# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2019030

## 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

Received  May 2018 Published  April 2019

Fund Project: 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.

Citation: Santiago Montaner, Arnaud Münch. Approximation of controls for linear wave equations: A first order mixed formulation. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019030
##### References:

show all references

##### References:
Regular meshes for $Q_T$; Left: uniform mesh - $h = 1.41\times10^{-1}$. Right: non uniform mesh - $h = 1.52\times 10^{-1}$
Non uniform mesh - Evolution of $\sqrt{hr}\delta_h$ with respect to $h$ (see Table 3) for $r = 1$ $({\bigcirc})$, $r = 10^{-1}$ $(\bigtriangledown)$, $r = 10^{-2}$ $({\bigtriangleup})$, $r = h$ $({\star})$, $r = h^2$ $({\circ)}$
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 $r = h$ $(\bigtriangledown)$
Iterative refinement of the triangular mesh over $Q_T$ with respect to the variable $\lambda_h$: $110$, $2\, 880$ and $8\, 636$ triangles
The primal variable $\lambda_h$ in $Q_T$ - Third adapted mesh in Figure 4, $r = 10^{-6}$.
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, $r = 10^{-6}$
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 $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$
 $N$ $10$ $20$ $40$ $80$ $160$ 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 $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$
$\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $W_h$-$M_h$ finite elements and non uniform mesh
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $0.264$ $0.197$ $0.132$ $0.099$ $0.070$ $r=10^{-1}$ $0.751$ $0.569$ $0.412$ $0.310$ $0.222$ $r=10^{-2}$ $1.881$ $1.478$ $1.112$ $0.839$ $0.627$ $r=h$ $0.652$ $0.660$ $0.660$ $0.679$ $0.661$ $r=h^2$ $1.397$ $1.934$ $2.642$ $3.636$ $5.031$
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $0.264$ $0.197$ $0.132$ $0.099$ $0.070$ $r=10^{-1}$ $0.751$ $0.569$ $0.412$ $0.310$ $0.222$ $r=10^{-2}$ $1.881$ $1.478$ $1.112$ $0.839$ $0.627$ $r=h$ $0.652$ $0.660$ $0.660$ $0.679$ $0.661$ $r=h^2$ $1.397$ $1.934$ $2.642$ $3.636$ $5.031$
$\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $W_h$-$M_h$ finite elements and non uniform mesh.
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$ $r=1$ $0.426$ $0.316$ $0.229$ $0.155$ $0.106$ $r=10^{-1}$ $0.991$ $0.868$ $0.698$ $0.489$ $0.339$ $r=10^{-2}$ $2.269$ $1.738$ $1.373$ $1.099$ $0.896$ $r=h$ $0.885$ $0.927$ $0.929$ $0.921$ $0.908$ $r=h^2$ $1.612$ $2.154$ $2.974$ $4.115$ $5.733$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$ $r=1$ $0.426$ $0.316$ $0.229$ $0.155$ $0.106$ $r=10^{-1}$ $0.991$ $0.868$ $0.698$ $0.489$ $0.339$ $r=10^{-2}$ $2.269$ $1.738$ $1.373$ $1.099$ $0.896$ $r=h$ $0.885$ $0.927$ $0.929$ $0.921$ $0.908$ $r=h^2$ $1.612$ $2.154$ $2.974$ $4.115$ $5.733$
$\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $\widetilde{W_h}$-$M_h$ finite elements and uniform mesh
 $h$ $1.41\times 10^{-1}$ $7.01\times 10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $5.77\times 10^{-5}$ $1.41\times 10^{-10}$ $1.8\times 10^{-10}$ $4.07\times 10^{-9}$ $1.97\times 10^{-10}$ $r=10^{-1}$ $2.45\times 10^{-9}$ $5.17\times 10^{-10}$ $2.23\times 10^{-10}$ $2.05\times 10^{-9}$ $1.63\times 10^{-9}$ $r=10^{-2}$ $1.51\times 10^{-8}$ $1.71\times 10^{-9}$ $3.88\times 10^{-9}$ $1.77\times 10^{-8}$ $8.11\times 10^{-9}$ $r=h$ $2.4\times 10^{-9}$ $1.05\times 10^{-9}$ $7.77\times 10^{-10}$ $6.73\times 10^{-9}$ $1.64\times 10^{-9}$ $r=h^2$ $4.92\times 10^{-9}$ $4.19\times 10^{-9}$ $2.6\times 10^{-9}$ $3.33\times 10^{-9}$ $1.44\times 10^{-9}$
 $h$ $1.41\times 10^{-1}$ $7.01\times 10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $5.77\times 10^{-5}$ $1.41\times 10^{-10}$ $1.8\times 10^{-10}$ $4.07\times 10^{-9}$ $1.97\times 10^{-10}$ $r=10^{-1}$ $2.45\times 10^{-9}$ $5.17\times 10^{-10}$ $2.23\times 10^{-10}$ $2.05\times 10^{-9}$ $1.63\times 10^{-9}$ $r=10^{-2}$ $1.51\times 10^{-8}$ $1.71\times 10^{-9}$ $3.88\times 10^{-9}$ $1.77\times 10^{-8}$ $8.11\times 10^{-9}$ $r=h$ $2.4\times 10^{-9}$ $1.05\times 10^{-9}$ $7.77\times 10^{-10}$ $6.73\times 10^{-9}$ $1.64\times 10^{-9}$ $r=h^2$ $4.92\times 10^{-9}$ $4.19\times 10^{-9}$ $2.6\times 10^{-9}$ $3.33\times 10^{-9}$ $1.44\times 10^{-9}$
$r = h$ - non uniform mesh
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.543$ $0.556$ $0.564$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.85\times 10^{-2}$ $2.49\times 10^{-2}$ $1.63\times 10^{-2}$ $1.06\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.538$ $0.555$ $0.564$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $5.27\times 10^{-2}$ $3.37\times 10^{-2}$ $2.18\times 10^{-2}$ $1.41\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.645$ $0.462$ $0.331$ $0.239$
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.543$ $0.556$ $0.564$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.85\times 10^{-2}$ $2.49\times 10^{-2}$ $1.63\times 10^{-2}$ $1.06\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.538$ $0.555$ $0.564$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $5.27\times 10^{-2}$ $3.37\times 10^{-2}$ $2.18\times 10^{-2}$ $1.41\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.645$ $0.462$ $0.331$ $0.239$
$r = h$ - uniform mesh
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.535$ $0.549$ $0.559$ $0.566$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.43\times 10^{-2}$ $2.22\times 10^{-2}$ $1.45\times 10^{-2}$ $9.43\times 10^{-3}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.545$ $0.558$ $0.566$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.89\times 10^{-2}$ $2.3\times 10^{-2}$ $1.46\times 10^{-2}$ $9.35\times 10^{-3}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.561$ $0.388$ $0.265$ $0.184$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.535$ $0.549$ $0.559$ $0.566$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.43\times 10^{-2}$ $2.22\times 10^{-2}$ $1.45\times 10^{-2}$ $9.43\times 10^{-3}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.545$ $0.558$ $0.566$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.89\times 10^{-2}$ $2.3\times 10^{-2}$ $1.46\times 10^{-2}$ $9.35\times 10^{-3}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.561$ $0.388$ $0.265$ $0.184$
$r = 10^{-6}$ - 4 adaptive meshes. Figure 4 displays the $1$st, $3$rd and $4th$ adaptive meshes used.
 $\sharp\ \text{triangles}$ $110$ $1197$ $2880$ $8636$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.46$ $0.57$ $0.574$ $0.577$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $8.24\times10^{-2}$ $1.55\times10^{-2}$ $3.72\times10^{-3}$ $5.18\times10^{-4}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.451$ $0.569$ $0.574$ $0.577$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $8.04\times10^{-2}$ $1.52\times10^{-2}$ $3.88\times10^{-3}$ $4.48\times10^{-4}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $1.13\times10^5$ $4.45\times10^4$ $1.48\times10^4$ $2.86\times10^3$
 $\sharp\ \text{triangles}$ $110$ $1197$ $2880$ $8636$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.46$ $0.57$ $0.574$ $0.577$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $8.24\times10^{-2}$ $1.55\times10^{-2}$ $3.72\times10^{-3}$ $5.18\times10^{-4}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.451$ $0.569$ $0.574$ $0.577$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $8.04\times10^{-2}$ $1.52\times10^{-2}$ $3.88\times10^{-3}$ $4.48\times10^{-4}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $1.13\times10^5$ $4.45\times10^4$ $1.48\times10^4$ $2.86\times10^3$
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 $36$ $43$ $56$ $80$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.566$ $0.574$ $0.573$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $2.39\times 10^{-1}$ $1.46\times 10^{-1}$ $1.19\times 10^{-1}$ $9.54\times 10^{-2}$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\sharp$ 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 $36$ $43$ $56$ $80$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.566$ $0.574$ $0.573$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $2.39\times 10^{-1}$ $1.46\times 10^{-1}$ $1.19\times 10^{-1}$ $9.54\times 10^{-2}$
$r = h$ - non uniform mesh - stabilized formulation with $\alpha = 0.5$.
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.444$ $0.494$ $0.522$ $0.539$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $7.47\times 10^{-2}$ $5.21\times 10^{-2}$ $3.65\times 10^{-2}$ $2.56\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.525$ $0.543$ $0.554$ $0.561$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $1.26\times 10^{-1}$ $6.5\times 10^{-2}$ $4.2\times 10^{-2}$ $2.79\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.423$ $0.343$ $0.281$ $0.235$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.444$ $0.494$ $0.522$ $0.539$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $7.47\times 10^{-2}$ $5.21\times 10^{-2}$ $3.65\times 10^{-2}$ $2.56\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.525$ $0.543$ $0.554$ $0.561$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $1.26\times 10^{-1}$ $6.5\times 10^{-2}$ $4.2\times 10^{-2}$ $2.79\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.423$ $0.343$ $0.281$ $0.235$
$r = h$ - non uniform mesh - stabilized formulation with $\alpha = h^2$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.456$ $0.498$ $0.523$ $0.54$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $6.99\times 10^{-2}$ $4.99\times 10^{-2}$ $3.56\times 10^{-2}$ $2.54\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.511$ $0.536$ $0.55$ $0.559$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $7.2\times 10^{-2}$ $5.06\times 10^{-2}$ $3.59\times 10^{-2}$ $2.55\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.474$ $0.363$ $0.29$ $0.238$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.456$ $0.498$ $0.523$ $0.54$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $6.99\times 10^{-2}$ $4.99\times 10^{-2}$ $3.56\times 10^{-2}$ $2.54\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.511$ $0.536$ $0.55$ $0.559$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $7.2\times 10^{-2}$ $5.06\times 10^{-2}$ $3.59\times 10^{-2}$ $2.55\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.474$ $0.363$ $0.29$ $0.238$
 [1] Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845 [2] Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353 [3] Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143 [4] Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216 [5] Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2019, 0 (0) : 1-32. doi: 10.3934/eect.2020011 [6] David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35 [7] Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257 [8] David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131 [9] Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019031 [10] Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607 [11] Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001 [12] Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212 [13] Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901 [14] Irina F. Sivergina, Michael P. Polis. About global null controllability of a quasi-static thermoelastic contact system. Conference Publications, 2005, 2005 (Special) : 816-823. doi: 10.3934/proc.2005.2005.816 [15] Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699 [16] Nicolas Hegoburu, Marius Tucsnak. Null controllability of the Lotka-McKendrick system with spatial diffusion. Mathematical Control & Related Fields, 2018, 8 (3&4) : 707-720. doi: 10.3934/mcrf.2018030 [17] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [18] Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91 [19] Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 [20] Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419

2018 Impact Factor: 1.292

## Tools

Article outline

Figures and Tables