American Institute of Mathematical Sciences

Advanced
July  2009, 23(3): 973-990. doi: 10.3934/dcds.2009.23.973

Dynamic materials for an optimal design problem under the two-dimensional wave equation

Faustino Maestre 1, and  Pablo Pedregal 2,

1. 

Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
2. 

E.T.S. Ingenieros Industriales, Universidad de Castilla La Mancha

Received  January 2008 Revised  August 2008 Published  November 2008

In this work, we analyze a 3-d dynamic optimal design problem in conductivity governed by the two-dimensional wave equation. Under this dynamic perspective, the optimal design problem consists in seeking the time-dependent optimal layout of two isotropic materials on a 2-d domain ($\Omega\subsetR^2$); this is done by minimizing a cost functional depending on the square of the gradient of the state function involving coefficients which can depend on time, space and design. The lack of classical solutions of this type of problem is well-known, so that a relaxation must be sought. We utilize a specially appropriate characterization of 3-d ($(t,x)\inR\timesR^2$) divergence free vector fields through Clebsh potentials; this lets us transform the optimal design problem into a typical non-convex vector variational problem, to which Young measure theory can be applied to compute explicitly the "constrained quasiconvexification" of the cost density. Moreover this relaxation is recovered by dynamic (time-space) first- or second-order laminates. There are two main concerns in this work: the 2-d hyperbolic state law, and the vector character of the problem. Though these two ingredients have been previously considered separately, we put them together in this work.
Keywords: quasiconvexification, Dynamic optimal design, relaxation, Young measure.
Mathematics Subject Classification: Primary: 49J45, 74P10.
Citation: Faustino Maestre, Pablo Pedregal. Dynamic materials for an optimal design problem under the two-dimensional wave equation. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 973-990. doi: 10.3934/dcds.2009.23.973
[1]

Pablo Pedregal. Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design. Electronic Research Announcements, 2001, 7: 72-78.

[2]

José C. Bellido, Pablo Pedregal. Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 967-982. doi: 10.3934/dcds.2002.8.967
[3]

Hyeon Je Cho, Ganguk Hwang. Optimal design for dynamic spectrum access in cognitive radio networks under Rayleigh fading. Journal of Industrial & Management Optimization, 2012, 8 (4) : 821-840. doi: 10.3934/jimo.2012.8.821
[4]

Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257
[5]

Alice Fiaschi. Young-measure quasi-static damage evolution: The nonconvex and the brittle cases. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 17-42. doi: 10.3934/dcdss.2013.6.17
[6]

Yan Chen, Kewei Zhang. Young measure solutions of the two-dimensional Perona-Malik equation in image processing. Communications on Pure & Applied Analysis, 2006, 5 (3) : 617-637. doi: 10.3934/cpaa.2006.5.617
[7]

H. T. Banks, R. A. Everett, Neha Murad, R. D. White, J. E. Banks, Bodil N. Cass, Jay A. Rosenheim. Optimal design for dynamical modeling of pest populations. Mathematical Biosciences & Engineering, 2018, 15 (4) : 993-1010. doi: 10.3934/mbe.2018044
[8]

K.F.C. Yiu, K.L. Mak, K. L. Teo. Airfoil design via optimal control theory. Journal of Industrial & Management Optimization, 2005, 1 (1) : 133-148. doi: 10.3934/jimo.2005.1.133
[9]

Boris P. Belinskiy. Optimal design of an optical length of a rod with the given mass. Conference Publications, 2007, 2007 (Special) : 85-91. doi: 10.3934/proc.2007.2007.85
[10]

Xueling Zhou, Bingo Wing-Kuen Ling, Hai Huyen Dam, Kok-Lay Teo. Optimal design of window functions for filter window bank. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1119-1145. doi: 10.3934/jimo.2020014
[11]

Yannick Privat, Emmanuel Trélat. Optimal design of sensors for a damped wave equation. Conference Publications, 2015, 2015 (special) : 936-944. doi: 10.3934/proc.2015.0936
[12]

Wei Xu, Liying Yu, Gui-Hua Lin, Zhi Guo Feng. Optimal switching signal design with a cost on switching action. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2531-2549. doi: 10.3934/jimo.2019068
[13]

Bin Li, Kok Lay Teo, Cheng-Chew Lim, Guang Ren Duan. An optimal PID controller design for nonlinear constrained optimal control problems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1101-1117. doi: 10.3934/dcdsb.2011.16.1101
[14]

Tomasz R. Bielecki, Igor Cialenco, Marcin Pitera. A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 3-. doi: 10.1186/s41546-017-0012-9
[15]

Leszek Gasiński, Nikolaos S. Papageorgiou. Relaxation of optimal control problems driven by nonlinear evolution equations. Evolution Equations & Control Theory, 2020, 9 (4) : 1027-1040. doi: 10.3934/eect.2020050
[16]

Qiang Du, Jingyan Zhang. Asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal partition problem. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1443-1461. doi: 10.3934/dcds.2011.29.1443
[17]

Lars Grüne, Manuela Sigurani. Numerical event-based ISS controller design via a dynamic game approach. Journal of Computational Dynamics, 2015, 2 (1) : 65-81. doi: 10.3934/jcd.2015.2.65
[18]

Stefano Luzzatto, Marks Ruziboev. Young towers for product systems. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1465-1491. doi: 10.3934/dcds.2016.36.1465
[19]

Alexis De Vos, Yvan Van Rentergem. Young subgroups for reversible computers. Advances in Mathematics of Communications, 2008, 2 (2) : 183-200. doi: 10.3934/amc.2008.2.183
[20]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy