A spatio-temporal design problem for a damped wave equation

SIAM Journal on Applied Mathematics

Volumn 68, Issue 1, 2007, Pages 109-122

  Maestre, Faustino ^a   Münch, Arnaud ^b   Pedregal, Pablo ^a  

^a UNIVERSITY OF CASTILLA LA MANCHA   (Spain)

^b UNIVERSITÉ DE FRANCHE COMTÉ   (France)

Author keywords

Optimal design; Relaxation; Wave equation; Young measure

Indexed keywords

NONLINEAR PROGRAMMING; OPTIMIZATION; RELAXATION PROCESSES;

OPTIMAL DESIGN PROBLEMS; YOUNG MEASURE;

WAVE EQUATIONS;

EID: 39849088235     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/07067965X     Document Type: Article

References (27)

1. G. ALLAIRE, Shape Optimization by the Homogenization Method, Springer, New York, 2002.
2. E. ARANDA AND P. PEDREGAL, Constrained envelope for a general class of design problems, Discrete Contin. Dynam. Systems, Supplement (2003), pp. 30-41.
3. J. C. BELLIDO AND A. DONOSO, On an Optimal Design Problem in Wave-Propagation, preprint 3-2007, Universidad Castilla-La-Mancha, Ciudad Real, Spain.
4. M. P. BENDSØE AND O. SIGMUND, Topology Optimization: Theory, Methods, and Applications, Springer, Berlin, Heidelberg, New York, 2003.
5. D. BUCUR AND G. BUTTAZZO, Variational Methods in Shape Optimization Problems, Progr. Nonlinear Differential Equations Appl. 65, Birkhäuser, Basel, 2005.
6. M. BURGER AND S. J. OSHER, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), pp. 263-301.
7. J. CAGNOL AND J-.P. ZOLÉSIO, Shape control in hyperbolic problems. Optimal control of partial differential equations, in Internat. Ser. Numer. Math. 133, Birkhäuser, Basel, 1999, pp. 77-88.
8. A. CHAMBOLLE AND F. SANTOSA, Control of the wave equation by time-dependent coefficient, ESAIM Control Optim. Calc. Var., 8 (2002), pp. 375-392.
9. S. J. COX, Designing for optimal energy absorption II, The damped wave equation, in Internat. Ser. Numer. Math. 126, Birkhäuser, Basel, 1998, pp. 103-109.
10. S. J. COX, I. NAKIC̀, A. RITTMANN, AND K. VESELIĆ, Lyapunov optimization of a damped system, Systems Control Lett., 53 (2004), pp. 187-194.
11. B. DACOROGNA, Direct Method in the Calculus of Variations, Springer, New York, 1989.
12. A. DONOSO AND P. PEDREGAL, Optimal design of 2-D conducting graded materials by minimizing quadratic functionals in the field, Struct. Multidisc. Optim., 30 (2005), pp. 360-367.
13. F. FAHROO AND K. ITO, Variational formulation of optimal damping designs, Contemp. Math., 209 (1997), pp. 95-114.
14. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes equations. Theory and Algorithms, Springer, New York, 1986.
15. P. HEBRARD AND A. HENROT, Optimal shape and position of the actuators for the stabilization of a string, Systems Control Lett., 48 (2003), pp. 199-209.
16. J.-L. LIONS AND E. MAGENES, Problèmes aux limites non homogènes et applications, Dunod, Paris, 1968.
17. K. LURIE, Control of the coefficients of linear hyperbolic equations via spatio-temporal composites, in Homogenization, V. Berdychevisky, V. Jikov, and G. Papanicolau, eds., World Scientific, Singapore, 1999, pp. 285-315.
18. K. LURIE, Some new advances in the theory of dynamic materials, J. Elasticity, 72 (2003), pp. 229-239.
19. F. MAESTRE AND P. PEDREGAL, Quasiconvexification in 3-D for a variational reformulation of an optimal design problem in conductivity, Nonlinear Anal., 64 (2006), pp. 1962-1976.
20. F. MAESTRE, A. MÜNCH, AND P. PEDREGAL, Optimal design under the one-dimensional wave equation, Interfaces Free Bound., to appear.
21. A. MÜNCH, P. PEDREGAL, AND F. PERIAGO, A variational approach to a shape design problem for the wave equation, C. R. Acad. Sci. Paris Sér. I, 343 (2006), pp. 371-376.
22. A. MÜNCH, P. PEDREGAL, AND F. PERIAGO, Optimal design of the damping set for the stabilization of the wave equation, J. Differential Equations, 231 (2006), pp. 331-358.
23. A. MÜNCH, P. PEDREGAL, AND F. PERIAGO, Optimal Internal Design Stabilization of the Linear System of Elasticity, preprint 2007/01, Université de Franche-Comté, Bensancon, France.
24. F. MURAT, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. Ser., 4 (1977), pp. 49-68.
25. P. PEDREGAL, Parametrized measures and variational principles, Birkhäuser, Berlin, 1997.
26. P. PEDREGAL, Vector variational problems and applications to optimal design, ESAIM Control Optim. Cale. Var., 15 (2005), pp. 357-381.
27. J. SOKOŁOWSKI AND A. ŻOCHOWSKI, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), pp. 1251-1272.