| Title:
|
Reliable solution of parabolic obstacle problems with respect to uncertain data (English) |
| Author:
|
Lovíšek, Ján |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
48 |
| Issue:
|
5 |
| Year:
|
2003 |
| Pages:
|
321-351 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A class of parabolic initial-boundary value problems is considered, where admissible coefficients are given in certain intervals. We are looking for maximal values of the solution with respect to the set of admissible coefficients. We give the abstract general scheme, proposing how to solve such problems with uncertain data. We formulate a general maximization problem and prove its solvability, provided all fundamental assumptions are fulfilled. We apply the theory to certain Fourier obstacle type maximization problem. (English) |
| Keyword:
|
uncertain data |
| Keyword:
|
optimal design approach |
| Keyword:
|
parabolic obstacle problems |
| Keyword:
|
penalization method |
| Keyword:
|
Fourier problem |
| MSC:
|
35B30 |
| MSC:
|
35K85 |
| MSC:
|
49J40 |
| idZBL:
|
Zbl 1099.35054 |
| idMR:
|
MR2008888 |
| DOI:
|
10.1023/B:APOM.0000024480.06960.ea |
| . |
| Date available:
|
2009-09-22T18:14:14Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134534 |
| . |
| Reference:
|
[1] J. P. Aubin: Un théorème de compacité.C. R. Acad. Sci. Paris 256 (1963), 5042–5044. Zbl 0195.13002, MR 0152860 |
| Reference:
|
[2] V. Barbu: Optimal Control of Variational Inequalities.Pitman, Boston, 1984. Zbl 0574.49005, MR 0742624 |
| Reference:
|
[3] I. Bock, J. Lovíšek: Optimal control of a viscoelastic plate bending.Math. Nachr. 125 (1968), 135–151. MR 0847355, 10.1002/mana.19861250109 |
| Reference:
|
[4] H. Brézis: Problèmes unilatéraux.J. Math. Pures Appl. 51 (1972), 1–168. |
| Reference:
|
[5] H. Brézis: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert.North Holland, Amsterdam, 1973. MR 0348562 |
| Reference:
|
[6] H. Brézis: Analyse fonctionelle.Masson, Paris, 1982. |
| Reference:
|
[7] H. Gajewski, K. Gröger and K. Zacharias: Nichtlineare Operatorgleichungen und Operator-Differentialgleichungen.Akademie, Berlin, 1974. MR 0636412 |
| Reference:
|
[8] I. Hlaváček: Reliable solutions of elliptic boundary value problems with respect to uncertain data.Nonlinear Anal. 30 (1997), 3879–3980. MR 1602891, 10.1016/S0362-546X(96)00236-2 |
| Reference:
|
[9] I. Hlaváček: Reliable solutions of linear parabolic problems with respect to uncertain coefficients.Z. Angew. Math. Mech. 79 (1999), 291–301. MR 1695286, 10.1002/(SICI)1521-4001(199905)79:5<291::AID-ZAMM291>3.0.CO;2-N |
| Reference:
|
[10] J. L. Lions: Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles.Dunod, Paris, 1968. MR 0244606 |
| Reference:
|
[11] J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires.Dunod, Paris, 1969. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[12] V. C. Litvinov: Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics.Birkhäuser-Verlag, Berlin, 2000. Zbl 0947.49001, MR 1774123 |
| Reference:
|
[13] F. Mignot, J.-P. Puel: Optimal control of some variational inequalities.SIAM J. Control Optim. 22 (1984), 466–478. MR 0739836, 10.1137/0322028 |
| Reference:
|
[14] U. Mosco: Convergence of convex sets and solutions of variational inequalities.Adv. Math. 3 (1969), 510–585. MR 0298508, 10.1016/0001-8708(69)90009-7 |
| Reference:
|
[15] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Masson, Paris, 1967. MR 0227584 |
| Reference:
|
[16] P. Neittaanmäki, D. Tiba: Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications. Pure and Applied Mathematics.Marcel Dekker inc., New York, 1994. MR 1275836 |
| Reference:
|
[17] T. I. Seidman, Hong Xing Zhou: Existence and uniqueness of optimal controls for a quasilinear parabolic equation.SIAM J. Control Optim. 20 (1982), 747–762. MR 0675567, 10.1137/0320054 |
| Reference:
|
[18] J. Sokolowski, J. P. Zolesio: Introduction to Shape Optimization. Shape Sensitivity Analysis.Springer-Verlag, New York, 1992. MR 1215733 |
| Reference:
|
[19] J. P. Yvon: Contrôle optimal de systèmes gouvernés par des inéquations variationnelles.Rapport de Recherche 22, INRIA, Paris, 1974. |
| Reference:
|
[20] J. Steinbach: A Variational Inequality Approach to Free Boundary Problems with Applications in Mould Filling.Birkhäuser-Verlag, Basel, 2002. Zbl 1011.35001, MR 1891393 |
| Reference:
|
[21] G. M. Troianiello: Elliptic Differential Equations and Obstacle Problems.Plenum Press, New York, 1987. Zbl 0655.35002, MR 1094820 |
| Reference:
|
[22] E. Zeidler: Nonlinear Functional Analysis and its Applications II/A, Linear Monotone Operators.Springer-Verlag, New York, 1990. Zbl 0684.47028, MR 1033497 |
| . |
