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Article Contents

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

• * Corresponding author: Arnaud Münch

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)

• 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.

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

 Citation:

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

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

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 $r = h$ $(\bigtriangledown)$

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

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

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, $r = 10^{-6}$

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 $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$

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

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

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

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

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

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

Table 5.  $r = h$ - non uniform mesh

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

Table 6.  $r = h$ - uniform mesh

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

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

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

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

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

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

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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