Approximation of controls for linear wave equations: A first order mixed formulation

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doi:10.3934/mcrf.2019030

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2019, Volume 9, Issue 4: 729-758. Doi: 10.3934/mcrf.2019030

This issue Previous Article On the null controllability of the Lotka-Mckendrick system Next Article Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

Approximation of controls for linear wave equations: A first order mixed formulation

Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, UMR CNRS 6620, Campus des Cézeaux, 63178 Aubière, France

^* Corresponding author: Arnaud Münch
^* Corresponding author: Arnaud Münch

Received: April 30, 2018

Published: November 2019

This work has been sponsored by the French government research program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25)

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Abstract

This paper deals with the numerical approximation of null controls for the wave equation posed in a bounded domain of $ \mathbb{R}^n $. 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 [Cindea & Münch, A mixed formulation for the direct approximation of the control of minimal $ L^2 $-norm for linear type wave equations], we have introduced a space-time variational approach ensuring strong convergent approximations with respect to the discretization parameter. The method, which relies on generalized observability inequality, requires $ H^2 $-finite element approximation both in time and space. Following a similar approach, we present and analyze a variational method still leading to strong convergent results but using simpler $ H^1 $-approximation. The main point is to preliminary restate the second order wave equation into a first order system and then prove an appropriate observability inequality.

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Mathematics Subject Classification: Primary: 35B37, 93B05; Secondary: 65N06.

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Figure 1. Regular meshes for $ Q_T $; Left: uniform mesh - $ h = 1.41\times10^{-1} $. Right: non uniform mesh - $ h = 1.52\times 10^{-1} $

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Figure 2. Non uniform mesh - Evolution of $ \sqrt{hr}\delta_h $ with respect to $ h $ (see Table 3) for $ r = 1 $ $ ({\bigcirc}) $, $ r = 10^{-1} $ $ (\bigtriangledown) $, $ r = 10^{-2} $ $ ({\bigtriangleup}) $, $ r = h $ $ ({\star}) $, $ r = h^2 $ $ ({\circ)} $

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Figure 3. Evolution of $ \|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)} $ w.r.t $ h $ for uniform mesh with $ r = 1 $ $ (\circ) $, $ r = h $ $ (\star) $ and non uniform mesh with $ r = 1 $ $ (\bigtriangleup) $ and $ r = h $ $ (\bigtriangledown) $

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Figure 4. Iterative refinement of the triangular mesh over $ Q_T $ with respect to the variable $ \lambda_h $: $ 110 $, $ 2\, 880 $ and $ 8\, 636 $ triangles

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Figure 5. The primal variable $ \lambda_h $ in $ Q_T $ - Third adapted mesh in Figure 4, $ r = 10^{-6} $.

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Figure 6. Control of minimal $ L^2 $-norm $ v $ (dashed blue line) and its approximation $ \lambda_h(1, \cdot) $ (red line) on $ (0, T) $. Third adapted mesh in Figure 4, $ r = 10^{-6} $

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Table 1. Number of elements for the uniform(u)/non uniform(nu) meshes and value of $h$ for each type of mesh w.r.t. $N$ with $T = 2$.

$N$	$10$	$20$	$40$	$80$	$160$
card$(\mathcal{T}_h)$-u	$400$	$1\, 600$	$6\, 400$	$25\, 600$	$102\, 400$
card$(\mathcal{T}_h)$-nu	$446$	$1\, 784$	$7\, 136$	$28\, 544$	$114\, 176$
$\sharp$ nodes-u	$861$	$3\, 321$	$13\, 041$	$56\, 681$	$205\, 761$
$\sharp$ nodes-nu	$953$	$3\, 689$	$14\, 513$	$57\, 569$	$229\, 313$
$h$-u	$1.41\times 10^{-1}$	$7.01\times 10^{-2}$	$3.53\times 10^{-2}$	$1.76\times 10^{-2}$	$8.83\times 10^{-3}$
$h$-nu	$1.52\times 10^{-1}$	$7.60\times 10^{-2}$	$3.80\times 10^{-2}$	$1.90\times 10^{-2}$	$9.50\times 10^{-3}$

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Table 2. $ \delta_h $ w.r.t. $ r $ and $ h $, $ T = 2 $, for the $ W_h $-$ M_h $ finite elements and non uniform mesh

$ h $	$ 1.41\times10^{-1} $	$ 7.01\times10^{-2} $	$ 3.53\times 10^{-2} $	$ 1.76\times 10^{-2} $	$ 8.83\times 10^{-3} $
$ r=1 $	$ 0.264 $	$ 0.197 $	$ 0.132 $	$ 0.099 $	$ 0.070 $
$ r=10^{-1} $	$ 0.751 $	$ 0.569 $	$ 0.412 $	$ 0.310 $	$ 0.222 $
$ r=10^{-2} $	$ 1.881 $	$ 1.478 $	$ 1.112 $	$ 0.839 $	$ 0.627 $
$ r=h $	$ 0.652 $	$ 0.660 $	$ 0.660 $	$ 0.679 $	$ 0.661 $
$ r=h^2 $	$ 1.397 $	$ 1.934 $	$ 2.642 $	$ 3.636 $	$ 5.031 $

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Table 3. $\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $W_h$-$M_h$ finite elements and non uniform mesh.

$h$	$1.52\times 10^{-1}$	$7.60\times 10^{-2}$	$3.80\times 10^{-2}$	$1.90\times 10^{-2}$	$9.50\times 10^{-3}$
$r=1$	$0.426$	$0.316$	$0.229$	$0.155$	$0.106$
$r=10^{-1}$	$0.991$	$0.868$	$0.698$	$0.489$	$0.339$
$r=10^{-2}$	$2.269$	$1.738$	$1.373$	$1.099$	$0.896$
$r=h$	$0.885$	$0.927$	$0.929$	$0.921$	$0.908$
$r=h^2$	$1.612$	$2.154$	$2.974$	$4.115$	$5.733$

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Table 4. $ \delta_h $ w.r.t. $ r $ and $ h $, $ T = 2 $, for the $ \widetilde{W_h} $-$ M_h $ finite elements and uniform mesh

$ h $	$ 1.41\times 10^{-1} $	$ 7.01\times 10^{-2} $	$ 3.53\times 10^{-2} $	$ 1.76\times 10^{-2} $	$ 8.83\times 10^{-3} $
$ r=1 $	$ 5.77\times 10^{-5} $	$ 1.41\times 10^{-10} $	$ 1.8\times 10^{-10} $	$ 4.07\times 10^{-9} $	$ 1.97\times 10^{-10} $
$ r=10^{-1} $	$ 2.45\times 10^{-9} $	$ 5.17\times 10^{-10} $	$ 2.23\times 10^{-10} $	$ 2.05\times 10^{-9} $	$ 1.63\times 10^{-9} $
$ r=10^{-2} $	$ 1.51\times 10^{-8} $	$ 1.71\times 10^{-9} $	$ 3.88\times 10^{-9} $	$ 1.77\times 10^{-8} $	$ 8.11\times 10^{-9} $
$ r=h $	$ 2.4\times 10^{-9} $	$ 1.05\times 10^{-9} $	$ 7.77\times 10^{-10} $	$ 6.73\times 10^{-9} $	$ 1.64\times 10^{-9} $
$ r=h^2 $	$ 4.92\times 10^{-9} $	$ 4.19\times 10^{-9} $	$ 2.6\times 10^{-9} $	$ 3.33\times 10^{-9} $	$ 1.44\times 10^{-9} $

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Table 5. $ r = h $ - non uniform mesh

$ h $	$ 1.41\times10^{-1} $	$ 7.01\times10^{-2} $	$ 3.53\times 10^{-2} $	$ 1.76\times 10^{-2} $
$ \\|\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)} $	$ 0.523 $	$ 0.543 $	$ 0.556 $	$ 0.564 $
$ \\|v-\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)} $	$ 3.85\times 10^{-2} $	$ 2.49\times 10^{-2} $	$ 1.63\times 10^{-2} $	$ 1.06\times 10^{-2} $
$ \\|\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 0.538 $	$ 0.555 $	$ 0.564 $	$ 0.57 $
$ \\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 5.27\times 10^{-2} $	$ 3.37\times 10^{-2} $	$ 2.18\times 10^{-2} $	$ 1.41\times 10^{-2} $
$ \\|M(w_h, \mathbf{q}_h)\\|_{L^2(Q_T)} $	$ 0.645 $	$ 0.462 $	$ 0.331 $	$ 0.239 $

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Table 6. $r = h$ - uniform mesh

$h$	$1.52\times 10^{-1}$	$7.60\times 10^{-2}$	$3.80\times 10^{-2}$	$1.90\times 10^{-2}$
$\\|\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)}$	$0.535$	$0.549$	$0.559$	$0.566$
$\\|v-\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)}$	$3.43\times 10^{-2}$	$2.22\times 10^{-2}$	$1.45\times 10^{-2}$	$9.43\times 10^{-3}$
$\\|\lambda_h(1, \cdot)\\|_{L^2(0, T)}$	$0.545$	$0.558$	$0.566$	$0.57$
$\\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)}$	$3.89\times 10^{-2}$	$2.3\times 10^{-2}$	$1.46\times 10^{-2}$	$9.35\times 10^{-3}$
$\\|M(w_h, \mathbf{q}_h)\\|_{L^2(Q_T)}$	$0.561$	$0.388$	$0.265$	$0.184$

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Table 7. $r = 10^{-6}$ - 4 adaptive meshes. Figure 4 displays the $1$st, $3$rd and $4th$ adaptive meshes used.

$\sharp\ \text{triangles}$	$110$	$1197$	$2880$	$8636$
$\\|\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)}$	$0.46$	$0.57$	$0.574$	$0.577$
$\\|v-\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)}$	$8.24\times10^{-2}$	$1.55\times10^{-2}$	$3.72\times10^{-3}$	$5.18\times10^{-4}$
$\\|\lambda_h(1, \cdot)\\|_{L^2(0, T)}$	$0.451$	$0.569$	$0.574$	$0.577$
$\\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)}$	$8.04\times10^{-2}$	$1.52\times10^{-2}$	$3.88\times10^{-3}$	$4.48\times10^{-4}$
$\\|M(w_h, \mathbf{q}_h)\\|_{L^2(Q_T)}$	$1.13\times10^5$	$4.45\times10^4$	$1.48\times10^4$	$2.86\times10^3$

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Table 8. Non uniform mesh - Conjugate gradient method - Number of iterates for $ r = 1 $ (top), $ r = 10^{-2} $ and $ r = h $ (bottom)

$ h $	$ 1.52\times 10^{-1} $	$ 7.60\times 10^{-2} $	$ 3.80\times 10^{-2} $	$ 1.90\times 10^{-2} $
$ \sharp $ iterates	$ 31 $	$ 41 $	$ 54 $	$ 77 $
$ \\|\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 0.469 $	$ 0.576 $	$ 0.589 $	$ 0.586 $
$ \\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 3.21\times 10^{-1} $	$ 1.72\times 10^{-1} $	$ 1.43\times 10^{-1} $	$ 1.25\times 10^{-1} $

$ h $	$ 1.52\times 10^{-1} $	$ 7.60\times 10^{-2} $	$ 3.80\times 10^{-2} $	$ 1.90\times 10^{-2} $
$ \sharp $ iterates	$ 46 $	$ 103 $	$ 125 $	$ 133 $
$ \\|\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 0.55 $	$ 0.566 $	$ 0.569 $	$ 0.571 $
$ \\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 2.05\times 10^{-1} $	$ 1.47\times 10^{-1} $	$ 1.12\times 10^{-1} $	$ 8.71\times 10^{-2} $

$ h $	$ 1.52\times 10^{-1} $	$ 7.60\times 10^{-2} $	$ 3.80\times 10^{-2} $	$ 1.90\times 10^{-2} $
$ \sharp $ iterates	$ 36 $	$ 43 $	$ 56 $	$ 80 $
$ \\|\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 0.523 $	$ 0.566 $	$ 0.574 $	$ 0.573 $
$ \\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 2.39\times 10^{-1} $	$ 1.46\times 10^{-1} $	$ 1.19\times 10^{-1} $	$ 9.54\times 10^{-2} $

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Table 9. $ r = h $ - non uniform mesh - stabilized formulation with $ \alpha = 0.5 $.

$ h $	$ 1.52\times 10^{-1} $	$ 7.60\times 10^{-2} $	$ 3.80\times 10^{-2} $	$ 1.90\times 10^{-2} $
$ \\|\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)} $	$ 0.444 $	$ 0.494 $	$ 0.522 $	$ 0.539 $
$ \\|v-\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)} $	$ 7.47\times 10^{-2} $	$ 5.21\times 10^{-2} $	$ 3.65\times 10^{-2} $	$ 2.56\times 10^{-2} $
$ \\|\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 0.525 $	$ 0.543 $	$ 0.554 $	$ 0.561 $
$ \\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)} $	$ 1.26\times 10^{-1} $	$ 6.5\times 10^{-2} $	$ 4.2\times 10^{-2} $	$ 2.79\times 10^{-2} $
$ \\|M(w_h, \mathbf{q}_h)\\|_{L^2(Q_T)} $	$ 0.423 $	$ 0.343 $	$ 0.281 $	$ 0.235 $

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Table 10. $ r = h $ - non uniform mesh - stabilized formulation with $ \alpha = h^2 $

$h$	$1.52\times 10^{-1}$	$7.60\times 10^{-2}$	$3.80\times 10^{-2}$	$1.90\times 10^{-2}$
$\\|\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)}$	$0.456$	$0.498$	$0.523$	$0.54$
$\\|v-\mathbf{q}_h(1, \cdot)\\|_{L^2(0, T)}$	$6.99\times 10^{-2}$	$4.99\times 10^{-2}$	$3.56\times 10^{-2}$	$2.54\times 10^{-2}$
$\\|\lambda_h(1, \cdot)\\|_{L^2(0, T)}$	$0.511$	$0.536$	$0.55$	$0.559$
$\\|v-\lambda_h(1, \cdot)\\|_{L^2(0, T)}$	$7.2\times 10^{-2}$	$5.06\times 10^{-2}$	$3.59\times 10^{-2}$	$2.55\times 10^{-2}$
$\\|M(w_h, \mathbf{q}_h)\\|_{L^2(Q_T)}$	$0.474$	$0.363$	$0.29$	$0.238$

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