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Volume 5, Issue 2
Optimal Design of the Support for the Control for the 2-D Wave Equation: A Numerical Method

Arnaud Münch

Int. J. Numer. Anal. Mod., 5 (2008), pp. 331-351.

Published online: 2008-05

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  • Abstract

We consider in this paper the homogeneous 2-D wave equation defined on $\Omega \subset \mathbb{R}^2$. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset $\omega \subset \Omega$, the control $v_{\omega}$ of minimal $L^2 (\omega \times (0, T))$-norm which drives to rest the system at a time $T>0$ large enough. We address the question of the optimal position of $\omega$ which minimize the functional $J : \omega \rightarrow ||v_{\omega}||_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^1(\Omega)$, we express the shape derivative of $J$ as a curvilinear integral on $∂\omega \times (0,T)$ independently of any adjoint solution. This expression leads to a descent direction and permits to define a gradient algorithm efficiently initialized by the topological derivative associated to $J$. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering its relaxation.

  • AMS Subject Headings

35L05, 49J20, 65K10, 65M60, 93B05

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-5-331, author = {Münch , Arnaud}, title = {Optimal Design of the Support for the Control for the 2-D Wave Equation: A Numerical Method}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2008}, volume = {5}, number = {2}, pages = {331--351}, abstract = {

We consider in this paper the homogeneous 2-D wave equation defined on $\Omega \subset \mathbb{R}^2$. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset $\omega \subset \Omega$, the control $v_{\omega}$ of minimal $L^2 (\omega \times (0, T))$-norm which drives to rest the system at a time $T>0$ large enough. We address the question of the optimal position of $\omega$ which minimize the functional $J : \omega \rightarrow ||v_{\omega}||_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^1(\Omega)$, we express the shape derivative of $J$ as a curvilinear integral on $∂\omega \times (0,T)$ independently of any adjoint solution. This expression leads to a descent direction and permits to define a gradient algorithm efficiently initialized by the topological derivative associated to $J$. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering its relaxation.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/815.html} }
TY - JOUR T1 - Optimal Design of the Support for the Control for the 2-D Wave Equation: A Numerical Method AU - Münch , Arnaud JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 331 EP - 351 PY - 2008 DA - 2008/05 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/815.html KW - optimal shape design, exact controllability of wave equation, level set method, numerical schemes, relaxation. AB -

We consider in this paper the homogeneous 2-D wave equation defined on $\Omega \subset \mathbb{R}^2$. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset $\omega \subset \Omega$, the control $v_{\omega}$ of minimal $L^2 (\omega \times (0, T))$-norm which drives to rest the system at a time $T>0$ large enough. We address the question of the optimal position of $\omega$ which minimize the functional $J : \omega \rightarrow ||v_{\omega}||_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^1(\Omega)$, we express the shape derivative of $J$ as a curvilinear integral on $∂\omega \times (0,T)$ independently of any adjoint solution. This expression leads to a descent direction and permits to define a gradient algorithm efficiently initialized by the topological derivative associated to $J$. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering its relaxation.

Arnaud Münch. (1970). Optimal Design of the Support for the Control for the 2-D Wave Equation: A Numerical Method. International Journal of Numerical Analysis and Modeling. 5 (2). 331-351. doi:
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