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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
• 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:

• 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

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
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