On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form ∂y ∂n +f(y)=0 and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is appro...

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Bibliographic Details
Published in:Journal of Differential Equations 2004-01, Vol.196 (2), p.385-417
Main Authors: Doubova, A., Fernández-Cara, E., González-Burgos, M.
Format: Article
Language:English
Subjects:
Controllability
Heat equation
Nonlinear boundary conditions
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Summary:In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form ∂y ∂n +f(y)=0 and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2003.09.002