinear gradient terms
Subjects:Sciences > Mathematics > Numerical analysis
ID Code:12178
Deposited On:07 Feb 2011 08:59
Last Modified:06 Jun 2011 09:41

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