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 - <p style="text-align: justify;">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 $T&gt;0$ 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.</p>