A mixed formulation for the direct approximation of?the?control of minimal \(L^2\)

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• 作者：Nicolae C?ndea ; Arnaud Münch
• 关键词：Linear wave equation ; Null controllability ; Finite elements methods ; Mixed formulation ; 35L10 ; 65M12 ; 93B40
• 刊名：Calcolo
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：52
• 期：3
• 页码：245-288
• 全文大小：2,187 KB
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• 作者单位：Nicolae C?ndea (1)
  Arnaud Münch (1)
  
  1. Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand 2), UMR CNRS 6620, Campus de Cézeaux, 63171?, Aubière, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Theory of Computation
  
• 出版者：Springer Milan
• ISSN：1126-5434

文摘

This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [C?ndea, Fernández-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013], a so called primal method is described leading to a strongly convergent approximation of boundary controls: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable 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 controllability. For simplicity, we present the approach in the one dimensional case. Keywords Linear wave equation Null controllability Finite elements methods Mixed formulation