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A mixed formulation for the direct approximation of the control of minimal \(L^2\)-norm for linear type wave equations

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Abstract

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 [Cîndea, 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.

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Notes

  1. The condition number \(\kappa (\mathcal {M}_h)\) of any square matrix \(\mathcal {M}_h\) is defined by \(\kappa (\mathcal {M}_h)=\vert \vert \vert \mathcal {M}_h\vert \vert \vert _2 \vert \vert \vert \mathcal {M}_h^{-1}\vert \vert \vert _2\) where the norm \(\vert \vert \vert \mathcal {M}_h\vert \vert \vert _2\) stands for the largest singular value of \(\mathcal {M}_h\).

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand 2), UMR CNRS 6620, Campus de Cézeaux, 63171 , Aubière, France

    Nicolae Cîndea & Arnaud Münch

Authors
  1. Nicolae Cîndea
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  2. Arnaud Münch
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Correspondence to Arnaud Münch.

Appendix: numerical tables

Appendix: numerical tables

See Tables 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, and 37.

Table 25 Example EX1—BFS element—\(r=10^{-4}\)
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Table 26 Example EX1—BFS element—\(r=h^2\)
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Table 27 Example EX1—BFS element—\(r=10^2\)
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Table 28 Example EX1—HCT element—uniform mesh—\(r=10\)
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Table 29 Example EX1—HCT element—uniform mesh—\(r=h^2\)
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Table 30 Example EX1—HCT element—uniform mesh—\(r=10^{-4}\)
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Table 31 Example EX1—HCT element—non uniform mesh—\(r=1\)
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Table 32 Example EX1—HCT element—non uniform mesh—\(r=10^{-2}\)
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Table 33 Example EX2—HCT element—uniform mesh—\(r=1\)
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Table 34 Example EX2—HCT element—uniform mesh—\(r=10^{-2}\)
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Table 35 Example EX3—BFS element—\(r=10^{-2}\)
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Table 36 Example EX3—HCT element—uniform mesh—\(r=10^{-2}\)
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Table 37 Example EX3—HCT element—non uniform mesh—\(r=10^{-2}\)
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Cîndea, N., Münch, A. A mixed formulation for the direct approximation of the control of minimal \(L^2\)-norm for linear type wave equations. Calcolo 52, 245–288 (2015). https://doi.org/10.1007/s10092-014-0116-x

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