Abstract
We consider in this paper the homogeneous 1-D wave equation defined on Ω⊂ℝ. Using the Hilbert Uniqueness Method, one may define, for each subset ω⊂Ω, the exact control v ω of minimal L 2(ω×(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 ω which minimizes the functional \(J:\omega \rightarrow \|v_{\omega}\|_{L^{2}(\omega \times (0,T))}\) . We express the shape derivative of J as an integral on ∂ ω×(0,T) independently of any adjoint solution. This expression leads to a descent direction for J and permits to define a gradient algorithm efficiently initialized by the topological derivative associated with 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 a relaxed formulation.
Similar content being viewed by others
References
Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and level-set methods. J. Comput. Phys. 194(1), 363–393 (2004)
Asch, M., Lebeau, G.: Geometrical aspects of exact controllability for the wave equation—a numerical study. Esaim: COCV 3, 163–212 (1998)
Barles, G., Sougadinis, P.E.: On the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 31(4), 925–939 (2000)
Burger, M., Osher, S.J.: A survey on level set methods for inverse problems and optimal design. Eur. J. Appl. Math. 16(2), 263–301 (2005)
Cagnol, J., Zolesio, J.P.: Shape derivative in the wave equation with Dirichlet boundary condition. J. Differ. Equ. 158, 175–210 (1999)
Castro, C., Micu, S., Münch, A.: Numerical approximation of the boundary control of the 2-D wave equation with mixed finite elements. IMA Numer. Anal. (in press), doi:10.1093/imanum/drm012
Cea, J.: Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. Math. Model. Numer. Anal. 20(3), 371–402 (1986)
Delfour, M.C., Zolesio, J.P.: Shapes and Geometries—Analysis, Differential Calculus and Optimization. Advances in Design and Control. SIAM, Philadelphia (2001)
Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39(6), 1756–1778 (2001)
Glowinski, R., Li, C.H., Lions, J.-L.: A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: description of the numerical methods. Int. J. Numer. Methods Eng. 27(3), 623–636 (1989)
Henrot, A., Pierre, M.: Variation et Optimisation de Formes: une Analyse Géométrique. Mathématiques et Applications, vol. 48. Springer, Berlin (2005)
Lions, J.L.: Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués, Tome 1. Masson, RMA 8, Paris (1988)
Münch, A.: A uniformly controllable and implicit scheme for the 1-D wave equation. Math. Model. Numer. Anal. 39(2), 377–418 (2005)
Münch, A.: Optimal design of the support of the control for the 2-D wave equation: a numerical method. Int. J. Numer. Anal. Model. 5(2), 331–351 (2008)
Münch, A.: Optimal internal stabilization of a damped wave equation by a level set approach. Prépublication du laboratoire de Mathématiques de Besancon, 2005/1. Downlable at www-math.univ-fcomte.fr/amunch/research.htm
Münch, A., Pedregal, P., Periago, F.: Optimal design of the damping set for the stabilization of the wave equation. J. Differ. Equ. 231(1), 331–358 (2006)
Sokolowski, J., Zochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999)
Osher, S.J., Fedkiw, R.: Level set Methods and Dynamics Implicit Surfaces. Applied Mathematics Sciences, vol. 153. Springer, Berlin (2002)
Sethian, J.A.: Level set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1996)
Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192, 227–246 (2003)
Zuazua, E.: Propagation, observation, control and numerical approximation of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Münch, A. Optimal location of the support of the control for the 1-D wave equation: numerical investigations. Comput Optim Appl 42, 443–470 (2009). https://doi.org/10.1007/s10589-007-9133-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-007-9133-x