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

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 $