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Controllability of the linear elasticity as a first-order system using a stabilized space-time mixed formulation

  • * Corresponding author: Arthur Bottois

    * Corresponding author: Arthur Bottois 
Abstract Full Text(HTML) Figure(9) / Table(2) Related Papers Cited by
  • The aim of this paper is to study the boundary controllability of the linear elasticity system as a first-order system in both space and time. Using the observability inequality known for the usual second-order elasticity system, we deduce an equivalent observability inequality for the associated first-order system. Then, the control of minimal $ L^2 $-norm can be found as the solution to a space-time mixed formulation. This first-order framework is particularly interesting from a numerical perspective since it is possible to solve the space-time mixed formulation using only piecewise linear $ C^0 $-finite elements. Numerical simulations illustrate the theoretical results.

    Mathematics Subject Classification: 35Q93, 49K20, 74B05.

    Citation:

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  • Figure 1.  Domains $ Q $ and associated meshes. (a) A structured mesh of $ Q = (0, 1)^2 \times (0, T) $. (b) An example of non-convex domain $ \Omega $ and (c) a mesh of $ Q = \Omega \times (0, T) $ associated to this domain

    Figure 2.  Initial datum $ (u^0, u^1) $ constructed in [12]

    Figure 3.  Results for initial datum $ (u^0, u^1) $ displayed in Figure 2. (a) Evolution of the norm residuals $ ( \boldsymbol{g}_{n}, \boldsymbol{G}_{n}) $. (b) Norm $ L^{2} $ of the control $ \boldsymbol{h}(t) $

    Figure 4.  Norm of the error between the exact and numerical controls for different values of $ h $ and two different values of $ \alpha_1 $

    Figure 5.  Solution $ (\zeta_{n}, {\bf{\Theta}}_{n}) $ obtained once the conjugate algorithm converged for the mesh 5. (a) $ \zeta_{n} $. (b) $ \Theta_{n, 1} $. (c) $ \Theta_{n, 2} $

    Figure 6.  Evolution with respect to the time $ t $ of the norms of primal and dual solutions for: (a) the wave equation; (b) the elasticity system and the initial data in Figure 2

    Figure 7.  Norm of the control $ \boldsymbol{h} $ for the five meshes described in Table 1 computed from $ ( \boldsymbol{\zeta}_{n}, {\bf{\Theta}}_{n}) $ (left) and from $ ( \boldsymbol{w}_{n}, \boldsymbol{Q}_{n}) $ (right), for $ \alpha_{1} = 10^{-3} $ (up) and $ \alpha_{1} = 9\times 10^{-1} $ (bottom), respectively

    Figure 8.  The six components of the solution for the initial datum in Figure 2 and the finest mesh in Table 1. (a) $ \zeta_{n, 1} $. (b) $ \Theta_{n, 11} $. (c) $ \Theta_{n, 12} $. (d) $ \zeta_{n, 2} $. (e) $ \Theta_{n, 21} $. (f) $ \Theta_{n, 22 } $

    Figure 9.  Norm of the control for initial data given by (74). (a) $ \alpha_2 = 10^{-3} $ and different meshes. (b) Computation on the mesh $ \sharp 5 $ and different values for $ \alpha_2 $

    Table 1.  Description of five meshes of the domain $ Q = (0, 1)^2 \times (0, T) $

    Mesh number 1 2 3 4 5
    Diameter $ h $ of elements $ \frac{1}{10} $ $ \frac{1}{20} $ $ \frac{1}{30} $ $ \frac{1}{40} $ $ \frac{1}{50} $
    Number of nodes 3 267 23 814 76 880 179 867 345 933
    Number of tetrahedra 15 600 127 200 426 600 1 017 600 1 980 000
     | Show Table
    DownLoad: CSV

    Table 2.  Description of five meshes of the domain $ Q = \Omega \times (0, T) $ for $ \Omega $ displayed in Figure 1 (b)

    Mesh number 1 2 3 4 5
    Diameter $ h $ of elements $ \frac{1}{10} $ $ \frac{1}{20} $ $ \frac{1}{30} $ $ \frac{1}{40} $ $ \frac{1}{50} $
    Number of nodes 4 557 29 707 99 094 212 234 406 945
    Number of tetrahedra 21 510 155 700 515 080 1 185 760 2 303 550
     | Show Table
    DownLoad: CSV
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