November  2002, 8(4): 967-982. doi: 10.3934/dcds.2002.8.967

Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing

1. 

Departamento de Matemáticas, Universidad de Castilla-La Mancha, c/ Campus Universitario s.n., 13.071-Ciudad Real, Spain

2. 

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

Received  April 2001 Revised  May 2002 Published  July 2002

We study relaxation for optimal design problems in conductivity in the two-dimensional situation. To this end, we reformulate the optimal design problem in an equivalent way as a genuine vector variational problem, and then analyze relaxation of this new variational problem. Our main achievement is to explicitly compute the quasiconvexification of the involved density in this problem for some interesting cases. We think the method given here could be generalized to compute quasiconvex envelopes in other situations. We restrict attention to the two-dimensional case.
Citation: José C. Bellido, Pablo Pedregal. Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 967-982. doi: 10.3934/dcds.2002.8.967
[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]

Massimiliano Berti, M. Matzeu, Enrico Valdinoci. On periodic elliptic equations with gradient dependence. Communications on Pure & Applied Analysis, 2008, 7 (3) : 601-615. doi: 10.3934/cpaa.2008.7.601

[3]

Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020

[4]

Dumitru Motreanu, Viorica V. Motreanu, Abdelkrim Moussaoui. Location of Nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 293-307. doi: 10.3934/dcdss.2018016

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019068

[10]

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

[11]

Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020

[12]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017

[13]

Bin Li, Hai Huyen Dam, Antonio Cantoni. A low-complexity zero-forcing Beamformer design for multiuser MIMO systems via a dual gradient method. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 297-304. doi: 10.3934/naco.2016012

[14]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021

[15]

Eleonora Catsigeras, Yun Zhao. Observable optimal state points of subadditive potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1375-1388. doi: 10.3934/dcds.2013.33.1375

[16]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[17]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241

[18]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[19]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425

[20]

K.H. Wong, C. Myburgh, L. Omari. A gradient flow approach for computing jump linear quadratic optimal feedback gains. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 803-808. doi: 10.3934/dcds.2000.6.803

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]