Artículo
Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods
Autor/es | Fernández Cara, Enrique
Münch, Arnaud |
Departamento | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Fecha de publicación | 2012-09 |
Fecha de depósito | 2016-07-07 |
Publicado en |
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Resumen | This paper deals with the numerical computation of distributed null controls for semilinear 1D heat equations, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has ... This paper deals with the numerical computation of distributed null controls for semilinear 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. |
Agencias financiadoras | Dirección General de Enseñanza Superior. España Agence Nationale de la Recherche. France |
Identificador del proyecto | MTM2006-07932
MTM2010-15592 ANR-07-JC-183284 |
Cita | Fernández Cara, E. y Münch, A. (2012). Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods. Mathematical Control and Related Fields, 2 (3), 217-246. |
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