Modelling and control in pseudoplate problem with discontinuous thickness

Ján Lovíšek

Applications of Mathematics (2009)

  • Volume: 54, Issue: 6, page 491-525
  • ISSN: 0862-7940

Abstract

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This paper concerns an obstacle control problem for an elastic (homogeneous) and isotropic) pseudoplate. The state problem is modelled by a coercive variational inequality, where control variable enters the coefficients of the linear operator. Here, the role of control variable is played by the thickness of the pseudoplate which need not belong to the set of continuous functions. Since in general problems of control in coefficients have no optimal solution, a class of the extended optimal control is introduced. Taking into account the results of G -convergence theory, we prove the existence of an optimal solution of extended control problem. Moreover, approximate optimization problem is introduced, making use of the finite element method. The solvability of the approximate problem is proved on the basis of a general theorem. When the mesh size tends to zero, a subsequence of any sequence of approximate solutions converges uniformly to a solution of the continuous problem.

How to cite

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Lovíšek, Ján. "Modelling and control in pseudoplate problem with discontinuous thickness." Applications of Mathematics 54.6 (2009): 491-525. <http://eudml.org/doc/37834>.

@article{Lovíšek2009,
abstract = {This paper concerns an obstacle control problem for an elastic (homogeneous) and isotropic) pseudoplate. The state problem is modelled by a coercive variational inequality, where control variable enters the coefficients of the linear operator. Here, the role of control variable is played by the thickness of the pseudoplate which need not belong to the set of continuous functions. Since in general problems of control in coefficients have no optimal solution, a class of the extended optimal control is introduced. Taking into account the results of $G$-convergence theory, we prove the existence of an optimal solution of extended control problem. Moreover, approximate optimization problem is introduced, making use of the finite element method. The solvability of the approximate problem is proved on the basis of a general theorem. When the mesh size tends to zero, a subsequence of any sequence of approximate solutions converges uniformly to a solution of the continuous problem.},
author = {Lovíšek, Ján},
journal = {Applications of Mathematics},
keywords = {control of variational inequalities; optimal design; minimization; pseudoplate with obstacles; cost functional; thickness; $G$-convergence; coercive variational inequality; approximate optimization problem; finite element; control of variational inequalities; optimal design; minimization; pseudoplate with obstacles; cost functional},
language = {eng},
number = {6},
pages = {491-525},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Modelling and control in pseudoplate problem with discontinuous thickness},
url = {http://eudml.org/doc/37834},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Lovíšek, Ján
TI - Modelling and control in pseudoplate problem with discontinuous thickness
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 6
SP - 491
EP - 525
AB - This paper concerns an obstacle control problem for an elastic (homogeneous) and isotropic) pseudoplate. The state problem is modelled by a coercive variational inequality, where control variable enters the coefficients of the linear operator. Here, the role of control variable is played by the thickness of the pseudoplate which need not belong to the set of continuous functions. Since in general problems of control in coefficients have no optimal solution, a class of the extended optimal control is introduced. Taking into account the results of $G$-convergence theory, we prove the existence of an optimal solution of extended control problem. Moreover, approximate optimization problem is introduced, making use of the finite element method. The solvability of the approximate problem is proved on the basis of a general theorem. When the mesh size tends to zero, a subsequence of any sequence of approximate solutions converges uniformly to a solution of the continuous problem.
LA - eng
KW - control of variational inequalities; optimal design; minimization; pseudoplate with obstacles; cost functional; thickness; $G$-convergence; coercive variational inequality; approximate optimization problem; finite element; control of variational inequalities; optimal design; minimization; pseudoplate with obstacles; cost functional
UR - http://eudml.org/doc/37834
ER -

References

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  1. Adams, R. A., Sobolev Spaces, Academic Press New York-London (1975). (1975) Zbl0314.46030MR0450957
  2. Armand, J. L., Application of the Theory of Optimal Control of Distributed Parameter Systems of Structural Optimization, NASA (1972). (1972) 
  3. Boccardo, L., Murat, F., Nouveaux résultats de convergence des problèmes unilateraux, Res. Notes Math. 60 (1982), 64-85 French. (1982) MR0652507
  4. Boccardo, L., Marcellini, F., 10.1007/BF02418003, Ann. Mat. Pura Appl., IV. Ser. 110 (1976), 137-159 Italian. (1976) Zbl0333.35030MR0425344DOI10.1007/BF02418003
  5. Boccardo, L., Dolcetta, J. C., G -convergenza e problema di Dirichlet unilaterale, Boll. Unione Math. Ital., IV. Ser. 12 (1975), 115-123 Italian. (1975) Zbl0337.35023MR0399988
  6. Brezis, H., Stampacchia, G., 10.24033/bsmf.1663, Bull. Soc. Math. Fr. 96 (1968), 153-180 French. (1968) Zbl0165.45601MR0239302DOI10.24033/bsmf.1663
  7. Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North Holland Amsterdam-New York-Oxford (1978). (1978) Zbl0383.65058MR0520174
  8. Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer New York (1984). (1984) Zbl0536.65054MR0737005
  9. Haslinger, J., Mäkinen, R. A. E., Introduction to Shape Optimization. Theory, Approximation and Computation Advance in Design and Control, SIAM Philadelphia (2003). (2003) MR1969772
  10. Hlaváček, I., Chleboun, J., Babuška, I., Uncertain Input Data Problems and the Worst Scenario Method, Elsevier Amsterdam (2004). (2004) Zbl1116.74003MR2285091
  11. Hlaváček, I., Lovíšek, J., 10.4064/am28-4-3, Applicationes Mathematicae 28 (2001), 407-426 Control in obstacle-pseudoplate problems with friction on the boundary. Approximate optimal design and worst scenario problems. Applicationes Mathematicae 29 (2002), 75-95. (2002) MR1873903DOI10.4064/am28-4-3
  12. Kinderlehrer, D., Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press New York (1980). (1980) Zbl0457.35001MR0567696
  13. Křížek, M., Neittaanmäki, P., Finite Element Approximation of Variational Problems and Applications, Longman Scientific & Technical/John Wiley & Sons Harlow/New York (1990). (1990) MR1066462
  14. Lurie, K. A., Cherkaev, A. V., 10.1007/BF00934300, J. Optimization Theory Appl. 42 (1984), 283-304. (1984) Zbl0504.73060MR0737972DOI10.1007/BF00934300
  15. Mignot, F., 10.1016/0022-1236(76)90017-3, J. Funct. Anal. 22 (1976), 130-185 French. (1976) Zbl0364.49003MR0423155DOI10.1016/0022-1236(76)90017-3
  16. Murat, F., H -convergence, Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger. Lecture Notes (1977-1978). (1978) 
  17. Nečas, J., Les Méthodes Directes en Théorie des Équations Elliptiques, Academia Prague (1967), French. (1967) MR0227584
  18. Petersson, J., 10.1090/qam/1402408, Q. Appl. Math. 54 (1996), 541-550. (1996) Zbl0871.73046MR1402408DOI10.1090/qam/1402408
  19. Raitum, U. E., Sufficient conditions for sets of solutions of linear elliptic equations to be weakly sequentially closed, Latvian Math. J. 24 (1980), 142-155. (1980) MR0616251
  20. Rodrigues, J.-F., Obstacle Problems in Mathematical Physics, North Holland Amsterdam (1987). (1987) Zbl0606.73017MR0880369
  21. Shillor, M., Sofonea, M., Telega, J. J., Models and Analysis of Quasistatic Contact. Variational Methods, Springer Berlin (2004). (2004) Zbl1069.74001
  22. Sokolowski, J., Optimal control in coefficients of boundary value problems with unilateral constraints, Bull. Pol. Acad. Sci., Tech. Sci. 31 (1983), 71-81. (1983) Zbl0544.49005
  23. Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat., III. Ser. 22 (1968), 571-597 Italian. (1968) MR0240443
  24. Zhikov, V. V., Kozlov, S. M., Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals, Springer Berlin (1994). (1994) MR1329546

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