A mixed formulation for the direct approximation of L 2-weighted controls for the linear heat equation

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• 作者：Arnaud Münch ; Diego A. Souza
• 关键词：Linear heat equation ; Null controllability ; Finite element methods ; Mixed formulation ; 35K35 ; 65M12 ; 93B40 ; 65K10
• 刊名：Advances in Computational Mathematics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：42
• 期：1
• 页码：85-125
• 全文大小：3,389 KB
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• 作者单位：Arnaud Münch (1)
  Diego A. Souza (2) (3)
  
  1. Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand 2), UMR CNRS 6620, Campus de Cézeaux, 63177, Aubière, France
  2. Departamento EDAN, University of Sevilla, 41080, Sevilla, Spain
  3. Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa–PB, Brazil
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Calculus of Variations and Optimal Control
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9044

文摘

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension. Keywords Linear heat equation Null controllability Finite element methods Mixed formulation