A mixed formulation for the direct approximation of $L^2$-weighted controls for the linear heat equation

TL;DR

The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint and the method is adapted to approximate the control of minimal square integrable-weighted norm.

Abstract

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara \& Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension.

Figures (6)

• Figure 1: $\omega=(0.2,0.5)$; $y_0(x)=\sin(\pi x)$: $\varepsilon=10^{-2}$. ; $\frac{\|\rho_0 (v_{\varepsilon} - v_{\varepsilon,h})\|_{L^2(q_T)}}{\Vert \rho_0 v_{\varepsilon} \Vert_{L^2(q_T)}}$ ( Left) and $\frac{\|y_{\varepsilon} - \lambda_{\varepsilon,h}\|_{L^2(Q_T)}}{\Vert y_{\varepsilon} \Vert_{L^2(Q_T)}}$ ( Right) vs. $h$ for $r=10^{2}$ ($\circ$), $r=1.$ ($\star$) and $r=10^{-2}$ ($\square$).
• Figure 2: $\omega=(0.2,0.5)$; $y_0(x)=\sin(\pi x)$: $\varepsilon=10^{-4}$. ; $\frac{\|\rho_0 (v_{\varepsilon} - v_{\varepsilon,h})\|_{L^2(q_T)}}{\Vert \rho_0 v_{\varepsilon} \Vert_{L^2(q_T)}}$ ( Left) and $\frac{\|y_{\varepsilon} - \lambda_{\varepsilon,h}\|_{L^2(Q_T)}}{\Vert y_{\varepsilon} \Vert_{L^2(Q_T)}}$ ( Right) vs. $h$ for $r=10^{2}$ ($\circ$), $r=1.$ ($\star$) and $r=10^{-2}$ ($\square$).
• Figure 3: $\omega=(0.2,0.5)$; $y_0(x)=\sin(\pi x)$: $\varepsilon=10^{-8}$. ; $\frac{\|\rho_0 (v_{\varepsilon} - v_{\varepsilon,h})\|_{L^2(q_T)}}{\Vert \rho_0 v_{\varepsilon} \Vert_{L^2(q_T)}}$ ( Left) and $\frac{\|y_{\varepsilon} - \lambda_{\varepsilon,h}\|_{L^2(Q_T)}}{\Vert y_{\varepsilon} \Vert_{L^2(Q_T)}}$ ( Right) vs. $h$ for $r=10^{2}$ ($\circ$), $r=1.$ ($\star$) and $r=10^{-2}$ ($\square$).
• Figure 4: $\omega=(0.2,0.5)$; $y_0(x)=\sin(\pi x)$: $\varepsilon=0$. ; $\frac{\|\rho_0 (v - v_h)\|_{L^2(q_T)}}{\Vert \rho_0 v \Vert_{L^2(q_T)}}$ ( Left) and $\frac{\|y - \rho^{-1}\lambda_h\|_{L^2(Q_T)}}{\Vert y \Vert_{L^2(Q_T)}}$ ( Right) vs. $h$ for $r=10^{2}$ ($\circ$), $r=1.$ ($\star$) and $r=10^{-2}$ ($\square$).
• Figure 5: $\omega=(0.2,0.5)$; Approximation $\rho^{-1} \lambda_h$ of the controlled state $y$ over $Q_T$ - $r=1$ and $h=8.83\times 10^{-3}$.

Theorems & Definitions (9)

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• Remark 1
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• Remark 3