# American Institute of Mathematical Sciences

September  2012, 2(3): 217-246. doi: 10.3934/mcrf.2012.2.217

## Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods

 1 Dpto. EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain 2 Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand), UMR CNRS 6620, Campus des Cézeaux, 63177 Aubière, France

Received  November 2011 Revised  May 2012 Published  August 2012

This paper deals with the numerical computation of distributed null controls for semi-linear 1D heat equations, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fernandez-Cara $\&$ Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani's Theorem together ensure the existence of solutions to fixed points for an equivalent fixed point reformulated problem. A nontrivial difficulty appears when we want to extract from the associated Picard iterates a convergent (sub)sequence. In this paper, we introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points. We also formulate and apply a Newton-Raphson algorithm in this context. Several numerical experiments that make it possible to test and compare these methods are performed.
Citation: Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217
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